author | wenzelm |
Thu, 01 Sep 2016 16:05:22 +0200 | |
changeset 63750 | 9c8a366778e1 |
parent 63649 | e690d6f2185b |
child 63882 | 018998c00003 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Polynomial.thy |
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Author: Brian Huffman |
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Author: Clemens Ballarin |
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Author: Amine Chaieb |
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Author: Florian Haftmann |
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*) |
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section \<open>Polynomials as type over a ring structure\<close> |
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theory Polynomial |
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imports Main "~~/src/HOL/Deriv" "~~/src/HOL/Library/More_List" |
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"~~/src/HOL/Library/Infinite_Set" |
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begin |
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subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close> |
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definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "##" 65) |
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where |
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"x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)" |
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lemma cCons_0_Nil_eq [simp]: |
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"0 ## [] = []" |
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by (simp add: cCons_def) |
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||
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lemma cCons_Cons_eq [simp]: |
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"x ## y # ys = x # y # ys" |
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by (simp add: cCons_def) |
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lemma cCons_append_Cons_eq [simp]: |
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"x ## xs @ y # ys = x # xs @ y # ys" |
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by (simp add: cCons_def) |
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lemma cCons_not_0_eq [simp]: |
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"x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs" |
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by (simp add: cCons_def) |
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||
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lemma strip_while_not_0_Cons_eq [simp]: |
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"strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs" |
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proof (cases "x = 0") |
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case False then show ?thesis by simp |
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next |
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case True show ?thesis |
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proof (induct xs rule: rev_induct) |
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case Nil with True show ?case by simp |
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next |
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case (snoc y ys) then show ?case |
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by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons) |
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qed |
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qed |
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lemma tl_cCons [simp]: |
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"tl (x ## xs) = xs" |
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by (simp add: cCons_def) |
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subsection \<open>Definition of type \<open>poly\<close>\<close> |
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typedef (overloaded) 'a poly = "{f :: nat \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n. f n = 0}" |
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morphisms coeff Abs_poly |
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by (auto intro!: ALL_MOST) |
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setup_lifting type_definition_poly |
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lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)" |
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by (simp add: coeff_inject [symmetric] fun_eq_iff) |
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lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q" |
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by (simp add: poly_eq_iff) |
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lemma MOST_coeff_eq_0: "\<forall>\<^sub>\<infinity> n. coeff p n = 0" |
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using coeff [of p] by simp |
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subsection \<open>Degree of a polynomial\<close> |
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|
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definition degree :: "'a::zero poly \<Rightarrow> nat" |
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where |
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"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)" |
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lemma coeff_eq_0: |
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assumes "degree p < n" |
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shows "coeff p n = 0" |
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proof - |
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have "\<exists>n. \<forall>i>n. coeff p i = 0" |
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using MOST_coeff_eq_0 by (simp add: MOST_nat) |
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then have "\<forall>i>degree p. coeff p i = 0" |
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unfolding degree_def by (rule LeastI_ex) |
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with assms show ?thesis by simp |
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qed |
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lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p" |
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by (erule contrapos_np, rule coeff_eq_0, simp) |
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lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n" |
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unfolding degree_def by (erule Least_le) |
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lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0" |
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unfolding degree_def by (drule not_less_Least, simp) |
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subsection \<open>The zero polynomial\<close> |
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instantiation poly :: (zero) zero |
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begin |
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lift_definition zero_poly :: "'a poly" |
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is "\<lambda>_. 0" by (rule MOST_I) simp |
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instance .. |
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end |
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lemma coeff_0 [simp]: |
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"coeff 0 n = 0" |
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by transfer rule |
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lemma degree_0 [simp]: |
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"degree 0 = 0" |
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by (rule order_antisym [OF degree_le le0]) simp |
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lemma leading_coeff_neq_0: |
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assumes "p \<noteq> 0" |
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shows "coeff p (degree p) \<noteq> 0" |
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proof (cases "degree p") |
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case 0 |
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from \<open>p \<noteq> 0\<close> have "\<exists>n. coeff p n \<noteq> 0" |
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by (simp add: poly_eq_iff) |
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then obtain n where "coeff p n \<noteq> 0" .. |
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hence "n \<le> degree p" by (rule le_degree) |
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with \<open>coeff p n \<noteq> 0\<close> and \<open>degree p = 0\<close> |
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show "coeff p (degree p) \<noteq> 0" by simp |
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next |
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case (Suc n) |
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from \<open>degree p = Suc n\<close> have "n < degree p" by simp |
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hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp) |
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then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast |
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from \<open>degree p = Suc n\<close> and \<open>n < i\<close> have "degree p \<le> i" by simp |
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also from \<open>coeff p i \<noteq> 0\<close> have "i \<le> degree p" by (rule le_degree) |
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finally have "degree p = i" . |
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with \<open>coeff p i \<noteq> 0\<close> show "coeff p (degree p) \<noteq> 0" by simp |
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qed |
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lemma leading_coeff_0_iff [simp]: |
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"coeff p (degree p) = 0 \<longleftrightarrow> p = 0" |
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by (cases "p = 0", simp, simp add: leading_coeff_neq_0) |
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subsection \<open>List-style constructor for polynomials\<close> |
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lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
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is "\<lambda>a p. case_nat a (coeff p)" |
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by (rule MOST_SucD) (simp add: MOST_coeff_eq_0) |
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lemmas coeff_pCons = pCons.rep_eq |
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lemma coeff_pCons_0 [simp]: |
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"coeff (pCons a p) 0 = a" |
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by transfer simp |
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lemma coeff_pCons_Suc [simp]: |
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"coeff (pCons a p) (Suc n) = coeff p n" |
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by (simp add: coeff_pCons) |
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lemma degree_pCons_le: |
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"degree (pCons a p) \<le> Suc (degree p)" |
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by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split) |
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lemma degree_pCons_eq: |
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"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)" |
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apply (rule order_antisym [OF degree_pCons_le]) |
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apply (rule le_degree, simp) |
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done |
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lemma degree_pCons_0: |
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"degree (pCons a 0) = 0" |
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apply (rule order_antisym [OF _ le0]) |
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apply (rule degree_le, simp add: coeff_pCons split: nat.split) |
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done |
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lemma degree_pCons_eq_if [simp]: |
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"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" |
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apply (cases "p = 0", simp_all) |
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apply (rule order_antisym [OF _ le0]) |
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apply (rule degree_le, simp add: coeff_pCons split: nat.split) |
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apply (rule order_antisym [OF degree_pCons_le]) |
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apply (rule le_degree, simp) |
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done |
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lemma pCons_0_0 [simp]: |
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"pCons 0 0 = 0" |
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by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) |
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lemma pCons_eq_iff [simp]: |
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"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q" |
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proof safe |
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assume "pCons a p = pCons b q" |
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then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp |
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then show "a = b" by simp |
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next |
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assume "pCons a p = pCons b q" |
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then have "\<forall>n. coeff (pCons a p) (Suc n) = |
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coeff (pCons b q) (Suc n)" by simp |
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then show "p = q" by (simp add: poly_eq_iff) |
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qed |
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lemma pCons_eq_0_iff [simp]: |
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"pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0" |
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using pCons_eq_iff [of a p 0 0] by simp |
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lemma pCons_cases [cases type: poly]: |
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obtains (pCons) a q where "p = pCons a q" |
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proof |
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show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))" |
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by transfer |
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(simp_all add: MOST_inj[where f=Suc and P="\<lambda>n. p n = 0" for p] fun_eq_iff Abs_poly_inverse |
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split: nat.split) |
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qed |
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lemma pCons_induct [case_names 0 pCons, induct type: poly]: |
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assumes zero: "P 0" |
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assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)" |
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shows "P p" |
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proof (induct p rule: measure_induct_rule [where f=degree]) |
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case (less p) |
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obtain a q where "p = pCons a q" by (rule pCons_cases) |
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have "P q" |
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proof (cases "q = 0") |
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case True |
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then show "P q" by (simp add: zero) |
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next |
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case False |
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then have "degree (pCons a q) = Suc (degree q)" |
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by (rule degree_pCons_eq) |
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then have "degree q < degree p" |
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using \<open>p = pCons a q\<close> by simp |
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then show "P q" |
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by (rule less.hyps) |
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qed |
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have "P (pCons a q)" |
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proof (cases "a \<noteq> 0 \<or> q \<noteq> 0") |
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case True |
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with \<open>P q\<close> show ?thesis by (auto intro: pCons) |
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next |
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case False |
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with zero show ?thesis by simp |
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qed |
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then show ?case |
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using \<open>p = pCons a q\<close> by simp |
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qed |
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lemma degree_eq_zeroE: |
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fixes p :: "'a::zero poly" |
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assumes "degree p = 0" |
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obtains a where "p = pCons a 0" |
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proof - |
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obtain a q where p: "p = pCons a q" by (cases p) |
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with assms have "q = 0" by (cases "q = 0") simp_all |
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with p have "p = pCons a 0" by simp |
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with that show thesis . |
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qed |
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subsection \<open>Quickcheck generator for polynomials\<close> |
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quickcheck_generator poly constructors: "0 :: _ poly", pCons |
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subsection \<open>List-style syntax for polynomials\<close> |
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syntax |
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"_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]") |
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translations |
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"[:x, xs:]" == "CONST pCons x [:xs:]" |
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"[:x:]" == "CONST pCons x 0" |
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"[:x:]" <= "CONST pCons x (_constrain 0 t)" |
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subsection \<open>Representation of polynomials by lists of coefficients\<close> |
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primrec Poly :: "'a::zero list \<Rightarrow> 'a poly" |
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where |
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[code_post]: "Poly [] = 0" |
283 |
| [code_post]: "Poly (a # as) = pCons a (Poly as)" |
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lemma Poly_replicate_0 [simp]: |
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"Poly (replicate n 0) = 0" |
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by (induct n) simp_all |
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lemma Poly_eq_0: |
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"Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)" |
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by (induct as) (auto simp add: Cons_replicate_eq) |
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lemma Poly_append_replicate_zero [simp]: |
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"Poly (as @ replicate n 0) = Poly as" |
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by (induct as) simp_all |
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lemma Poly_snoc_zero [simp]: |
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"Poly (as @ [0]) = Poly as" |
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using Poly_append_replicate_zero [of as 1] by simp |
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lemma Poly_cCons_eq_pCons_Poly [simp]: |
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"Poly (a ## p) = pCons a (Poly p)" |
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by (simp add: cCons_def) |
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lemma Poly_on_rev_starting_with_0 [simp]: |
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assumes "hd as = 0" |
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shows "Poly (rev (tl as)) = Poly (rev as)" |
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using assms by (cases as) simp_all |
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lemma degree_Poly: "degree (Poly xs) \<le> length xs" |
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by (induction xs) simp_all |
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lemma coeff_Poly_eq [simp]: |
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62422
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changeset
|
314 |
"coeff (Poly xs) = nth_default 0 xs" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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62422
diff
changeset
|
315 |
by (induct xs) (simp_all add: fun_eq_iff coeff_pCons split: nat.splits) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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parents:
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diff
changeset
|
316 |
|
52380 | 317 |
definition coeffs :: "'a poly \<Rightarrow> 'a::zero list" |
318 |
where |
|
319 |
"coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])" |
|
320 |
||
321 |
lemma coeffs_eq_Nil [simp]: |
|
322 |
"coeffs p = [] \<longleftrightarrow> p = 0" |
|
323 |
by (simp add: coeffs_def) |
|
324 |
||
325 |
lemma not_0_coeffs_not_Nil: |
|
326 |
"p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []" |
|
327 |
by simp |
|
328 |
||
329 |
lemma coeffs_0_eq_Nil [simp]: |
|
330 |
"coeffs 0 = []" |
|
331 |
by simp |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
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|
332 |
|
52380 | 333 |
lemma coeffs_pCons_eq_cCons [simp]: |
334 |
"coeffs (pCons a p) = a ## coeffs p" |
|
335 |
proof - |
|
336 |
{ fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a" |
|
337 |
assume "\<forall>m\<in>set ms. m > 0" |
|
55415 | 338 |
then have "map (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)" |
58199
5fbe474b5da8
explicit theory with additional, less commonly used list operations
haftmann
parents:
57862
diff
changeset
|
339 |
by (induct ms) (auto split: nat.split) |
5fbe474b5da8
explicit theory with additional, less commonly used list operations
haftmann
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diff
changeset
|
340 |
} |
52380 | 341 |
note * = this |
342 |
show ?thesis |
|
60570 | 343 |
by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc) |
52380 | 344 |
qed |
345 |
||
62065 | 346 |
lemma length_coeffs: "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = degree p + 1" |
347 |
by (simp add: coeffs_def) |
|
348 |
||
349 |
lemma coeffs_nth: |
|
350 |
assumes "p \<noteq> 0" "n \<le> degree p" |
|
351 |
shows "coeffs p ! n = coeff p n" |
|
352 |
using assms unfolding coeffs_def by (auto simp del: upt_Suc) |
|
353 |
||
52380 | 354 |
lemma not_0_cCons_eq [simp]: |
355 |
"p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p" |
|
356 |
by (simp add: cCons_def) |
|
357 |
||
358 |
lemma Poly_coeffs [simp, code abstype]: |
|
359 |
"Poly (coeffs p) = p" |
|
54856 | 360 |
by (induct p) auto |
52380 | 361 |
|
362 |
lemma coeffs_Poly [simp]: |
|
363 |
"coeffs (Poly as) = strip_while (HOL.eq 0) as" |
|
364 |
proof (induct as) |
|
365 |
case Nil then show ?case by simp |
|
366 |
next |
|
367 |
case (Cons a as) |
|
368 |
have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)" |
|
369 |
using replicate_length_same [of as 0] by (auto dest: sym [of _ as]) |
|
370 |
with Cons show ?case by auto |
|
371 |
qed |
|
372 |
||
373 |
lemma last_coeffs_not_0: |
|
374 |
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0" |
|
375 |
by (induct p) (auto simp add: cCons_def) |
|
376 |
||
377 |
lemma strip_while_coeffs [simp]: |
|
378 |
"strip_while (HOL.eq 0) (coeffs p) = coeffs p" |
|
379 |
by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last) |
|
380 |
||
381 |
lemma coeffs_eq_iff: |
|
382 |
"p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q") |
|
383 |
proof |
|
384 |
assume ?P then show ?Q by simp |
|
385 |
next |
|
386 |
assume ?Q |
|
387 |
then have "Poly (coeffs p) = Poly (coeffs q)" by simp |
|
388 |
then show ?P by simp |
|
389 |
qed |
|
390 |
||
391 |
lemma nth_default_coeffs_eq: |
|
392 |
"nth_default 0 (coeffs p) = coeff p" |
|
393 |
by (simp add: fun_eq_iff coeff_Poly_eq [symmetric]) |
|
394 |
||
395 |
lemma [code]: |
|
396 |
"coeff p = nth_default 0 (coeffs p)" |
|
397 |
by (simp add: nth_default_coeffs_eq) |
|
398 |
||
399 |
lemma coeffs_eqI: |
|
400 |
assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n" |
|
401 |
assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" |
|
402 |
shows "coeffs p = xs" |
|
403 |
proof - |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
404 |
from coeff have "p = Poly xs" by (simp add: poly_eq_iff) |
52380 | 405 |
with zero show ?thesis by simp (cases xs, simp_all) |
406 |
qed |
|
407 |
||
408 |
lemma degree_eq_length_coeffs [code]: |
|
409 |
"degree p = length (coeffs p) - 1" |
|
410 |
by (simp add: coeffs_def) |
|
411 |
||
412 |
lemma length_coeffs_degree: |
|
413 |
"p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)" |
|
414 |
by (induct p) (auto simp add: cCons_def) |
|
415 |
||
416 |
lemma [code abstract]: |
|
417 |
"coeffs 0 = []" |
|
418 |
by (fact coeffs_0_eq_Nil) |
|
419 |
||
420 |
lemma [code abstract]: |
|
421 |
"coeffs (pCons a p) = a ## coeffs p" |
|
422 |
by (fact coeffs_pCons_eq_cCons) |
|
423 |
||
424 |
instantiation poly :: ("{zero, equal}") equal |
|
425 |
begin |
|
426 |
||
427 |
definition |
|
428 |
[code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)" |
|
429 |
||
60679 | 430 |
instance |
431 |
by standard (simp add: equal equal_poly_def coeffs_eq_iff) |
|
52380 | 432 |
|
433 |
end |
|
434 |
||
60679 | 435 |
lemma [code nbe]: "HOL.equal (p :: _ poly) p \<longleftrightarrow> True" |
52380 | 436 |
by (fact equal_refl) |
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
437 |
|
52380 | 438 |
definition is_zero :: "'a::zero poly \<Rightarrow> bool" |
439 |
where |
|
440 |
[code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)" |
|
441 |
||
442 |
lemma is_zero_null [code_abbrev]: |
|
443 |
"is_zero p \<longleftrightarrow> p = 0" |
|
444 |
by (simp add: is_zero_def null_def) |
|
445 |
||
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
446 |
subsubsection \<open>Reconstructing the polynomial from the list\<close> |
63145 | 447 |
\<comment> \<open>contributed by Sebastiaan J.C. Joosten and René Thiemann\<close> |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
448 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
449 |
definition poly_of_list :: "'a::comm_monoid_add list \<Rightarrow> 'a poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
450 |
where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
451 |
[simp]: "poly_of_list = Poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
452 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
453 |
lemma poly_of_list_impl [code abstract]: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
454 |
"coeffs (poly_of_list as) = strip_while (HOL.eq 0) as" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
455 |
by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
456 |
|
52380 | 457 |
|
60500 | 458 |
subsection \<open>Fold combinator for polynomials\<close> |
52380 | 459 |
|
460 |
definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b" |
|
461 |
where |
|
462 |
"fold_coeffs f p = foldr f (coeffs p)" |
|
463 |
||
464 |
lemma fold_coeffs_0_eq [simp]: |
|
465 |
"fold_coeffs f 0 = id" |
|
466 |
by (simp add: fold_coeffs_def) |
|
467 |
||
468 |
lemma fold_coeffs_pCons_eq [simp]: |
|
469 |
"f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
470 |
by (simp add: fold_coeffs_def cCons_def fun_eq_iff) |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
471 |
|
52380 | 472 |
lemma fold_coeffs_pCons_0_0_eq [simp]: |
473 |
"fold_coeffs f (pCons 0 0) = id" |
|
474 |
by (simp add: fold_coeffs_def) |
|
475 |
||
476 |
lemma fold_coeffs_pCons_coeff_not_0_eq [simp]: |
|
477 |
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
478 |
by (simp add: fold_coeffs_def) |
|
479 |
||
480 |
lemma fold_coeffs_pCons_not_0_0_eq [simp]: |
|
481 |
"p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
482 |
by (simp add: fold_coeffs_def) |
|
483 |
||
60500 | 484 |
subsection \<open>Canonical morphism on polynomials -- evaluation\<close> |
52380 | 485 |
|
486 |
definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" |
|
487 |
where |
|
61585 | 488 |
"poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" \<comment> \<open>The Horner Schema\<close> |
52380 | 489 |
|
490 |
lemma poly_0 [simp]: |
|
491 |
"poly 0 x = 0" |
|
492 |
by (simp add: poly_def) |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
493 |
|
52380 | 494 |
lemma poly_pCons [simp]: |
495 |
"poly (pCons a p) x = a + x * poly p x" |
|
496 |
by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def) |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
497 |
|
62065 | 498 |
lemma poly_altdef: |
499 |
"poly p (x :: 'a :: {comm_semiring_0, semiring_1}) = (\<Sum>i\<le>degree p. coeff p i * x ^ i)" |
|
500 |
proof (induction p rule: pCons_induct) |
|
501 |
case (pCons a p) |
|
502 |
show ?case |
|
503 |
proof (cases "p = 0") |
|
504 |
case False |
|
505 |
let ?p' = "pCons a p" |
|
506 |
note poly_pCons[of a p x] |
|
507 |
also note pCons.IH |
|
508 |
also have "a + x * (\<Sum>i\<le>degree p. coeff p i * x ^ i) = |
|
509 |
coeff ?p' 0 * x^0 + (\<Sum>i\<le>degree p. coeff ?p' (Suc i) * x^Suc i)" |
|
510 |
by (simp add: field_simps setsum_right_distrib coeff_pCons) |
|
511 |
also note setsum_atMost_Suc_shift[symmetric] |
|
62072 | 512 |
also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric] |
62065 | 513 |
finally show ?thesis . |
514 |
qed simp |
|
515 |
qed simp |
|
516 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
517 |
lemma poly_0_coeff_0: "poly p 0 = coeff p 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
518 |
by (cases p) (auto simp: poly_altdef) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
519 |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
520 |
|
60500 | 521 |
subsection \<open>Monomials\<close> |
29451 | 522 |
|
52380 | 523 |
lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" |
524 |
is "\<lambda>a m n. if m = n then a else 0" |
|
59983
cd2efd7d06bd
replace almost_everywhere_zero by Infinite_Set.MOST
hoelzl
parents:
59815
diff
changeset
|
525 |
by (simp add: MOST_iff_cofinite) |
52380 | 526 |
|
527 |
lemma coeff_monom [simp]: |
|
528 |
"coeff (monom a m) n = (if m = n then a else 0)" |
|
529 |
by transfer rule |
|
29451 | 530 |
|
52380 | 531 |
lemma monom_0: |
532 |
"monom a 0 = pCons a 0" |
|
533 |
by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) |
|
29451 | 534 |
|
52380 | 535 |
lemma monom_Suc: |
536 |
"monom a (Suc n) = pCons 0 (monom a n)" |
|
537 |
by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) |
|
29451 | 538 |
|
539 |
lemma monom_eq_0 [simp]: "monom 0 n = 0" |
|
52380 | 540 |
by (rule poly_eqI) simp |
29451 | 541 |
|
542 |
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0" |
|
52380 | 543 |
by (simp add: poly_eq_iff) |
29451 | 544 |
|
545 |
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b" |
|
52380 | 546 |
by (simp add: poly_eq_iff) |
29451 | 547 |
|
548 |
lemma degree_monom_le: "degree (monom a n) \<le> n" |
|
549 |
by (rule degree_le, simp) |
|
550 |
||
551 |
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n" |
|
552 |
apply (rule order_antisym [OF degree_monom_le]) |
|
553 |
apply (rule le_degree, simp) |
|
554 |
done |
|
555 |
||
52380 | 556 |
lemma coeffs_monom [code abstract]: |
557 |
"coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])" |
|
558 |
by (induct n) (simp_all add: monom_0 monom_Suc) |
|
559 |
||
560 |
lemma fold_coeffs_monom [simp]: |
|
561 |
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a" |
|
562 |
by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff) |
|
563 |
||
564 |
lemma poly_monom: |
|
565 |
fixes a x :: "'a::{comm_semiring_1}" |
|
566 |
shows "poly (monom a n) x = a * x ^ n" |
|
567 |
by (cases "a = 0", simp_all) |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
568 |
(induct n, simp_all add: mult.left_commute poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
569 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
570 |
lemma monom_eq_iff': "monom c n = monom d m \<longleftrightarrow> c = d \<and> (c = 0 \<or> n = m)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
571 |
by (auto simp: poly_eq_iff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
572 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
573 |
lemma monom_eq_const_iff: "monom c n = [:d:] \<longleftrightarrow> c = d \<and> (c = 0 \<or> n = 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
574 |
using monom_eq_iff'[of c n d 0] by (simp add: monom_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
575 |
|
62065 | 576 |
|
60500 | 577 |
subsection \<open>Addition and subtraction\<close> |
29451 | 578 |
|
579 |
instantiation poly :: (comm_monoid_add) comm_monoid_add |
|
580 |
begin |
|
581 |
||
52380 | 582 |
lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
583 |
is "\<lambda>p q n. coeff p n + coeff q n" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
584 |
proof - |
60679 | 585 |
fix q p :: "'a poly" |
586 |
show "\<forall>\<^sub>\<infinity>n. coeff p n + coeff q n = 0" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
587 |
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp |
52380 | 588 |
qed |
29451 | 589 |
|
60679 | 590 |
lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n" |
52380 | 591 |
by (simp add: plus_poly.rep_eq) |
29451 | 592 |
|
60679 | 593 |
instance |
594 |
proof |
|
29451 | 595 |
fix p q r :: "'a poly" |
596 |
show "(p + q) + r = p + (q + r)" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57482
diff
changeset
|
597 |
by (simp add: poly_eq_iff add.assoc) |
29451 | 598 |
show "p + q = q + p" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57482
diff
changeset
|
599 |
by (simp add: poly_eq_iff add.commute) |
29451 | 600 |
show "0 + p = p" |
52380 | 601 |
by (simp add: poly_eq_iff) |
29451 | 602 |
qed |
603 |
||
604 |
end |
|
605 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
606 |
instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
607 |
begin |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
608 |
|
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
609 |
lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
610 |
is "\<lambda>p q n. coeff p n - coeff q n" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
611 |
proof - |
60679 | 612 |
fix q p :: "'a poly" |
613 |
show "\<forall>\<^sub>\<infinity>n. coeff p n - coeff q n = 0" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
614 |
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
615 |
qed |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
616 |
|
60679 | 617 |
lemma coeff_diff [simp]: "coeff (p - q) n = coeff p n - coeff q n" |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
618 |
by (simp add: minus_poly.rep_eq) |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
619 |
|
60679 | 620 |
instance |
621 |
proof |
|
29540 | 622 |
fix p q r :: "'a poly" |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
623 |
show "p + q - p = q" |
52380 | 624 |
by (simp add: poly_eq_iff) |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
625 |
show "p - q - r = p - (q + r)" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
626 |
by (simp add: poly_eq_iff diff_diff_eq) |
29540 | 627 |
qed |
628 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
629 |
end |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
630 |
|
29451 | 631 |
instantiation poly :: (ab_group_add) ab_group_add |
632 |
begin |
|
633 |
||
52380 | 634 |
lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly" |
635 |
is "\<lambda>p n. - coeff p n" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
636 |
proof - |
60679 | 637 |
fix p :: "'a poly" |
638 |
show "\<forall>\<^sub>\<infinity>n. - coeff p n = 0" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
639 |
using MOST_coeff_eq_0 by simp |
52380 | 640 |
qed |
29451 | 641 |
|
642 |
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" |
|
52380 | 643 |
by (simp add: uminus_poly.rep_eq) |
29451 | 644 |
|
60679 | 645 |
instance |
646 |
proof |
|
29451 | 647 |
fix p q :: "'a poly" |
648 |
show "- p + p = 0" |
|
52380 | 649 |
by (simp add: poly_eq_iff) |
29451 | 650 |
show "p - q = p + - q" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
52380
diff
changeset
|
651 |
by (simp add: poly_eq_iff) |
29451 | 652 |
qed |
653 |
||
654 |
end |
|
655 |
||
656 |
lemma add_pCons [simp]: |
|
657 |
"pCons a p + pCons b q = pCons (a + b) (p + q)" |
|
52380 | 658 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 659 |
|
660 |
lemma minus_pCons [simp]: |
|
661 |
"- pCons a p = pCons (- a) (- p)" |
|
52380 | 662 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 663 |
|
664 |
lemma diff_pCons [simp]: |
|
665 |
"pCons a p - pCons b q = pCons (a - b) (p - q)" |
|
52380 | 666 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 667 |
|
29539 | 668 |
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)" |
29451 | 669 |
by (rule degree_le, auto simp add: coeff_eq_0) |
670 |
||
29539 | 671 |
lemma degree_add_le: |
672 |
"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n" |
|
673 |
by (auto intro: order_trans degree_add_le_max) |
|
674 |
||
29453 | 675 |
lemma degree_add_less: |
676 |
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n" |
|
29539 | 677 |
by (auto intro: le_less_trans degree_add_le_max) |
29453 | 678 |
|
29451 | 679 |
lemma degree_add_eq_right: |
680 |
"degree p < degree q \<Longrightarrow> degree (p + q) = degree q" |
|
681 |
apply (cases "q = 0", simp) |
|
682 |
apply (rule order_antisym) |
|
29539 | 683 |
apply (simp add: degree_add_le) |
29451 | 684 |
apply (rule le_degree) |
685 |
apply (simp add: coeff_eq_0) |
|
686 |
done |
|
687 |
||
688 |
lemma degree_add_eq_left: |
|
689 |
"degree q < degree p \<Longrightarrow> degree (p + q) = degree p" |
|
690 |
using degree_add_eq_right [of q p] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57482
diff
changeset
|
691 |
by (simp add: add.commute) |
29451 | 692 |
|
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
693 |
lemma degree_minus [simp]: |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
694 |
"degree (- p) = degree p" |
29451 | 695 |
unfolding degree_def by simp |
696 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
697 |
lemma degree_diff_le_max: |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
698 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
699 |
shows "degree (p - q) \<le> max (degree p) (degree q)" |
29451 | 700 |
using degree_add_le [where p=p and q="-q"] |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
52380
diff
changeset
|
701 |
by simp |
29451 | 702 |
|
29539 | 703 |
lemma degree_diff_le: |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
704 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
705 |
assumes "degree p \<le> n" and "degree q \<le> n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
706 |
shows "degree (p - q) \<le> n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
707 |
using assms degree_add_le [of p n "- q"] by simp |
29539 | 708 |
|
29453 | 709 |
lemma degree_diff_less: |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
710 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
711 |
assumes "degree p < n" and "degree q < n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
712 |
shows "degree (p - q) < n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
713 |
using assms degree_add_less [of p n "- q"] by simp |
29453 | 714 |
|
29451 | 715 |
lemma add_monom: "monom a n + monom b n = monom (a + b) n" |
52380 | 716 |
by (rule poly_eqI) simp |
29451 | 717 |
|
718 |
lemma diff_monom: "monom a n - monom b n = monom (a - b) n" |
|
52380 | 719 |
by (rule poly_eqI) simp |
29451 | 720 |
|
721 |
lemma minus_monom: "- monom a n = monom (-a) n" |
|
52380 | 722 |
by (rule poly_eqI) simp |
29451 | 723 |
|
724 |
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)" |
|
725 |
by (cases "finite A", induct set: finite, simp_all) |
|
726 |
||
727 |
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)" |
|
52380 | 728 |
by (rule poly_eqI) (simp add: coeff_setsum) |
729 |
||
730 |
fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
|
731 |
where |
|
732 |
"plus_coeffs xs [] = xs" |
|
733 |
| "plus_coeffs [] ys = ys" |
|
734 |
| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys" |
|
735 |
||
736 |
lemma coeffs_plus_eq_plus_coeffs [code abstract]: |
|
737 |
"coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)" |
|
738 |
proof - |
|
739 |
{ fix xs ys :: "'a list" and n |
|
740 |
have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n" |
|
741 |
proof (induct xs ys arbitrary: n rule: plus_coeffs.induct) |
|
60679 | 742 |
case (3 x xs y ys n) |
743 |
then show ?case by (cases n) (auto simp add: cCons_def) |
|
52380 | 744 |
qed simp_all } |
745 |
note * = this |
|
746 |
{ fix xs ys :: "'a list" |
|
747 |
assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0" |
|
748 |
moreover assume "plus_coeffs xs ys \<noteq> []" |
|
749 |
ultimately have "last (plus_coeffs xs ys) \<noteq> 0" |
|
750 |
proof (induct xs ys rule: plus_coeffs.induct) |
|
751 |
case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis |
|
752 |
qed simp_all } |
|
753 |
note ** = this |
|
754 |
show ?thesis |
|
755 |
apply (rule coeffs_eqI) |
|
756 |
apply (simp add: * nth_default_coeffs_eq) |
|
757 |
apply (rule **) |
|
758 |
apply (auto dest: last_coeffs_not_0) |
|
759 |
done |
|
760 |
qed |
|
761 |
||
762 |
lemma coeffs_uminus [code abstract]: |
|
763 |
"coeffs (- p) = map (\<lambda>a. - a) (coeffs p)" |
|
764 |
by (rule coeffs_eqI) |
|
765 |
(simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) |
|
766 |
||
767 |
lemma [code]: |
|
768 |
fixes p q :: "'a::ab_group_add poly" |
|
769 |
shows "p - q = p + - q" |
|
59557 | 770 |
by (fact diff_conv_add_uminus) |
52380 | 771 |
|
772 |
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" |
|
773 |
apply (induct p arbitrary: q, simp) |
|
774 |
apply (case_tac q, simp, simp add: algebra_simps) |
|
775 |
done |
|
776 |
||
777 |
lemma poly_minus [simp]: |
|
778 |
fixes x :: "'a::comm_ring" |
|
779 |
shows "poly (- p) x = - poly p x" |
|
780 |
by (induct p) simp_all |
|
781 |
||
782 |
lemma poly_diff [simp]: |
|
783 |
fixes x :: "'a::comm_ring" |
|
784 |
shows "poly (p - q) x = poly p x - poly q x" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
52380
diff
changeset
|
785 |
using poly_add [of p "- q" x] by simp |
52380 | 786 |
|
787 |
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" |
|
788 |
by (induct A rule: infinite_finite_induct) simp_all |
|
29451 | 789 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
790 |
lemma degree_setsum_le: "finite S \<Longrightarrow> (\<And> p . p \<in> S \<Longrightarrow> degree (f p) \<le> n) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
791 |
\<Longrightarrow> degree (setsum f S) \<le> n" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
792 |
proof (induct S rule: finite_induct) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
793 |
case (insert p S) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
794 |
hence "degree (setsum f S) \<le> n" "degree (f p) \<le> n" by auto |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
795 |
thus ?case unfolding setsum.insert[OF insert(1-2)] by (metis degree_add_le) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
796 |
qed simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
797 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
798 |
lemma poly_as_sum_of_monoms': |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
799 |
assumes n: "degree p \<le> n" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
800 |
shows "(\<Sum>i\<le>n. monom (coeff p i) i) = p" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
801 |
proof - |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
802 |
have eq: "\<And>i. {..n} \<inter> {i} = (if i \<le> n then {i} else {})" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
803 |
by auto |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
804 |
show ?thesis |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
805 |
using n by (simp add: poly_eq_iff coeff_setsum coeff_eq_0 setsum.If_cases eq |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
806 |
if_distrib[where f="\<lambda>x. x * a" for a]) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
807 |
qed |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
808 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
809 |
lemma poly_as_sum_of_monoms: "(\<Sum>i\<le>degree p. monom (coeff p i) i) = p" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
810 |
by (intro poly_as_sum_of_monoms' order_refl) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
811 |
|
62065 | 812 |
lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)" |
813 |
by (induction xs) (simp_all add: monom_0 monom_Suc) |
|
814 |
||
29451 | 815 |
|
60500 | 816 |
subsection \<open>Multiplication by a constant, polynomial multiplication and the unit polynomial\<close> |
29451 | 817 |
|
52380 | 818 |
lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
819 |
is "\<lambda>a p n. a * coeff p n" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
820 |
proof - |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
821 |
fix a :: 'a and p :: "'a poly" show "\<forall>\<^sub>\<infinity> i. a * coeff p i = 0" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
822 |
using MOST_coeff_eq_0[of p] by eventually_elim simp |
52380 | 823 |
qed |
29451 | 824 |
|
52380 | 825 |
lemma coeff_smult [simp]: |
826 |
"coeff (smult a p) n = a * coeff p n" |
|
827 |
by (simp add: smult.rep_eq) |
|
29451 | 828 |
|
829 |
lemma degree_smult_le: "degree (smult a p) \<le> degree p" |
|
830 |
by (rule degree_le, simp add: coeff_eq_0) |
|
831 |
||
29472 | 832 |
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57482
diff
changeset
|
833 |
by (rule poly_eqI, simp add: mult.assoc) |
29451 | 834 |
|
835 |
lemma smult_0_right [simp]: "smult a 0 = 0" |
|
52380 | 836 |
by (rule poly_eqI, simp) |
29451 | 837 |
|
838 |
lemma smult_0_left [simp]: "smult 0 p = 0" |
|
52380 | 839 |
by (rule poly_eqI, simp) |
29451 | 840 |
|
841 |
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" |
|
52380 | 842 |
by (rule poly_eqI, simp) |
29451 | 843 |
|
844 |
lemma smult_add_right: |
|
845 |
"smult a (p + q) = smult a p + smult a q" |
|
52380 | 846 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 847 |
|
848 |
lemma smult_add_left: |
|
849 |
"smult (a + b) p = smult a p + smult b p" |
|
52380 | 850 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 851 |
|
29457 | 852 |
lemma smult_minus_right [simp]: |
29451 | 853 |
"smult (a::'a::comm_ring) (- p) = - smult a p" |
52380 | 854 |
by (rule poly_eqI, simp) |
29451 | 855 |
|
29457 | 856 |
lemma smult_minus_left [simp]: |
29451 | 857 |
"smult (- a::'a::comm_ring) p = - smult a p" |
52380 | 858 |
by (rule poly_eqI, simp) |
29451 | 859 |
|
860 |
lemma smult_diff_right: |
|
861 |
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" |
|
52380 | 862 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 863 |
|
864 |
lemma smult_diff_left: |
|
865 |
"smult (a - b::'a::comm_ring) p = smult a p - smult b p" |
|
52380 | 866 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 867 |
|
29472 | 868 |
lemmas smult_distribs = |
869 |
smult_add_left smult_add_right |
|
870 |
smult_diff_left smult_diff_right |
|
871 |
||
29451 | 872 |
lemma smult_pCons [simp]: |
873 |
"smult a (pCons b p) = pCons (a * b) (smult a p)" |
|
52380 | 874 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 875 |
|
876 |
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" |
|
877 |
by (induct n, simp add: monom_0, simp add: monom_Suc) |
|
878 |
||
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
879 |
lemma smult_Poly: "smult c (Poly xs) = Poly (map (op * c) xs)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
880 |
by (auto simp add: poly_eq_iff coeff_Poly_eq nth_default_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
881 |
|
29659 | 882 |
lemma degree_smult_eq [simp]: |
63498 | 883 |
fixes a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" |
29659 | 884 |
shows "degree (smult a p) = (if a = 0 then 0 else degree p)" |
885 |
by (cases "a = 0", simp, simp add: degree_def) |
|
886 |
||
887 |
lemma smult_eq_0_iff [simp]: |
|
63498 | 888 |
fixes a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" |
29659 | 889 |
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0" |
52380 | 890 |
by (simp add: poly_eq_iff) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
891 |
|
52380 | 892 |
lemma coeffs_smult [code abstract]: |
63498 | 893 |
fixes p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
52380 | 894 |
shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))" |
895 |
by (rule coeffs_eqI) |
|
896 |
(auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) |
|
63498 | 897 |
|
29451 | 898 |
instantiation poly :: (comm_semiring_0) comm_semiring_0 |
899 |
begin |
|
900 |
||
901 |
definition |
|
52380 | 902 |
"p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0" |
29474 | 903 |
|
904 |
lemma mult_poly_0_left: "(0::'a poly) * q = 0" |
|
52380 | 905 |
by (simp add: times_poly_def) |
29474 | 906 |
|
907 |
lemma mult_pCons_left [simp]: |
|
908 |
"pCons a p * q = smult a q + pCons 0 (p * q)" |
|
52380 | 909 |
by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def) |
29474 | 910 |
|
911 |
lemma mult_poly_0_right: "p * (0::'a poly) = 0" |
|
52380 | 912 |
by (induct p) (simp add: mult_poly_0_left, simp) |
29451 | 913 |
|
29474 | 914 |
lemma mult_pCons_right [simp]: |
915 |
"p * pCons a q = smult a p + pCons 0 (p * q)" |
|
52380 | 916 |
by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps) |
29474 | 917 |
|
918 |
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right |
|
919 |
||
52380 | 920 |
lemma mult_smult_left [simp]: |
921 |
"smult a p * q = smult a (p * q)" |
|
922 |
by (induct p) (simp add: mult_poly_0, simp add: smult_add_right) |
|
29474 | 923 |
|
52380 | 924 |
lemma mult_smult_right [simp]: |
925 |
"p * smult a q = smult a (p * q)" |
|
926 |
by (induct q) (simp add: mult_poly_0, simp add: smult_add_right) |
|
29474 | 927 |
|
928 |
lemma mult_poly_add_left: |
|
929 |
fixes p q r :: "'a poly" |
|
930 |
shows "(p + q) * r = p * r + q * r" |
|
52380 | 931 |
by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps) |
29451 | 932 |
|
60679 | 933 |
instance |
934 |
proof |
|
29451 | 935 |
fix p q r :: "'a poly" |
936 |
show 0: "0 * p = 0" |
|
29474 | 937 |
by (rule mult_poly_0_left) |
29451 | 938 |
show "p * 0 = 0" |
29474 | 939 |
by (rule mult_poly_0_right) |
29451 | 940 |
show "(p + q) * r = p * r + q * r" |
29474 | 941 |
by (rule mult_poly_add_left) |
29451 | 942 |
show "(p * q) * r = p * (q * r)" |
29474 | 943 |
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) |
29451 | 944 |
show "p * q = q * p" |
29474 | 945 |
by (induct p, simp add: mult_poly_0, simp) |
29451 | 946 |
qed |
947 |
||
948 |
end |
|
949 |
||
63498 | 950 |
lemma coeff_mult_degree_sum: |
951 |
"coeff (p * q) (degree p + degree q) = |
|
952 |
coeff p (degree p) * coeff q (degree q)" |
|
953 |
by (induct p, simp, simp add: coeff_eq_0) |
|
954 |
||
955 |
instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors |
|
956 |
proof |
|
957 |
fix p q :: "'a poly" |
|
958 |
assume "p \<noteq> 0" and "q \<noteq> 0" |
|
959 |
have "coeff (p * q) (degree p + degree q) = |
|
960 |
coeff p (degree p) * coeff q (degree q)" |
|
961 |
by (rule coeff_mult_degree_sum) |
|
962 |
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" |
|
963 |
using \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> by simp |
|
964 |
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. |
|
965 |
thus "p * q \<noteq> 0" by (simp add: poly_eq_iff) |
|
966 |
qed |
|
967 |
||
29540 | 968 |
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
969 |
||
29474 | 970 |
lemma coeff_mult: |
971 |
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))" |
|
972 |
proof (induct p arbitrary: n) |
|
973 |
case 0 show ?case by simp |
|
974 |
next |
|
975 |
case (pCons a p n) thus ?case |
|
976 |
by (cases n, simp, simp add: setsum_atMost_Suc_shift |
|
977 |
del: setsum_atMost_Suc) |
|
978 |
qed |
|
29451 | 979 |
|
29474 | 980 |
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" |
981 |
apply (rule degree_le) |
|
982 |
apply (induct p) |
|
983 |
apply simp |
|
984 |
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) |
|
29451 | 985 |
done |
986 |
||
987 |
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" |
|
60679 | 988 |
by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc) |
29451 | 989 |
|
990 |
instantiation poly :: (comm_semiring_1) comm_semiring_1 |
|
991 |
begin |
|
992 |
||
60679 | 993 |
definition one_poly_def: "1 = pCons 1 0" |
29451 | 994 |
|
60679 | 995 |
instance |
996 |
proof |
|
997 |
show "1 * p = p" for p :: "'a poly" |
|
52380 | 998 |
unfolding one_poly_def by simp |
29451 | 999 |
show "0 \<noteq> (1::'a poly)" |
1000 |
unfolding one_poly_def by simp |
|
1001 |
qed |
|
1002 |
||
1003 |
end |
|
1004 |
||
63498 | 1005 |
instance poly :: ("{comm_semiring_1,semiring_1_no_zero_divisors}") semiring_1_no_zero_divisors .. |
1006 |
||
52380 | 1007 |
instance poly :: (comm_ring) comm_ring .. |
1008 |
||
1009 |
instance poly :: (comm_ring_1) comm_ring_1 .. |
|
1010 |
||
63498 | 1011 |
instance poly :: (comm_ring_1) comm_semiring_1_cancel .. |
1012 |
||
29451 | 1013 |
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" |
1014 |
unfolding one_poly_def |
|
1015 |
by (simp add: coeff_pCons split: nat.split) |
|
1016 |
||
60570 | 1017 |
lemma monom_eq_1 [simp]: |
1018 |
"monom 1 0 = 1" |
|
1019 |
by (simp add: monom_0 one_poly_def) |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1020 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1021 |
lemma monom_eq_1_iff: "monom c n = 1 \<longleftrightarrow> c = 1 \<and> n = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1022 |
using monom_eq_const_iff[of c n 1] by (auto simp: one_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1023 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1024 |
lemma monom_altdef: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1025 |
"monom c n = smult c ([:0, 1:]^n)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1026 |
by (induction n) (simp_all add: monom_0 monom_Suc one_poly_def) |
60570 | 1027 |
|
29451 | 1028 |
lemma degree_1 [simp]: "degree 1 = 0" |
1029 |
unfolding one_poly_def |
|
1030 |
by (rule degree_pCons_0) |
|
1031 |
||
52380 | 1032 |
lemma coeffs_1_eq [simp, code abstract]: |
1033 |
"coeffs 1 = [1]" |
|
1034 |
by (simp add: one_poly_def) |
|
1035 |
||
1036 |
lemma degree_power_le: |
|
1037 |
"degree (p ^ n) \<le> degree p * n" |
|
1038 |
by (induct n) (auto intro: order_trans degree_mult_le) |
|
1039 |
||
1040 |
lemma poly_smult [simp]: |
|
1041 |
"poly (smult a p) x = a * poly p x" |
|
1042 |
by (induct p, simp, simp add: algebra_simps) |
|
1043 |
||
1044 |
lemma poly_mult [simp]: |
|
1045 |
"poly (p * q) x = poly p x * poly q x" |
|
1046 |
by (induct p, simp_all, simp add: algebra_simps) |
|
1047 |
||
1048 |
lemma poly_1 [simp]: |
|
1049 |
"poly 1 x = 1" |
|
1050 |
by (simp add: one_poly_def) |
|
1051 |
||
1052 |
lemma poly_power [simp]: |
|
1053 |
fixes p :: "'a::{comm_semiring_1} poly" |
|
1054 |
shows "poly (p ^ n) x = poly p x ^ n" |
|
1055 |
by (induct n) simp_all |
|
1056 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1057 |
lemma poly_setprod: "poly (\<Prod>k\<in>A. p k) x = (\<Prod>k\<in>A. poly (p k) x)" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1058 |
by (induct A rule: infinite_finite_induct) simp_all |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1059 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1060 |
lemma degree_setprod_setsum_le: "finite S \<Longrightarrow> degree (setprod f S) \<le> setsum (degree o f) S" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1061 |
proof (induct S rule: finite_induct) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1062 |
case (insert a S) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1063 |
show ?case unfolding setprod.insert[OF insert(1-2)] setsum.insert[OF insert(1-2)] |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1064 |
by (rule le_trans[OF degree_mult_le], insert insert, auto) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1065 |
qed simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1066 |
|
62065 | 1067 |
subsection \<open>Conversions from natural numbers\<close> |
1068 |
||
1069 |
lemma of_nat_poly: "of_nat n = [:of_nat n :: 'a :: comm_semiring_1:]" |
|
1070 |
proof (induction n) |
|
1071 |
case (Suc n) |
|
1072 |
hence "of_nat (Suc n) = 1 + (of_nat n :: 'a poly)" |
|
1073 |
by simp |
|
1074 |
also have "(of_nat n :: 'a poly) = [: of_nat n :]" |
|
1075 |
by (subst Suc) (rule refl) |
|
1076 |
also have "1 = [:1:]" by (simp add: one_poly_def) |
|
1077 |
finally show ?case by (subst (asm) add_pCons) simp |
|
1078 |
qed simp |
|
1079 |
||
1080 |
lemma degree_of_nat [simp]: "degree (of_nat n) = 0" |
|
1081 |
by (simp add: of_nat_poly) |
|
1082 |
||
1083 |
lemma degree_numeral [simp]: "degree (numeral n) = 0" |
|
1084 |
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp |
|
1085 |
||
1086 |
lemma numeral_poly: "numeral n = [:numeral n:]" |
|
1087 |
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp |
|
52380 | 1088 |
|
60500 | 1089 |
subsection \<open>Lemmas about divisibility\<close> |
29979 | 1090 |
|
1091 |
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q" |
|
1092 |
proof - |
|
1093 |
assume "p dvd q" |
|
1094 |
then obtain k where "q = p * k" .. |
|
1095 |
then have "smult a q = p * smult a k" by simp |
|
1096 |
then show "p dvd smult a q" .. |
|
1097 |
qed |
|
1098 |
||
1099 |
lemma dvd_smult_cancel: |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1100 |
fixes a :: "'a :: field" |
29979 | 1101 |
shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q" |
1102 |
by (drule dvd_smult [where a="inverse a"]) simp |
|
1103 |
||
1104 |
lemma dvd_smult_iff: |
|
1105 |
fixes a :: "'a::field" |
|
1106 |
shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q" |
|
1107 |
by (safe elim!: dvd_smult dvd_smult_cancel) |
|
1108 |
||
31663 | 1109 |
lemma smult_dvd_cancel: |
1110 |
"smult a p dvd q \<Longrightarrow> p dvd q" |
|
1111 |
proof - |
|
1112 |
assume "smult a p dvd q" |
|
1113 |
then obtain k where "q = smult a p * k" .. |
|
1114 |
then have "q = p * smult a k" by simp |
|
1115 |
then show "p dvd q" .. |
|
1116 |
qed |
|
1117 |
||
1118 |
lemma smult_dvd: |
|
1119 |
fixes a :: "'a::field" |
|
1120 |
shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q" |
|
1121 |
by (rule smult_dvd_cancel [where a="inverse a"]) simp |
|
1122 |
||
1123 |
lemma smult_dvd_iff: |
|
1124 |
fixes a :: "'a::field" |
|
1125 |
shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)" |
|
1126 |
by (auto elim: smult_dvd smult_dvd_cancel) |
|
1127 |
||
29451 | 1128 |
|
60500 | 1129 |
subsection \<open>Polynomials form an integral domain\<close> |
29451 | 1130 |
|
63498 | 1131 |
instance poly :: (idom) idom .. |
29451 | 1132 |
|
1133 |
lemma degree_mult_eq: |
|
63498 | 1134 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
29451 | 1135 |
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" |
1136 |
apply (rule order_antisym [OF degree_mult_le le_degree]) |
|
1137 |
apply (simp add: coeff_mult_degree_sum) |
|
1138 |
done |
|
1139 |
||
60570 | 1140 |
lemma degree_mult_right_le: |
63498 | 1141 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
60570 | 1142 |
assumes "q \<noteq> 0" |
1143 |
shows "degree p \<le> degree (p * q)" |
|
1144 |
using assms by (cases "p = 0") (simp_all add: degree_mult_eq) |
|
1145 |
||
1146 |
lemma coeff_degree_mult: |
|
63498 | 1147 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
60570 | 1148 |
shows "coeff (p * q) (degree (p * q)) = |
1149 |
coeff q (degree q) * coeff p (degree p)" |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1150 |
by (cases "p = 0 \<or> q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum mult_ac) |
60570 | 1151 |
|
29451 | 1152 |
lemma dvd_imp_degree_le: |
63498 | 1153 |
fixes p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" |
29451 | 1154 |
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1155 |
by (erule dvdE, hypsubst, subst degree_mult_eq) auto |
29451 | 1156 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1157 |
lemma divides_degree: |
63498 | 1158 |
assumes pq: "p dvd (q :: 'a ::{comm_semiring_1,semiring_no_zero_divisors} poly)" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1159 |
shows "degree p \<le> degree q \<or> q = 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1160 |
by (metis dvd_imp_degree_le pq) |
63498 | 1161 |
|
1162 |
lemma const_poly_dvd_iff: |
|
1163 |
fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}" |
|
1164 |
shows "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)" |
|
1165 |
proof (cases "c = 0 \<or> p = 0") |
|
1166 |
case False |
|
1167 |
show ?thesis |
|
1168 |
proof |
|
1169 |
assume "[:c:] dvd p" |
|
1170 |
thus "\<forall>n. c dvd coeff p n" by (auto elim!: dvdE simp: coeffs_def) |
|
1171 |
next |
|
1172 |
assume *: "\<forall>n. c dvd coeff p n" |
|
1173 |
define mydiv where "mydiv = (\<lambda>x y :: 'a. SOME z. x = y * z)" |
|
1174 |
have mydiv: "x = y * mydiv x y" if "y dvd x" for x y |
|
1175 |
using that unfolding mydiv_def dvd_def by (rule someI_ex) |
|
1176 |
define q where "q = Poly (map (\<lambda>a. mydiv a c) (coeffs p))" |
|
1177 |
from False * have "p = q * [:c:]" |
|
1178 |
by (intro poly_eqI) (auto simp: q_def nth_default_def not_less length_coeffs_degree |
|
1179 |
coeffs_nth intro!: coeff_eq_0 mydiv) |
|
1180 |
thus "[:c:] dvd p" by (simp only: dvd_triv_right) |
|
1181 |
qed |
|
1182 |
qed (auto intro!: poly_eqI) |
|
1183 |
||
1184 |
lemma const_poly_dvd_const_poly_iff [simp]: |
|
1185 |
"[:a::'a::{comm_semiring_1,semiring_no_zero_divisors}:] dvd [:b:] \<longleftrightarrow> a dvd b" |
|
1186 |
by (subst const_poly_dvd_iff) (auto simp: coeff_pCons split: nat.splits) |
|
1187 |
||
29451 | 1188 |
|
60500 | 1189 |
subsection \<open>Polynomials form an ordered integral domain\<close> |
29878 | 1190 |
|
63498 | 1191 |
definition pos_poly :: "'a::linordered_semidom poly \<Rightarrow> bool" |
29878 | 1192 |
where |
1193 |
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)" |
|
1194 |
||
1195 |
lemma pos_poly_pCons: |
|
1196 |
"pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)" |
|
1197 |
unfolding pos_poly_def by simp |
|
1198 |
||
1199 |
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0" |
|
1200 |
unfolding pos_poly_def by simp |
|
1201 |
||
1202 |
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)" |
|
1203 |
apply (induct p arbitrary: q, simp) |
|
1204 |
apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos) |
|
1205 |
done |
|
1206 |
||
1207 |
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)" |
|
1208 |
unfolding pos_poly_def |
|
1209 |
apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0") |
|
56544 | 1210 |
apply (simp add: degree_mult_eq coeff_mult_degree_sum) |
29878 | 1211 |
apply auto |
1212 |
done |
|
1213 |
||
63498 | 1214 |
lemma pos_poly_total: "(p :: 'a :: linordered_idom poly) = 0 \<or> pos_poly p \<or> pos_poly (- p)" |
29878 | 1215 |
by (induct p) (auto simp add: pos_poly_pCons) |
1216 |
||
52380 | 1217 |
lemma last_coeffs_eq_coeff_degree: |
1218 |
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)" |
|
1219 |
by (simp add: coeffs_def) |
|
1220 |
||
1221 |
lemma pos_poly_coeffs [code]: |
|
1222 |
"pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q") |
|
1223 |
proof |
|
1224 |
assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree) |
|
1225 |
next |
|
1226 |
assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def) |
|
1227 |
then have "p \<noteq> 0" by auto |
|
1228 |
with * show ?Q by (simp add: last_coeffs_eq_coeff_degree) |
|
1229 |
qed |
|
1230 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1231 |
instantiation poly :: (linordered_idom) linordered_idom |
29878 | 1232 |
begin |
1233 |
||
1234 |
definition |
|
37765 | 1235 |
"x < y \<longleftrightarrow> pos_poly (y - x)" |
29878 | 1236 |
|
1237 |
definition |
|
37765 | 1238 |
"x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)" |
29878 | 1239 |
|
1240 |
definition |
|
61945 | 1241 |
"\<bar>x::'a poly\<bar> = (if x < 0 then - x else x)" |
29878 | 1242 |
|
1243 |
definition |
|
37765 | 1244 |
"sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
29878 | 1245 |
|
60679 | 1246 |
instance |
1247 |
proof |
|
1248 |
fix x y z :: "'a poly" |
|
29878 | 1249 |
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
1250 |
unfolding less_eq_poly_def less_poly_def |
|
1251 |
apply safe |
|
1252 |
apply simp |
|
1253 |
apply (drule (1) pos_poly_add) |
|
1254 |
apply simp |
|
1255 |
done |
|
60679 | 1256 |
show "x \<le> x" |
29878 | 1257 |
unfolding less_eq_poly_def by simp |
60679 | 1258 |
show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
29878 | 1259 |
unfolding less_eq_poly_def |
1260 |
apply safe |
|
1261 |
apply (drule (1) pos_poly_add) |
|
1262 |
apply (simp add: algebra_simps) |
|
1263 |
done |
|
60679 | 1264 |
show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
29878 | 1265 |
unfolding less_eq_poly_def |
1266 |
apply safe |
|
1267 |
apply (drule (1) pos_poly_add) |
|
1268 |
apply simp |
|
1269 |
done |
|
60679 | 1270 |
show "x \<le> y \<Longrightarrow> z + x \<le> z + y" |
29878 | 1271 |
unfolding less_eq_poly_def |
1272 |
apply safe |
|
1273 |
apply (simp add: algebra_simps) |
|
1274 |
done |
|
1275 |
show "x \<le> y \<or> y \<le> x" |
|
1276 |
unfolding less_eq_poly_def |
|
1277 |
using pos_poly_total [of "x - y"] |
|
1278 |
by auto |
|
60679 | 1279 |
show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y" |
29878 | 1280 |
unfolding less_poly_def |
1281 |
by (simp add: right_diff_distrib [symmetric] pos_poly_mult) |
|
1282 |
show "\<bar>x\<bar> = (if x < 0 then - x else x)" |
|
1283 |
by (rule abs_poly_def) |
|
1284 |
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
|
1285 |
by (rule sgn_poly_def) |
|
1286 |
qed |
|
1287 |
||
1288 |
end |
|
1289 |
||
60500 | 1290 |
text \<open>TODO: Simplification rules for comparisons\<close> |
29878 | 1291 |
|
1292 |
||
60500 | 1293 |
subsection \<open>Synthetic division and polynomial roots\<close> |
52380 | 1294 |
|
60500 | 1295 |
text \<open> |
52380 | 1296 |
Synthetic division is simply division by the linear polynomial @{term "x - c"}. |
60500 | 1297 |
\<close> |
52380 | 1298 |
|
1299 |
definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" |
|
1300 |
where |
|
1301 |
"synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)" |
|
1302 |
||
1303 |
definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" |
|
1304 |
where |
|
1305 |
"synthetic_div p c = fst (synthetic_divmod p c)" |
|
1306 |
||
1307 |
lemma synthetic_divmod_0 [simp]: |
|
1308 |
"synthetic_divmod 0 c = (0, 0)" |
|
1309 |
by (simp add: synthetic_divmod_def) |
|
1310 |
||
1311 |
lemma synthetic_divmod_pCons [simp]: |
|
1312 |
"synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" |
|
1313 |
by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def) |
|
1314 |
||
1315 |
lemma synthetic_div_0 [simp]: |
|
1316 |
"synthetic_div 0 c = 0" |
|
1317 |
unfolding synthetic_div_def by simp |
|
1318 |
||
1319 |
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" |
|
1320 |
by (induct p arbitrary: a) simp_all |
|
1321 |
||
1322 |
lemma snd_synthetic_divmod: |
|
1323 |
"snd (synthetic_divmod p c) = poly p c" |
|
1324 |
by (induct p, simp, simp add: split_def) |
|
1325 |
||
1326 |
lemma synthetic_div_pCons [simp]: |
|
1327 |
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" |
|
1328 |
unfolding synthetic_div_def |
|
1329 |
by (simp add: split_def snd_synthetic_divmod) |
|
1330 |
||
1331 |
lemma synthetic_div_eq_0_iff: |
|
1332 |
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0" |
|
63649 | 1333 |
proof (induct p) |
1334 |
case 0 |
|
1335 |
then show ?case by simp |
|
1336 |
next |
|
1337 |
case (pCons a p) |
|
1338 |
then show ?case by (cases p) simp |
|
1339 |
qed |
|
52380 | 1340 |
|
1341 |
lemma degree_synthetic_div: |
|
1342 |
"degree (synthetic_div p c) = degree p - 1" |
|
63649 | 1343 |
by (induct p) (simp_all add: synthetic_div_eq_0_iff) |
52380 | 1344 |
|
1345 |
lemma synthetic_div_correct: |
|
1346 |
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" |
|
1347 |
by (induct p) simp_all |
|
1348 |
||
1349 |
lemma synthetic_div_unique: |
|
1350 |
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" |
|
1351 |
apply (induct p arbitrary: q r) |
|
1352 |
apply (simp, frule synthetic_div_unique_lemma, simp) |
|
1353 |
apply (case_tac q, force) |
|
1354 |
done |
|
1355 |
||
1356 |
lemma synthetic_div_correct': |
|
1357 |
fixes c :: "'a::comm_ring_1" |
|
1358 |
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" |
|
1359 |
using synthetic_div_correct [of p c] |
|
1360 |
by (simp add: algebra_simps) |
|
1361 |
||
1362 |
lemma poly_eq_0_iff_dvd: |
|
63498 | 1363 |
fixes c :: "'a::{comm_ring_1}" |
52380 | 1364 |
shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p" |
1365 |
proof |
|
1366 |
assume "poly p c = 0" |
|
1367 |
with synthetic_div_correct' [of c p] |
|
1368 |
have "p = [:-c, 1:] * synthetic_div p c" by simp |
|
1369 |
then show "[:-c, 1:] dvd p" .. |
|
1370 |
next |
|
1371 |
assume "[:-c, 1:] dvd p" |
|
1372 |
then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) |
|
1373 |
then show "poly p c = 0" by simp |
|
1374 |
qed |
|
1375 |
||
1376 |
lemma dvd_iff_poly_eq_0: |
|
63498 | 1377 |
fixes c :: "'a::{comm_ring_1}" |
52380 | 1378 |
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0" |
1379 |
by (simp add: poly_eq_0_iff_dvd) |
|
1380 |
||
1381 |
lemma poly_roots_finite: |
|
63498 | 1382 |
fixes p :: "'a::{comm_ring_1,ring_no_zero_divisors} poly" |
52380 | 1383 |
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" |
1384 |
proof (induct n \<equiv> "degree p" arbitrary: p) |
|
1385 |
case (0 p) |
|
1386 |
then obtain a where "a \<noteq> 0" and "p = [:a:]" |
|
1387 |
by (cases p, simp split: if_splits) |
|
1388 |
then show "finite {x. poly p x = 0}" by simp |
|
1389 |
next |
|
1390 |
case (Suc n p) |
|
1391 |
show "finite {x. poly p x = 0}" |
|
1392 |
proof (cases "\<exists>x. poly p x = 0") |
|
1393 |
case False |
|
1394 |
then show "finite {x. poly p x = 0}" by simp |
|
1395 |
next |
|
1396 |
case True |
|
1397 |
then obtain a where "poly p a = 0" .. |
|
1398 |
then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) |
|
1399 |
then obtain k where k: "p = [:-a, 1:] * k" .. |
|
60500 | 1400 |
with \<open>p \<noteq> 0\<close> have "k \<noteq> 0" by auto |
52380 | 1401 |
with k have "degree p = Suc (degree k)" |
1402 |
by (simp add: degree_mult_eq del: mult_pCons_left) |
|
60500 | 1403 |
with \<open>Suc n = degree p\<close> have "n = degree k" by simp |
1404 |
then have "finite {x. poly k x = 0}" using \<open>k \<noteq> 0\<close> by (rule Suc.hyps) |
|
52380 | 1405 |
then have "finite (insert a {x. poly k x = 0})" by simp |
1406 |
then show "finite {x. poly p x = 0}" |
|
57862 | 1407 |
by (simp add: k Collect_disj_eq del: mult_pCons_left) |
52380 | 1408 |
qed |
1409 |
qed |
|
1410 |
||
1411 |
lemma poly_eq_poly_eq_iff: |
|
63498 | 1412 |
fixes p q :: "'a::{comm_ring_1,ring_no_zero_divisors,ring_char_0} poly" |
52380 | 1413 |
shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q") |
1414 |
proof |
|
1415 |
assume ?Q then show ?P by simp |
|
1416 |
next |
|
63498 | 1417 |
{ fix p :: "'a poly" |
52380 | 1418 |
have "poly p = poly 0 \<longleftrightarrow> p = 0" |
1419 |
apply (cases "p = 0", simp_all) |
|
1420 |
apply (drule poly_roots_finite) |
|
1421 |
apply (auto simp add: infinite_UNIV_char_0) |
|
1422 |
done |
|
1423 |
} note this [of "p - q"] |
|
1424 |
moreover assume ?P |
|
1425 |
ultimately show ?Q by auto |
|
1426 |
qed |
|
1427 |
||
1428 |
lemma poly_all_0_iff_0: |
|
63498 | 1429 |
fixes p :: "'a::{ring_char_0, comm_ring_1,ring_no_zero_divisors} poly" |
52380 | 1430 |
shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0" |
1431 |
by (auto simp add: poly_eq_poly_eq_iff [symmetric]) |
|
1432 |
||
1433 |
||
60500 | 1434 |
subsection \<open>Long division of polynomials\<close> |
29451 | 1435 |
|
52380 | 1436 |
definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" |
29451 | 1437 |
where |
29537 | 1438 |
"pdivmod_rel x y q r \<longleftrightarrow> |
29451 | 1439 |
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" |
1440 |
||
29537 | 1441 |
lemma pdivmod_rel_0: |
1442 |
"pdivmod_rel 0 y 0 0" |
|
1443 |
unfolding pdivmod_rel_def by simp |
|
29451 | 1444 |
|
29537 | 1445 |
lemma pdivmod_rel_by_0: |
1446 |
"pdivmod_rel x 0 0 x" |
|
1447 |
unfolding pdivmod_rel_def by simp |
|
29451 | 1448 |
|
1449 |
lemma eq_zero_or_degree_less: |
|
1450 |
assumes "degree p \<le> n" and "coeff p n = 0" |
|
1451 |
shows "p = 0 \<or> degree p < n" |
|
1452 |
proof (cases n) |
|
1453 |
case 0 |
|
60500 | 1454 |
with \<open>degree p \<le> n\<close> and \<open>coeff p n = 0\<close> |
29451 | 1455 |
have "coeff p (degree p) = 0" by simp |
1456 |
then have "p = 0" by simp |
|
1457 |
then show ?thesis .. |
|
1458 |
next |
|
1459 |
case (Suc m) |
|
1460 |
have "\<forall>i>n. coeff p i = 0" |
|
60500 | 1461 |
using \<open>degree p \<le> n\<close> by (simp add: coeff_eq_0) |
29451 | 1462 |
then have "\<forall>i\<ge>n. coeff p i = 0" |
60500 | 1463 |
using \<open>coeff p n = 0\<close> by (simp add: le_less) |
29451 | 1464 |
then have "\<forall>i>m. coeff p i = 0" |
60500 | 1465 |
using \<open>n = Suc m\<close> by (simp add: less_eq_Suc_le) |
29451 | 1466 |
then have "degree p \<le> m" |
1467 |
by (rule degree_le) |
|
1468 |
then have "degree p < n" |
|
60500 | 1469 |
using \<open>n = Suc m\<close> by (simp add: less_Suc_eq_le) |
29451 | 1470 |
then show ?thesis .. |
1471 |
qed |
|
1472 |
||
29537 | 1473 |
lemma pdivmod_rel_pCons: |
1474 |
assumes rel: "pdivmod_rel x y q r" |
|
29451 | 1475 |
assumes y: "y \<noteq> 0" |
1476 |
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" |
|
29537 | 1477 |
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)" |
1478 |
(is "pdivmod_rel ?x y ?q ?r") |
|
29451 | 1479 |
proof - |
1480 |
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" |
|
29537 | 1481 |
using assms unfolding pdivmod_rel_def by simp_all |
29451 | 1482 |
|
1483 |
have 1: "?x = ?q * y + ?r" |
|
1484 |
using b x by simp |
|
1485 |
||
1486 |
have 2: "?r = 0 \<or> degree ?r < degree y" |
|
1487 |
proof (rule eq_zero_or_degree_less) |
|
29539 | 1488 |
show "degree ?r \<le> degree y" |
1489 |
proof (rule degree_diff_le) |
|
29451 | 1490 |
show "degree (pCons a r) \<le> degree y" |
29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1491 |
using r by auto |
29451 | 1492 |
show "degree (smult b y) \<le> degree y" |
1493 |
by (rule degree_smult_le) |
|
1494 |
qed |
|
1495 |
next |
|
1496 |
show "coeff ?r (degree y) = 0" |
|
60500 | 1497 |
using \<open>y \<noteq> 0\<close> unfolding b by simp |
29451 | 1498 |
qed |
1499 |
||
1500 |
from 1 2 show ?thesis |
|
29537 | 1501 |
unfolding pdivmod_rel_def |
60500 | 1502 |
using \<open>y \<noteq> 0\<close> by simp |
29451 | 1503 |
qed |
1504 |
||
29537 | 1505 |
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r" |
29451 | 1506 |
apply (cases "y = 0") |
29537 | 1507 |
apply (fast intro!: pdivmod_rel_by_0) |
29451 | 1508 |
apply (induct x) |
29537 | 1509 |
apply (fast intro!: pdivmod_rel_0) |
1510 |
apply (fast intro!: pdivmod_rel_pCons) |
|
29451 | 1511 |
done |
1512 |
||
29537 | 1513 |
lemma pdivmod_rel_unique: |
1514 |
assumes 1: "pdivmod_rel x y q1 r1" |
|
1515 |
assumes 2: "pdivmod_rel x y q2 r2" |
|
29451 | 1516 |
shows "q1 = q2 \<and> r1 = r2" |
1517 |
proof (cases "y = 0") |
|
1518 |
assume "y = 0" with assms show ?thesis |
|
29537 | 1519 |
by (simp add: pdivmod_rel_def) |
29451 | 1520 |
next |
1521 |
assume [simp]: "y \<noteq> 0" |
|
1522 |
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" |
|
29537 | 1523 |
unfolding pdivmod_rel_def by simp_all |
29451 | 1524 |
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" |
29537 | 1525 |
unfolding pdivmod_rel_def by simp_all |
29451 | 1526 |
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" |
29667 | 1527 |
by (simp add: algebra_simps) |
29451 | 1528 |
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y" |
29453 | 1529 |
by (auto intro: degree_diff_less) |
29451 | 1530 |
|
1531 |
show "q1 = q2 \<and> r1 = r2" |
|
1532 |
proof (rule ccontr) |
|
1533 |
assume "\<not> (q1 = q2 \<and> r1 = r2)" |
|
1534 |
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto |
|
1535 |
with r3 have "degree (r2 - r1) < degree y" by simp |
|
1536 |
also have "degree y \<le> degree (q1 - q2) + degree y" by simp |
|
1537 |
also have "\<dots> = degree ((q1 - q2) * y)" |
|
60500 | 1538 |
using \<open>q1 \<noteq> q2\<close> by (simp add: degree_mult_eq) |
29451 | 1539 |
also have "\<dots> = degree (r2 - r1)" |
1540 |
using q3 by simp |
|
1541 |
finally have "degree (r2 - r1) < degree (r2 - r1)" . |
|
1542 |
then show "False" by simp |
|
1543 |
qed |
|
1544 |
qed |
|
1545 |
||
29660 | 1546 |
lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0" |
1547 |
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0) |
|
1548 |
||
1549 |
lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x" |
|
1550 |
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0) |
|
1551 |
||
45605 | 1552 |
lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1] |
29451 | 1553 |
|
45605 | 1554 |
lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2] |
29451 | 1555 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1556 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1557 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1558 |
subsection\<open>Pseudo-Division and Division of Polynomials\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1559 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1560 |
text\<open>This part is by René Thiemann and Akihisa Yamada.\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1561 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1562 |
fun pseudo_divmod_main :: "'a :: comm_ring_1 \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1563 |
\<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly \<times> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1564 |
"pseudo_divmod_main lc q r d dr (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1565 |
rr = smult lc r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1566 |
qq = coeff r dr; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1567 |
rrr = rr - monom qq n * d; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1568 |
qqq = smult lc q + monom qq n |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1569 |
in pseudo_divmod_main lc qqq rrr d (dr - 1) n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1570 |
| "pseudo_divmod_main lc q r d dr 0 = (q,r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1571 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1572 |
definition pseudo_divmod :: "'a :: comm_ring_1 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1573 |
"pseudo_divmod p q \<equiv> if q = 0 then (0,p) else |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1574 |
pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1575 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1576 |
lemma coeff_0_degree_minus_1: "coeff rrr dr = 0 \<Longrightarrow> degree rrr \<le> dr \<Longrightarrow> degree rrr \<le> dr - 1" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1577 |
using eq_zero_or_degree_less by fastforce |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1578 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1579 |
lemma pseudo_divmod_main: assumes d: "d \<noteq> 0" "lc = coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1580 |
and *: "degree r \<le> dr" "pseudo_divmod_main lc q r d dr n = (q',r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1581 |
"n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> r = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1582 |
shows "(r' = 0 \<or> degree r' < degree d) \<and> smult (lc^n) (d * q + r) = d * q' + r'" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1583 |
using * |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1584 |
proof (induct n arbitrary: q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1585 |
case (Suc n q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1586 |
let ?rr = "smult lc r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1587 |
let ?qq = "coeff r dr" |
63040 | 1588 |
define b where [simp]: "b = monom ?qq n" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1589 |
let ?rrr = "?rr - b * d" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1590 |
let ?qqq = "smult lc q + b" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1591 |
note res = Suc(3) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1592 |
from res[unfolded pseudo_divmod_main.simps[of lc q] Let_def] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1593 |
have res: "pseudo_divmod_main lc ?qqq ?rrr d (dr - 1) n = (q',r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1594 |
by (simp del: pseudo_divmod_main.simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1595 |
have dr: "dr = n + degree d" using Suc(4) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1596 |
have "coeff (b * d) dr = coeff b n * coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1597 |
proof (cases "?qq = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1598 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1599 |
hence n: "n = degree b" by (simp add: degree_monom_eq) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1600 |
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1601 |
qed auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1602 |
also have "\<dots> = lc * coeff b n" unfolding d by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1603 |
finally have "coeff (b * d) dr = lc * coeff b n" . |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1604 |
moreover have "coeff ?rr dr = lc * coeff r dr" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1605 |
ultimately have c0: "coeff ?rrr dr = 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1606 |
have dr: "dr = n + degree d" using Suc(4) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1607 |
have deg_rr: "degree ?rr \<le> dr" using Suc(2) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1608 |
using degree_smult_le dual_order.trans by blast |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1609 |
have deg_bd: "degree (b * d) \<le> dr" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1610 |
unfolding dr |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1611 |
by(rule order.trans[OF degree_mult_le], auto simp: degree_monom_le) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1612 |
have "degree ?rrr \<le> dr" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1613 |
using degree_diff_le[OF deg_rr deg_bd] by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1614 |
with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)" by (rule coeff_0_degree_minus_1) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1615 |
have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1616 |
proof (cases dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1617 |
case 0 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1618 |
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1619 |
with deg_rrr have "degree ?rrr = 0" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1620 |
hence "\<exists> a. ?rrr = [: a :]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1621 |
from this obtain a where rrr: "?rrr = [:a:]" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1622 |
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1623 |
qed (insert Suc(4), auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1624 |
note IH = Suc(1)[OF deg_rrr res this] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1625 |
show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1626 |
proof (intro conjI) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1627 |
show "r' = 0 \<or> degree r' < degree d" using IH by blast |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1628 |
show "smult (lc ^ Suc n) (d * q + r) = d * q' + r'" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1629 |
unfolding IH[THEN conjunct2,symmetric] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1630 |
by (simp add: field_simps smult_add_right) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1631 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1632 |
qed auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1633 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1634 |
lemma pseudo_divmod: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1635 |
assumes g: "g \<noteq> 0" and *: "pseudo_divmod f g = (q,r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1636 |
shows "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" (is ?A) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1637 |
and "r = 0 \<or> degree r < degree g" (is ?B) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1638 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1639 |
from *[unfolded pseudo_divmod_def Let_def] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1640 |
have "pseudo_divmod_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)) = (q, r)" by (auto simp: g) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1641 |
note main = pseudo_divmod_main[OF _ _ _ this, OF g refl le_refl] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1642 |
have "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g \<or> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1643 |
degree f = 0 \<and> 1 + length (coeffs f) - length (coeffs g) = 0 \<and> f = 0" using g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1644 |
by (cases "f = 0"; cases "coeffs g", auto simp: degree_eq_length_coeffs) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1645 |
note main = main[OF this] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1646 |
from main show "r = 0 \<or> degree r < degree g" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1647 |
show "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1648 |
by (subst main[THEN conjunct2, symmetric], simp add: degree_eq_length_coeffs, |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1649 |
insert g, cases "f = 0"; cases "coeffs g", auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1650 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1651 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1652 |
definition "pseudo_mod_main lc r d dr n = snd (pseudo_divmod_main lc 0 r d dr n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1653 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1654 |
lemma snd_pseudo_divmod_main: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1655 |
"snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1656 |
by (induct n arbitrary: q q' lc r d dr; simp add: Let_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1657 |
|
63498 | 1658 |
definition pseudo_mod |
1659 |
:: "'a :: {comm_ring_1,semiring_1_no_zero_divisors} poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1660 |
"pseudo_mod f g = snd (pseudo_divmod f g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1661 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1662 |
lemma pseudo_mod: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1663 |
fixes f g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1664 |
defines "r \<equiv> pseudo_mod f g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1665 |
assumes g: "g \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1666 |
shows "\<exists> a q. a \<noteq> 0 \<and> smult a f = g * q + r" "r = 0 \<or> degree r < degree g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1667 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1668 |
let ?cg = "coeff g (degree g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1669 |
let ?cge = "?cg ^ (Suc (degree f) - degree g)" |
63040 | 1670 |
define a where "a = ?cge" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1671 |
obtain q where pdm: "pseudo_divmod f g = (q,r)" using r_def[unfolded pseudo_mod_def] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1672 |
by (cases "pseudo_divmod f g", auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1673 |
from pseudo_divmod[OF g pdm] have id: "smult a f = g * q + r" and "r = 0 \<or> degree r < degree g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1674 |
unfolding a_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1675 |
show "r = 0 \<or> degree r < degree g" by fact |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1676 |
from g have "a \<noteq> 0" unfolding a_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1677 |
thus "\<exists> a q. a \<noteq> 0 \<and> smult a f = g * q + r" using id by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1678 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1679 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1680 |
instantiation poly :: (idom_divide) idom_divide |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1681 |
begin |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1682 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1683 |
fun divide_poly_main :: "'a \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1684 |
\<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1685 |
"divide_poly_main lc q r d dr (Suc n) = (let cr = coeff r dr; a = cr div lc; mon = monom a n in |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1686 |
if False \<or> a * lc = cr then (* False \<or> is only because of problem in function-package *) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1687 |
divide_poly_main |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1688 |
lc |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1689 |
(q + mon) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1690 |
(r - mon * d) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1691 |
d (dr - 1) n else 0)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1692 |
| "divide_poly_main lc q r d dr 0 = q" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1693 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1694 |
definition divide_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1695 |
"divide_poly f g = (if g = 0 then 0 else |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1696 |
divide_poly_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1697 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1698 |
lemma divide_poly_main: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1699 |
assumes d: "d \<noteq> 0" "lc = coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1700 |
and *: "degree (d * r) \<le> dr" "divide_poly_main lc q (d * r) d dr n = q'" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1701 |
"n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> d * r = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1702 |
shows "q' = q + r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1703 |
using * |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1704 |
proof (induct n arbitrary: q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1705 |
case (Suc n q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1706 |
let ?rr = "d * r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1707 |
let ?a = "coeff ?rr dr" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1708 |
let ?qq = "?a div lc" |
63040 | 1709 |
define b where [simp]: "b = monom ?qq n" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1710 |
let ?rrr = "d * (r - b)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1711 |
let ?qqq = "q + b" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1712 |
note res = Suc(3) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1713 |
have dr: "dr = n + degree d" using Suc(4) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1714 |
have lc: "lc \<noteq> 0" using d by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1715 |
have "coeff (b * d) dr = coeff b n * coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1716 |
proof (cases "?qq = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1717 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1718 |
hence n: "n = degree b" by (simp add: degree_monom_eq) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1719 |
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1720 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1721 |
also have "\<dots> = lc * coeff b n" unfolding d by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1722 |
finally have c2: "coeff (b * d) dr = lc * coeff b n" . |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1723 |
have rrr: "?rrr = ?rr - b * d" by (simp add: field_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1724 |
have c1: "coeff (d * r) dr = lc * coeff r n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1725 |
proof (cases "degree r = n") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1726 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1727 |
thus ?thesis using Suc(2) unfolding dr using coeff_mult_degree_sum[of d r] d by (auto simp: ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1728 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1729 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1730 |
have "degree r \<le> n" using dr Suc(2) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1731 |
(metis add.commute add_le_cancel_left d(1) degree_0 degree_mult_eq diff_is_0_eq diff_zero le_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1732 |
with False have r_n: "degree r < n" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1733 |
hence right: "lc * coeff r n = 0" by (simp add: coeff_eq_0) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1734 |
have "coeff (d * r) dr = coeff (d * r) (degree d + n)" unfolding dr by (simp add: ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1735 |
also have "\<dots> = 0" using r_n |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1736 |
by (metis False Suc.prems(1) add.commute add_left_imp_eq coeff_degree_mult coeff_eq_0 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1737 |
coeff_mult_degree_sum degree_mult_le dr le_eq_less_or_eq) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1738 |
finally show ?thesis unfolding right . |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1739 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1740 |
have c0: "coeff ?rrr dr = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1741 |
and id: "lc * (coeff (d * r) dr div lc) = coeff (d * r) dr" unfolding rrr coeff_diff c2 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1742 |
unfolding b_def coeff_monom coeff_smult c1 using lc by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1743 |
from res[unfolded divide_poly_main.simps[of lc q] Let_def] id |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1744 |
have res: "divide_poly_main lc ?qqq ?rrr d (dr - 1) n = q'" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1745 |
by (simp del: divide_poly_main.simps add: field_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1746 |
note IH = Suc(1)[OF _ res] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1747 |
have dr: "dr = n + degree d" using Suc(4) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1748 |
have deg_rr: "degree ?rr \<le> dr" using Suc(2) by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1749 |
have deg_bd: "degree (b * d) \<le> dr" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1750 |
unfolding dr b_def by (rule order.trans[OF degree_mult_le], auto simp: degree_monom_le) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1751 |
have "degree ?rrr \<le> dr" unfolding rrr by (rule degree_diff_le[OF deg_rr deg_bd]) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1752 |
with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)" by (rule coeff_0_degree_minus_1) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1753 |
have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1754 |
proof (cases dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1755 |
case 0 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1756 |
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1757 |
with deg_rrr have "degree ?rrr = 0" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1758 |
from degree_eq_zeroE[OF this] obtain a where rrr: "?rrr = [:a:]" by metis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1759 |
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1760 |
qed (insert Suc(4), auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1761 |
note IH = IH[OF deg_rrr this] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1762 |
show ?case using IH by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1763 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1764 |
case (0 q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1765 |
show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1766 |
proof (cases "r = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1767 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1768 |
thus ?thesis using 0 by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1769 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1770 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1771 |
have "degree (d * r) = degree d + degree r" using d False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1772 |
by (subst degree_mult_eq, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1773 |
thus ?thesis using 0 d by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1774 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1775 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1776 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1777 |
lemma fst_pseudo_divmod_main_as_divide_poly_main: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1778 |
assumes d: "d \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1779 |
defines lc: "lc \<equiv> coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1780 |
shows "fst (pseudo_divmod_main lc q r d dr n) = divide_poly_main lc (smult (lc^n) q) (smult (lc^n) r) d dr n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1781 |
proof(induct n arbitrary: q r dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1782 |
case 0 then show ?case by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1783 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1784 |
case (Suc n) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1785 |
note lc0 = leading_coeff_neq_0[OF d, folded lc] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1786 |
then have "pseudo_divmod_main lc q r d dr (Suc n) = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1787 |
pseudo_divmod_main lc (smult lc q + monom (coeff r dr) n) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1788 |
(smult lc r - monom (coeff r dr) n * d) d (dr - 1) n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1789 |
by (simp add: Let_def ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1790 |
also have "fst ... = divide_poly_main lc |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1791 |
(smult (lc^n) (smult lc q + monom (coeff r dr) n)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1792 |
(smult (lc^n) (smult lc r - monom (coeff r dr) n * d)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1793 |
d (dr - 1) n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1794 |
unfolding Suc[unfolded divide_poly_main.simps Let_def].. |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1795 |
also have "... = divide_poly_main lc (smult (lc ^ Suc n) q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1796 |
(smult (lc ^ Suc n) r) d dr (Suc n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1797 |
unfolding smult_monom smult_distribs mult_smult_left[symmetric] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1798 |
using lc0 by (simp add: Let_def ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1799 |
finally show ?case. |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1800 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1801 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1802 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1803 |
lemma divide_poly_main_0: "divide_poly_main 0 0 r d dr n = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1804 |
proof (induct n arbitrary: r d dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1805 |
case (Suc n r d dr) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1806 |
show ?case unfolding divide_poly_main.simps[of _ _ r] Let_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1807 |
by (simp add: Suc del: divide_poly_main.simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1808 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1809 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1810 |
lemma divide_poly: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1811 |
assumes g: "g \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1812 |
shows "(f * g) div g = (f :: 'a poly)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1813 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1814 |
have "divide_poly_main (coeff g (degree g)) 0 (g * f) g (degree (g * f)) (1 + length (coeffs (g * f)) - length (coeffs g)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1815 |
= (f * g) div g" unfolding divide_poly_def Let_def by (simp add: ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1816 |
note main = divide_poly_main[OF g refl le_refl this] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1817 |
{ |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1818 |
fix f :: "'a poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1819 |
assume "f \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1820 |
hence "length (coeffs f) = Suc (degree f)" unfolding degree_eq_length_coeffs by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1821 |
} note len = this |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1822 |
have "(f * g) div g = 0 + f" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1823 |
proof (rule main, goal_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1824 |
case 1 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1825 |
show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1826 |
proof (cases "f = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1827 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1828 |
with g show ?thesis by (auto simp: degree_eq_length_coeffs) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1829 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1830 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1831 |
with g have fg: "g * f \<noteq> 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1832 |
show ?thesis unfolding len[OF fg] len[OF g] by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1833 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1834 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1835 |
thus ?thesis by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1836 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1837 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1838 |
lemma divide_poly_0: "f div 0 = (0 :: 'a poly)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1839 |
by (simp add: divide_poly_def Let_def divide_poly_main_0) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1840 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1841 |
instance by (standard, auto simp: divide_poly divide_poly_0) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1842 |
end |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1843 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1844 |
|
63498 | 1845 |
instance poly :: (idom_divide) algebraic_semidom .. |
1846 |
||
1847 |
||
1848 |
||
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1849 |
subsubsection\<open>Division in Field Polynomials\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1850 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1851 |
text\<open> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1852 |
This part connects the above result to the division of field polynomials. |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1853 |
Mainly imported from Isabelle 2016. |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1854 |
\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1855 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1856 |
lemma pseudo_divmod_field: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1857 |
assumes g: "(g::'a::field poly) \<noteq> 0" and *: "pseudo_divmod f g = (q,r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1858 |
defines "c \<equiv> coeff g (degree g) ^ (Suc (degree f) - degree g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1859 |
shows "f = g * smult (1/c) q + smult (1/c) r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1860 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1861 |
from leading_coeff_neq_0[OF g] have c0: "c \<noteq> 0" unfolding c_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1862 |
from pseudo_divmod(1)[OF g *, folded c_def] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1863 |
have "smult c f = g * q + r" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1864 |
also have "smult (1/c) ... = g * smult (1/c) q + smult (1/c) r" by (simp add: smult_add_right) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1865 |
finally show ?thesis using c0 by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1866 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1867 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1868 |
lemma divide_poly_main_field: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1869 |
assumes d: "(d::'a::field poly) \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1870 |
defines lc: "lc \<equiv> coeff d (degree d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1871 |
shows "divide_poly_main lc q r d dr n = fst (pseudo_divmod_main lc (smult ((1/lc)^n) q) (smult ((1/lc)^n) r) d dr n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1872 |
unfolding lc |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1873 |
by(subst fst_pseudo_divmod_main_as_divide_poly_main, auto simp: d power_one_over) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1874 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1875 |
lemma divide_poly_field: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1876 |
fixes f g :: "'a :: field poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1877 |
defines "f' \<equiv> smult ((1 / coeff g (degree g)) ^ (Suc (degree f) - degree g)) f" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1878 |
shows "(f::'a::field poly) div g = fst (pseudo_divmod f' g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1879 |
proof (cases "g = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1880 |
case True show ?thesis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1881 |
unfolding divide_poly_def pseudo_divmod_def Let_def f'_def True by (simp add: divide_poly_main_0) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1882 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1883 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1884 |
from leading_coeff_neq_0[OF False] have "degree f' = degree f" unfolding f'_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1885 |
then show ?thesis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1886 |
using length_coeffs_degree[of f'] length_coeffs_degree[of f] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1887 |
unfolding divide_poly_def pseudo_divmod_def Let_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1888 |
divide_poly_main_field[OF False] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1889 |
length_coeffs_degree[OF False] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1890 |
f'_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1891 |
by force |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1892 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1893 |
|
29451 | 1894 |
instantiation poly :: (field) ring_div |
1895 |
begin |
|
1896 |
||
1897 |
definition mod_poly where |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1898 |
"f mod g \<equiv> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1899 |
if g = 0 then f |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1900 |
else pseudo_mod (smult ((1/coeff g (degree g)) ^ (Suc (degree f) - degree g)) f) g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1901 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1902 |
lemma pdivmod_rel: "pdivmod_rel (x::'a::field poly) y (x div y) (x mod y)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1903 |
unfolding pdivmod_rel_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1904 |
proof (intro conjI) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1905 |
show "x = x div y * y + x mod y" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1906 |
proof(cases "y = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1907 |
case True show ?thesis by(simp add: True divide_poly_def divide_poly_0 mod_poly_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1908 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1909 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1910 |
then have "pseudo_divmod (smult ((1 / coeff y (degree y)) ^ (Suc (degree x) - degree y)) x) y = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1911 |
(x div y, x mod y)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1912 |
unfolding divide_poly_field mod_poly_def pseudo_mod_def by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1913 |
from pseudo_divmod[OF False this] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1914 |
show ?thesis using False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1915 |
by (simp add: power_mult_distrib[symmetric] mult.commute) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1916 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1917 |
show "if y = 0 then x div y = 0 else x mod y = 0 \<or> degree (x mod y) < degree y" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1918 |
proof (cases "y = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1919 |
case True then show ?thesis by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1920 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1921 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1922 |
with pseudo_mod[OF this] show ?thesis unfolding mod_poly_def by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1923 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1924 |
qed |
29451 | 1925 |
|
1926 |
lemma div_poly_eq: |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1927 |
"pdivmod_rel (x::'a::field poly) y q r \<Longrightarrow> x div y = q" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1928 |
by(rule pdivmod_rel_unique_div[OF pdivmod_rel]) |
29451 | 1929 |
|
1930 |
lemma mod_poly_eq: |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1931 |
"pdivmod_rel (x::'a::field poly) y q r \<Longrightarrow> x mod y = r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1932 |
by (rule pdivmod_rel_unique_mod[OF pdivmod_rel]) |
29451 | 1933 |
|
60679 | 1934 |
instance |
1935 |
proof |
|
29451 | 1936 |
fix x y :: "'a poly" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1937 |
from pdivmod_rel[of x y,unfolded pdivmod_rel_def] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1938 |
show "x div y * y + x mod y = x" by auto |
29451 | 1939 |
next |
1940 |
fix x :: "'a poly" |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1941 |
show "x div 0 = 0" by simp |
29451 | 1942 |
next |
1943 |
fix y :: "'a poly" |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
1944 |
show "0 div y = 0" by simp |
29451 | 1945 |
next |
1946 |
fix x y z :: "'a poly" |
|
1947 |
assume "y \<noteq> 0" |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1948 |
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" |
29537 | 1949 |
using pdivmod_rel [of x y] |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49834
diff
changeset
|
1950 |
by (simp add: pdivmod_rel_def distrib_right) |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1951 |
thus "(x + z * y) div y = z + x div y" |
29451 | 1952 |
by (rule div_poly_eq) |
30930 | 1953 |
next |
1954 |
fix x y z :: "'a poly" |
|
1955 |
assume "x \<noteq> 0" |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1956 |
show "(x * y) div (x * z) = y div z" |
30930 | 1957 |
proof (cases "y \<noteq> 0 \<and> z \<noteq> 0") |
1958 |
have "\<And>x::'a poly. pdivmod_rel x 0 0 x" |
|
1959 |
by (rule pdivmod_rel_by_0) |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1960 |
then have [simp]: "\<And>x::'a poly. x div 0 = 0" |
30930 | 1961 |
by (rule div_poly_eq) |
1962 |
have "\<And>x::'a poly. pdivmod_rel 0 x 0 0" |
|
1963 |
by (rule pdivmod_rel_0) |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1964 |
then have [simp]: "\<And>x::'a poly. 0 div x = 0" |
30930 | 1965 |
by (rule div_poly_eq) |
1966 |
case False then show ?thesis by auto |
|
1967 |
next |
|
1968 |
case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto |
|
60500 | 1969 |
with \<open>x \<noteq> 0\<close> |
30930 | 1970 |
have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)" |
1971 |
by (auto simp add: pdivmod_rel_def algebra_simps) |
|
1972 |
(rule classical, simp add: degree_mult_eq) |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1973 |
moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" . |
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1974 |
ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" . |
30930 | 1975 |
then show ?thesis by (simp add: div_poly_eq) |
1976 |
qed |
|
29451 | 1977 |
qed |
1978 |
||
1979 |
end |
|
1980 |
||
60570 | 1981 |
lemma is_unit_monom_0: |
1982 |
fixes a :: "'a::field" |
|
1983 |
assumes "a \<noteq> 0" |
|
1984 |
shows "is_unit (monom a 0)" |
|
1985 |
proof |
|
62351 | 1986 |
from assms show "1 = monom a 0 * monom (inverse a) 0" |
60570 | 1987 |
by (simp add: mult_monom) |
1988 |
qed |
|
1989 |
||
1990 |
lemma is_unit_triv: |
|
1991 |
fixes a :: "'a::field" |
|
1992 |
assumes "a \<noteq> 0" |
|
1993 |
shows "is_unit [:a:]" |
|
1994 |
using assms by (simp add: is_unit_monom_0 monom_0 [symmetric]) |
|
1995 |
||
1996 |
lemma is_unit_iff_degree: |
|
63498 | 1997 |
assumes "p \<noteq> (0 :: _ :: field poly)" |
60570 | 1998 |
shows "is_unit p \<longleftrightarrow> degree p = 0" (is "?P \<longleftrightarrow> ?Q") |
1999 |
proof |
|
2000 |
assume ?Q |
|
2001 |
then obtain a where "p = [:a:]" by (rule degree_eq_zeroE) |
|
2002 |
with assms show ?P by (simp add: is_unit_triv) |
|
2003 |
next |
|
2004 |
assume ?P |
|
2005 |
then obtain q where "q \<noteq> 0" "p * q = 1" .. |
|
2006 |
then have "degree (p * q) = degree 1" |
|
2007 |
by simp |
|
2008 |
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0" |
|
2009 |
by (simp add: degree_mult_eq) |
|
2010 |
then show ?Q by simp |
|
2011 |
qed |
|
2012 |
||
2013 |
lemma is_unit_pCons_iff: |
|
63498 | 2014 |
"is_unit (pCons (a::_::field) p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0" |
60570 | 2015 |
by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree) |
2016 |
||
2017 |
lemma is_unit_monom_trival: |
|
2018 |
fixes p :: "'a::field poly" |
|
2019 |
assumes "is_unit p" |
|
2020 |
shows "monom (coeff p (degree p)) 0 = p" |
|
2021 |
using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff) |
|
2022 |
||
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2023 |
lemma is_unit_polyE: |
63498 | 2024 |
assumes "is_unit (p::_::field poly)" |
60685
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2025 |
obtains a where "p = monom a 0" and "a \<noteq> 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2026 |
proof - |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2027 |
obtain a q where "p = pCons a q" by (cases p) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2028 |
with assms have "p = [:a:]" and "a \<noteq> 0" |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2029 |
by (simp_all add: is_unit_pCons_iff) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2030 |
with that show thesis by (simp add: monom_0) |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2031 |
qed |
cb21b7022b00
moved normalization and unit_factor into Main HOL corpus
haftmann
parents:
60679
diff
changeset
|
2032 |
|
29451 | 2033 |
lemma degree_mod_less: |
2034 |
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" |
|
29537 | 2035 |
using pdivmod_rel [of x y] |
2036 |
unfolding pdivmod_rel_def by simp |
|
29451 | 2037 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2038 |
lemma div_poly_less: "degree (x::'a::field poly) < degree y \<Longrightarrow> x div y = 0" |
29451 | 2039 |
proof - |
2040 |
assume "degree x < degree y" |
|
29537 | 2041 |
hence "pdivmod_rel x y 0 x" |
2042 |
by (simp add: pdivmod_rel_def) |
|
29451 | 2043 |
thus "x div y = 0" by (rule div_poly_eq) |
2044 |
qed |
|
2045 |
||
2046 |
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" |
|
2047 |
proof - |
|
2048 |
assume "degree x < degree y" |
|
29537 | 2049 |
hence "pdivmod_rel x y 0 x" |
2050 |
by (simp add: pdivmod_rel_def) |
|
29451 | 2051 |
thus "x mod y = x" by (rule mod_poly_eq) |
2052 |
qed |
|
2053 |
||
29659 | 2054 |
lemma pdivmod_rel_smult_left: |
2055 |
"pdivmod_rel x y q r |
|
2056 |
\<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)" |
|
2057 |
unfolding pdivmod_rel_def by (simp add: smult_add_right) |
|
2058 |
||
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2059 |
lemma div_smult_left: "(smult (a::'a::field) x) div y = smult a (x div y)" |
29659 | 2060 |
by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) |
2061 |
||
2062 |
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" |
|
2063 |
by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) |
|
2064 |
||
30072 | 2065 |
lemma poly_div_minus_left [simp]: |
2066 |
fixes x y :: "'a::field poly" |
|
2067 |
shows "(- x) div y = - (x div y)" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2068 |
using div_smult_left [of "- 1::'a"] by simp |
30072 | 2069 |
|
2070 |
lemma poly_mod_minus_left [simp]: |
|
2071 |
fixes x y :: "'a::field poly" |
|
2072 |
shows "(- x) mod y = - (x mod y)" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2073 |
using mod_smult_left [of "- 1::'a"] by simp |
30072 | 2074 |
|
57482
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2075 |
lemma pdivmod_rel_add_left: |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2076 |
assumes "pdivmod_rel x y q r" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2077 |
assumes "pdivmod_rel x' y q' r'" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2078 |
shows "pdivmod_rel (x + x') y (q + q') (r + r')" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2079 |
using assms unfolding pdivmod_rel_def |
59557 | 2080 |
by (auto simp add: algebra_simps degree_add_less) |
57482
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2081 |
|
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2082 |
lemma poly_div_add_left: |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2083 |
fixes x y z :: "'a::field poly" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2084 |
shows "(x + y) div z = x div z + y div z" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2085 |
using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel] |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2086 |
by (rule div_poly_eq) |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2087 |
|
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2088 |
lemma poly_mod_add_left: |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2089 |
fixes x y z :: "'a::field poly" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2090 |
shows "(x + y) mod z = x mod z + y mod z" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2091 |
using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel] |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2092 |
by (rule mod_poly_eq) |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2093 |
|
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2094 |
lemma poly_div_diff_left: |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2095 |
fixes x y z :: "'a::field poly" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2096 |
shows "(x - y) div z = x div z - y div z" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2097 |
by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left) |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2098 |
|
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2099 |
lemma poly_mod_diff_left: |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2100 |
fixes x y z :: "'a::field poly" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2101 |
shows "(x - y) mod z = x mod z - y mod z" |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2102 |
by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left) |
60459c3853af
add lemmas: polynomial div/mod distribute over addition
huffman
parents:
56544
diff
changeset
|
2103 |
|
29659 | 2104 |
lemma pdivmod_rel_smult_right: |
2105 |
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk> |
|
2106 |
\<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r" |
|
2107 |
unfolding pdivmod_rel_def by simp |
|
2108 |
||
2109 |
lemma div_smult_right: |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2110 |
"(a::'a::field) \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)" |
29659 | 2111 |
by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) |
2112 |
||
2113 |
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y" |
|
2114 |
by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) |
|
2115 |
||
30072 | 2116 |
lemma poly_div_minus_right [simp]: |
2117 |
fixes x y :: "'a::field poly" |
|
2118 |
shows "x div (- y) = - (x div y)" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2119 |
using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq) |
30072 | 2120 |
|
2121 |
lemma poly_mod_minus_right [simp]: |
|
2122 |
fixes x y :: "'a::field poly" |
|
2123 |
shows "x mod (- y) = x mod y" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2124 |
using mod_smult_right [of "- 1::'a"] by simp |
30072 | 2125 |
|
29660 | 2126 |
lemma pdivmod_rel_mult: |
2127 |
"\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk> |
|
2128 |
\<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)" |
|
2129 |
apply (cases "z = 0", simp add: pdivmod_rel_def) |
|
2130 |
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff) |
|
2131 |
apply (cases "r = 0") |
|
2132 |
apply (cases "r' = 0") |
|
2133 |
apply (simp add: pdivmod_rel_def) |
|
36350 | 2134 |
apply (simp add: pdivmod_rel_def field_simps degree_mult_eq) |
29660 | 2135 |
apply (cases "r' = 0") |
2136 |
apply (simp add: pdivmod_rel_def degree_mult_eq) |
|
36350 | 2137 |
apply (simp add: pdivmod_rel_def field_simps) |
29660 | 2138 |
apply (simp add: degree_mult_eq degree_add_less) |
2139 |
done |
|
2140 |
||
2141 |
lemma poly_div_mult_right: |
|
2142 |
fixes x y z :: "'a::field poly" |
|
2143 |
shows "x div (y * z) = (x div y) div z" |
|
2144 |
by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) |
|
2145 |
||
2146 |
lemma poly_mod_mult_right: |
|
2147 |
fixes x y z :: "'a::field poly" |
|
2148 |
shows "x mod (y * z) = y * (x div y mod z) + x mod y" |
|
2149 |
by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) |
|
2150 |
||
29451 | 2151 |
lemma mod_pCons: |
2152 |
fixes a and x |
|
2153 |
assumes y: "y \<noteq> 0" |
|
2154 |
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" |
|
2155 |
shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" |
|
2156 |
unfolding b |
|
2157 |
apply (rule mod_poly_eq) |
|
29537 | 2158 |
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) |
29451 | 2159 |
done |
2160 |
||
52380 | 2161 |
definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" |
2162 |
where |
|
2163 |
"pdivmod p q = (p div q, p mod q)" |
|
31663 | 2164 |
|
52380 | 2165 |
lemma pdivmod_0: |
2166 |
"pdivmod 0 q = (0, 0)" |
|
2167 |
by (simp add: pdivmod_def) |
|
31663 | 2168 |
|
52380 | 2169 |
lemma pdivmod_pCons: |
2170 |
"pdivmod (pCons a p) q = |
|
2171 |
(if q = 0 then (0, pCons a p) else |
|
2172 |
(let (s, r) = pdivmod p q; |
|
2173 |
b = coeff (pCons a r) (degree q) / coeff q (degree q) |
|
2174 |
in (pCons b s, pCons a r - smult b q)))" |
|
2175 |
apply (simp add: pdivmod_def Let_def, safe) |
|
2176 |
apply (rule div_poly_eq) |
|
2177 |
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) |
|
2178 |
apply (rule mod_poly_eq) |
|
2179 |
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) |
|
29451 | 2180 |
done |
2181 |
||
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2182 |
lemma pdivmod_fold_coeffs: |
52380 | 2183 |
"pdivmod p q = (if q = 0 then (0, p) |
2184 |
else fold_coeffs (\<lambda>a (s, r). |
|
2185 |
let b = coeff (pCons a r) (degree q) / coeff q (degree q) |
|
2186 |
in (pCons b s, pCons a r - smult b q) |
|
2187 |
) p (0, 0))" |
|
2188 |
apply (cases "q = 0") |
|
2189 |
apply (simp add: pdivmod_def) |
|
2190 |
apply (rule sym) |
|
2191 |
apply (induct p) |
|
2192 |
apply (simp_all add: pdivmod_0 pdivmod_pCons) |
|
2193 |
apply (case_tac "a = 0 \<and> p = 0") |
|
2194 |
apply (auto simp add: pdivmod_def) |
|
2195 |
done |
|
29980 | 2196 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2197 |
subsection \<open>List-based versions for fast implementation\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2198 |
(* Subsection by: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2199 |
Sebastiaan Joosten |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2200 |
René Thiemann |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2201 |
Akihisa Yamada |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2202 |
*) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2203 |
fun minus_poly_rev_list :: "'a :: group_add list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2204 |
"minus_poly_rev_list (x # xs) (y # ys) = (x - y) # (minus_poly_rev_list xs ys)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2205 |
| "minus_poly_rev_list xs [] = xs" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2206 |
| "minus_poly_rev_list [] (y # ys) = []" |
63035
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2207 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2208 |
fun pseudo_divmod_main_list :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2209 |
\<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2210 |
"pseudo_divmod_main_list lc q r d (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2211 |
rr = map (op * lc) r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2212 |
a = hd r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2213 |
qqq = cCons a (map (op * lc) q); |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2214 |
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2215 |
in pseudo_divmod_main_list lc qqq rrr d n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2216 |
| "pseudo_divmod_main_list lc q r d 0 = (q,r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2217 |
|
63035
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2218 |
fun pseudo_mod_main_list :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2219 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2220 |
"pseudo_mod_main_list lc r d (Suc n) = (let |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2221 |
rr = map (op * lc) r; |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2222 |
a = hd r; |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2223 |
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d)) |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2224 |
in pseudo_mod_main_list lc rrr d n)" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2225 |
| "pseudo_mod_main_list lc r d 0 = r" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2226 |
|
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2227 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2228 |
fun divmod_poly_one_main_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2229 |
\<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2230 |
"divmod_poly_one_main_list q r d (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2231 |
a = hd r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2232 |
qqq = cCons a q; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2233 |
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2234 |
in divmod_poly_one_main_list qqq rr d n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2235 |
| "divmod_poly_one_main_list q r d 0 = (q,r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2236 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2237 |
fun mod_poly_one_main_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2238 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2239 |
"mod_poly_one_main_list r d (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2240 |
a = hd r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2241 |
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2242 |
in mod_poly_one_main_list rr d n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2243 |
| "mod_poly_one_main_list r d 0 = r" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2244 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2245 |
definition pseudo_divmod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list \<times> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2246 |
"pseudo_divmod_list p q = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2247 |
(if q = [] then ([],p) else |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2248 |
(let rq = rev q; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2249 |
(qu,re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2250 |
(qu,rev re)))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2251 |
|
63035
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2252 |
definition pseudo_mod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2253 |
"pseudo_mod_list p q = |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2254 |
(if q = [] then p else |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2255 |
(let rq = rev q; |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2256 |
re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q) in |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2257 |
(rev re)))" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2258 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2259 |
lemma minus_zero_does_nothing: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2260 |
"(minus_poly_rev_list x (map (op * 0) y)) = (x :: 'a :: ring list)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2261 |
by(induct x y rule: minus_poly_rev_list.induct, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2262 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2263 |
lemma length_minus_poly_rev_list[simp]: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2264 |
"length (minus_poly_rev_list xs ys) = length xs" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2265 |
by(induct xs ys rule: minus_poly_rev_list.induct, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2266 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2267 |
lemma if_0_minus_poly_rev_list: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2268 |
"(if a = 0 then x else minus_poly_rev_list x (map (op * a) y)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2269 |
= minus_poly_rev_list x (map (op * (a :: 'a :: ring)) y)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2270 |
by(cases "a=0",simp_all add:minus_zero_does_nothing) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2271 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2272 |
lemma Poly_append: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2273 |
"Poly ((a::'a::comm_semiring_1 list) @ b) = Poly a + monom 1 (length a) * Poly b" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2274 |
by (induct a,auto simp: monom_0 monom_Suc) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2275 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2276 |
lemma minus_poly_rev_list: "length p \<ge> length (q :: 'a :: comm_ring_1 list) \<Longrightarrow> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2277 |
Poly (rev (minus_poly_rev_list (rev p) (rev q))) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2278 |
= Poly p - monom 1 (length p - length q) * Poly q" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2279 |
proof (induct "rev p" "rev q" arbitrary: p q rule: minus_poly_rev_list.induct) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2280 |
case (1 x xs y ys) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2281 |
have "length (rev q) \<le> length (rev p)" using 1 by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2282 |
from this[folded 1(2,3)] have ys_xs:"length ys \<le> length xs" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2283 |
hence a:"Poly (rev (minus_poly_rev_list xs ys)) = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2284 |
Poly (rev xs) - monom 1 (length xs - length ys) * Poly (rev ys)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2285 |
by(subst "1.hyps"(1)[of "rev xs" "rev ys", unfolded rev_rev_ident length_rev],auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2286 |
have "Poly p - monom 1 (length p - length q) * Poly q |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2287 |
= Poly (rev (rev p)) - monom 1 (length (rev (rev p)) - length (rev (rev q))) * Poly (rev (rev q))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2288 |
by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2289 |
also have "\<dots> = Poly (rev (x # xs)) - monom 1 (length (x # xs) - length (y # ys)) * Poly (rev (y # ys))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2290 |
unfolding 1(2,3) by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2291 |
also have "\<dots> = Poly (rev xs) + monom x (length xs) - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2292 |
(monom 1 (length xs - length ys) * Poly (rev ys) + monom y (length xs))" using ys_xs |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2293 |
by (simp add:Poly_append distrib_left mult_monom smult_monom) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2294 |
also have "\<dots> = Poly (rev (minus_poly_rev_list xs ys)) + monom (x - y) (length xs)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2295 |
unfolding a diff_monom[symmetric] by(simp) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2296 |
finally show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2297 |
unfolding 1(2,3)[symmetric] by (simp add: smult_monom Poly_append) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2298 |
qed auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2299 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2300 |
lemma smult_monom_mult: "smult a (monom b n * f) = monom (a * b) n * f" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2301 |
using smult_monom [of a _ n] by (metis mult_smult_left) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2302 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2303 |
lemma head_minus_poly_rev_list: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2304 |
"length d \<le> length r \<Longrightarrow> d\<noteq>[] \<Longrightarrow> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2305 |
hd (minus_poly_rev_list (map (op * (last d :: 'a :: comm_ring)) r) (map (op * (hd r)) (rev d))) = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2306 |
proof(induct r) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2307 |
case (Cons a rs) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2308 |
thus ?case by(cases "rev d", simp_all add:ac_simps) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2309 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2310 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2311 |
lemma Poly_map: "Poly (map (op * a) p) = smult a (Poly p)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2312 |
proof (induct p) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2313 |
case(Cons x xs) thus ?case by (cases "Poly xs = 0",auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2314 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2315 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2316 |
lemma last_coeff_is_hd: "xs \<noteq> [] \<Longrightarrow> coeff (Poly xs) (length xs - 1) = hd (rev xs)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2317 |
by (simp_all add: hd_conv_nth rev_nth nth_default_nth nth_append) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2318 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2319 |
lemma pseudo_divmod_main_list_invar : |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2320 |
assumes leading_nonzero:"last d \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2321 |
and lc:"last d = lc" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2322 |
and dNonempty:"d \<noteq> []" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2323 |
and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q',rev r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2324 |
and "n = (1 + length r - length d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2325 |
shows |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2326 |
"pseudo_divmod_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2327 |
(Poly q', Poly r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2328 |
using assms(4-) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2329 |
proof(induct "n" arbitrary: r q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2330 |
case (Suc n r q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2331 |
have ifCond: "\<not> Suc (length r) \<le> length d" using Suc.prems by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2332 |
have rNonempty:"r \<noteq> []" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2333 |
using ifCond dNonempty using Suc_leI length_greater_0_conv list.size(3) by fastforce |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2334 |
let ?a = "(hd (rev r))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2335 |
let ?rr = "map (op * lc) (rev r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2336 |
let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (op * ?a) (rev d))))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2337 |
let ?qq = "cCons ?a (map (op * lc) q)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2338 |
have n: "n = (1 + length r - length d - 1)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2339 |
using ifCond Suc(3) by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2340 |
have rr_val:"(length ?rrr) = (length r - 1)" using ifCond by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2341 |
hence rr_smaller: "(1 + length r - length d - 1) = (1 + length ?rrr - length d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2342 |
using rNonempty ifCond unfolding One_nat_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2343 |
from ifCond have id: "Suc (length r) - length d = Suc (length r - length d)" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2344 |
have "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) (1 + length r - length d - 1) = (q', rev r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2345 |
using Suc.prems ifCond by (simp add:Let_def if_0_minus_poly_rev_list id) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2346 |
hence v:"pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) n = (q', rev r')" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2347 |
using n by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2348 |
have sucrr:"Suc (length r) - length d = Suc (length r - length d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2349 |
using Suc_diff_le ifCond not_less_eq_eq by blast |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2350 |
have n_ok : "n = 1 + (length ?rrr) - length d" using Suc(3) rNonempty by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2351 |
have cong: "\<And> x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2352 |
pseudo_divmod_main lc x1 x2 x3 x4 n = pseudo_divmod_main lc y1 y2 y3 y4 n" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2353 |
have hd_rev:"coeff (Poly r) (length r - Suc 0) = hd (rev r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2354 |
using last_coeff_is_hd[OF rNonempty] by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2355 |
show ?case unfolding Suc.hyps(1)[OF v n_ok, symmetric] pseudo_divmod_main.simps Let_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2356 |
proof (rule cong[OF _ _ refl], goal_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2357 |
case 1 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2358 |
show ?case unfolding monom_Suc hd_rev[symmetric] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2359 |
by (simp add: smult_monom Poly_map) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2360 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2361 |
case 2 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2362 |
show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2363 |
proof (subst Poly_on_rev_starting_with_0, goal_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2364 |
show "hd (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))) = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2365 |
by (fold lc, subst head_minus_poly_rev_list, insert ifCond dNonempty,auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2366 |
from ifCond have "length d \<le> length r" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2367 |
then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2368 |
Poly (rev (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2369 |
by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2370 |
minus_poly_rev_list) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2371 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2372 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2373 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2374 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2375 |
lemma pseudo_divmod_impl[code]: "pseudo_divmod (f::'a::comm_ring_1 poly) g = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2376 |
map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2377 |
proof (cases "g=0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2378 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2379 |
hence coeffs_g_nonempty:"(coeffs g) \<noteq> []" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2380 |
hence lastcoeffs:"last (coeffs g) = coeff g (degree g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2381 |
by (simp add: hd_rev last_coeffs_eq_coeff_degree not_0_coeffs_not_Nil) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2382 |
obtain q r where qr: "pseudo_divmod_main_list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2383 |
(last (coeffs g)) (rev []) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2384 |
(rev (coeffs f)) (rev (coeffs g)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2385 |
(1 + length (coeffs f) - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2386 |
length (coeffs g)) = (q,rev (rev r))" by force |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2387 |
then have qr': "pseudo_divmod_main_list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2388 |
(hd (rev (coeffs g))) [] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2389 |
(rev (coeffs f)) (rev (coeffs g)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2390 |
(1 + length (coeffs f) - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2391 |
length (coeffs g)) = (q,r)" using hd_rev[OF coeffs_g_nonempty] by(auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2392 |
from False have cg: "(coeffs g = []) = False" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2393 |
have last_non0:"last (coeffs g) \<noteq> 0" using False by (simp add:last_coeffs_not_0) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2394 |
show ?thesis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2395 |
unfolding pseudo_divmod_def pseudo_divmod_list_def Let_def qr' map_prod_def split cg if_False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2396 |
pseudo_divmod_main_list_invar[OF last_non0 _ _ qr,unfolded lastcoeffs,simplified,symmetric,OF False] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2397 |
poly_of_list_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2398 |
using False by (auto simp: degree_eq_length_coeffs) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2399 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2400 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2401 |
show ?thesis unfolding True unfolding pseudo_divmod_def pseudo_divmod_list_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2402 |
by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2403 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2404 |
|
63035
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2405 |
lemma pseudo_mod_main_list: "snd (pseudo_divmod_main_list l q |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2406 |
xs ys n) = pseudo_mod_main_list l xs ys n" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2407 |
by (induct n arbitrary: l q xs ys, auto simp: Let_def) |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2408 |
|
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2409 |
lemma pseudo_mod_impl[code]: "pseudo_mod f g = |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2410 |
poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2411 |
proof - |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2412 |
have snd_case: "\<And> f g p. snd ((\<lambda> (x,y). (f x, g y)) p) = g (snd p)" |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2413 |
by auto |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2414 |
show ?thesis |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2415 |
unfolding pseudo_mod_def pseudo_divmod_impl pseudo_divmod_list_def |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2416 |
pseudo_mod_list_def Let_def |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2417 |
by (simp add: snd_case pseudo_mod_main_list) |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2418 |
qed |
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2419 |
|
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2420 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2421 |
(* *************** *) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2422 |
subsubsection \<open>Improved Code-Equations for Polynomial (Pseudo) Division\<close> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2423 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2424 |
lemma pdivmod_pdivmodrel: "pdivmod_rel p q r s = (pdivmod p q = (r, s))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2425 |
by (metis pdivmod_def pdivmod_rel pdivmod_rel_unique prod.sel) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2426 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2427 |
lemma pdivmod_via_pseudo_divmod: "pdivmod f g = (if g = 0 then (0,f) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2428 |
else let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2429 |
ilc = inverse (coeff g (degree g)); |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2430 |
h = smult ilc g; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2431 |
(q,r) = pseudo_divmod f h |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2432 |
in (smult ilc q, r))" (is "?l = ?r") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2433 |
proof (cases "g = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2434 |
case False |
63040 | 2435 |
define lc where "lc = inverse (coeff g (degree g))" |
2436 |
define h where "h = smult lc g" |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2437 |
from False have h1: "coeff h (degree h) = 1" and lc: "lc \<noteq> 0" unfolding h_def lc_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2438 |
hence h0: "h \<noteq> 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2439 |
obtain q r where p: "pseudo_divmod f h = (q,r)" by force |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2440 |
from False have id: "?r = (smult lc q, r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2441 |
unfolding Let_def h_def[symmetric] lc_def[symmetric] p by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2442 |
from pseudo_divmod[OF h0 p, unfolded h1] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2443 |
have f: "f = h * q + r" and r: "r = 0 \<or> degree r < degree h" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2444 |
have "pdivmod_rel f h q r" unfolding pdivmod_rel_def using f r h0 by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2445 |
hence "pdivmod f h = (q,r)" by (simp add: pdivmod_pdivmodrel) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2446 |
hence "pdivmod f g = (smult lc q, r)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2447 |
unfolding pdivmod_def h_def div_smult_right[OF lc] mod_smult_right[OF lc] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2448 |
using lc by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2449 |
with id show ?thesis by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2450 |
qed (auto simp: pdivmod_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2451 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2452 |
lemma pdivmod_via_pseudo_divmod_list: "pdivmod f g = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2453 |
cg = coeffs g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2454 |
in if cg = [] then (0,f) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2455 |
else let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2456 |
cf = coeffs f; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2457 |
ilc = inverse (last cg); |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2458 |
ch = map (op * ilc) cg; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2459 |
(q,r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg) |
63035
6c018eb1e177
fixed code equation for pdivmod, added improved code equation for pseudo_mod
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
63034
diff
changeset
|
2460 |
in (poly_of_list (map (op * ilc) q), poly_of_list (rev r)))" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2461 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2462 |
note d = pdivmod_via_pseudo_divmod |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2463 |
pseudo_divmod_impl pseudo_divmod_list_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2464 |
show ?thesis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2465 |
proof (cases "g = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2466 |
case True thus ?thesis unfolding d by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2467 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2468 |
case False |
63040 | 2469 |
define ilc where "ilc = inverse (coeff g (degree g))" |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2470 |
from False have ilc: "ilc \<noteq> 0" unfolding ilc_def by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2471 |
with False have id: "(g = 0) = False" "(coeffs g = []) = False" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2472 |
"last (coeffs g) = coeff g (degree g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2473 |
"(coeffs (smult ilc g) = []) = False" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2474 |
by (auto simp: last_coeffs_eq_coeff_degree) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2475 |
have id2: "hd (rev (coeffs (smult ilc g))) = 1" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2476 |
by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2477 |
have id3: "length (coeffs (smult ilc g)) = length (coeffs g)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2478 |
"rev (coeffs (smult ilc g)) = rev (map (op * ilc) (coeffs g))" unfolding coeffs_smult using ilc by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2479 |
obtain q r where pair: "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (op * ilc) (coeffs g))) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2480 |
(1 + length (coeffs f) - length (coeffs g)) = (q,r)" by force |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2481 |
show ?thesis unfolding d Let_def id if_False ilc_def[symmetric] map_prod_def[symmetric] id2 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2482 |
unfolding id3 pair map_prod_def split by (auto simp: Poly_map) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2483 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2484 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2485 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2486 |
lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2487 |
proof (intro ext, goal_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2488 |
case (1 q r d n) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2489 |
{ |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2490 |
fix xs :: "'a list" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2491 |
have "map (op * 1) xs = xs" by (induct xs, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2492 |
} note [simp] = this |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2493 |
show ?case by (induct n arbitrary: q r d, auto simp: Let_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2494 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2495 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2496 |
fun divide_poly_main_list :: "'a::idom_divide \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2497 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2498 |
"divide_poly_main_list lc q r d (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2499 |
cr = hd r |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2500 |
in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2501 |
a = cr div lc; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2502 |
qq = cCons a q; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2503 |
rr = minus_poly_rev_list r (map (op * a) d) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2504 |
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2505 |
| "divide_poly_main_list lc q r d 0 = q" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2506 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2507 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2508 |
lemma divide_poly_main_list_simp[simp]: "divide_poly_main_list lc q r d (Suc n) = (let |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2509 |
cr = hd r; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2510 |
a = cr div lc; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2511 |
qq = cCons a q; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2512 |
rr = minus_poly_rev_list r (map (op * a) d) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2513 |
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2514 |
by (simp add: Let_def minus_zero_does_nothing) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2515 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2516 |
declare divide_poly_main_list.simps(1)[simp del] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2517 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2518 |
definition divide_poly_list :: "'a::idom_divide poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2519 |
"divide_poly_list f g = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2520 |
(let cg = coeffs g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2521 |
in if cg = [] then g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2522 |
else let cf = coeffs f; cgr = rev cg |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2523 |
in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2524 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2525 |
lemmas pdivmod_via_divmod_list[code] = pdivmod_via_pseudo_divmod_list[unfolded pseudo_divmod_main_list_1] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2526 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2527 |
lemma mod_poly_one_main_list: "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2528 |
by (induct n arbitrary: q r d, auto simp: Let_def) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2529 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2530 |
lemma mod_poly_code[code]: "f mod g = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2531 |
(let cg = coeffs g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2532 |
in if cg = [] then f |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2533 |
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2534 |
r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2535 |
in poly_of_list (rev r))" (is "?l = ?r") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2536 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2537 |
have "?l = snd (pdivmod f g)" unfolding pdivmod_def by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2538 |
also have "\<dots> = ?r" unfolding pdivmod_via_divmod_list Let_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2539 |
mod_poly_one_main_list[symmetric, of _ _ _ Nil] by (auto split: prod.splits) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2540 |
finally show ?thesis . |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2541 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2542 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2543 |
definition div_field_poly_impl :: "'a :: field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2544 |
"div_field_poly_impl f g = ( |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2545 |
let cg = coeffs g |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2546 |
in if cg = [] then 0 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2547 |
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2548 |
q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg)) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2549 |
in poly_of_list ((map (op * ilc) q)))" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2550 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2551 |
text \<open>We do not declare the following lemma as code equation, since then polynomial division |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2552 |
on non-fields will no longer be executable. However, a code-unfold is possible, since |
63034 | 2553 |
\<open>div_field_poly_impl\<close> is a bit more efficient than the generic polynomial division.\<close> |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2554 |
lemma div_field_poly_impl[code_unfold]: "op div = div_field_poly_impl" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2555 |
proof (intro ext) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2556 |
fix f g :: "'a poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2557 |
have "f div g = fst (pdivmod f g)" unfolding pdivmod_def by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2558 |
also have "\<dots> = div_field_poly_impl f g" unfolding |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2559 |
div_field_poly_impl_def pdivmod_via_divmod_list Let_def by (auto split: prod.splits) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2560 |
finally show "f div g = div_field_poly_impl f g" . |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2561 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2562 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2563 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2564 |
lemma divide_poly_main_list: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2565 |
assumes lc0: "lc \<noteq> 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2566 |
and lc:"last d = lc" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2567 |
and d:"d \<noteq> []" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2568 |
and "n = (1 + length r - length d)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2569 |
shows |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2570 |
"Poly (divide_poly_main_list lc q (rev r) (rev d) n) = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2571 |
divide_poly_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2572 |
using assms(4-) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2573 |
proof(induct "n" arbitrary: r q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2574 |
case (Suc n r q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2575 |
have ifCond: "\<not> Suc (length r) \<le> length d" using Suc.prems by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2576 |
have r: "r \<noteq> []" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2577 |
using ifCond d using Suc_leI length_greater_0_conv list.size(3) by fastforce |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2578 |
then obtain rr lcr where r: "r = rr @ [lcr]" by (cases r rule: rev_cases, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2579 |
from d lc obtain dd where d: "d = dd @ [lc]" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2580 |
by (cases d rule: rev_cases, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2581 |
from Suc(2) ifCond have n: "n = 1 + length rr - length d" by (auto simp: r) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2582 |
from ifCond have len: "length dd \<le> length rr" by (simp add: r d) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2583 |
show ?case |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2584 |
proof (cases "lcr div lc * lc = lcr") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2585 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2586 |
thus ?thesis unfolding Suc(2)[symmetric] using r d |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2587 |
by (auto simp add: Let_def nth_default_append) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2588 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2589 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2590 |
hence id: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2591 |
"?thesis = (Poly (divide_poly_main_list lc (cCons (lcr div lc) q) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2592 |
(rev (rev (minus_poly_rev_list (rev rr) (rev (map (op * (lcr div lc)) dd))))) (rev d) n) = |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2593 |
divide_poly_main lc |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2594 |
(monom 1 (Suc n) * Poly q + monom (lcr div lc) n) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2595 |
(Poly r - monom (lcr div lc) n * Poly d) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2596 |
(Poly d) (length rr - 1) n)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2597 |
using r d |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2598 |
by (cases r rule: rev_cases; cases "d" rule: rev_cases; |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2599 |
auto simp add: Let_def rev_map nth_default_append) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2600 |
have cong: "\<And> x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow> |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2601 |
divide_poly_main lc x1 x2 x3 x4 n = divide_poly_main lc y1 y2 y3 y4 n" by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2602 |
show ?thesis unfolding id |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2603 |
proof (subst Suc(1), simp add: n, |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2604 |
subst minus_poly_rev_list, force simp: len, rule cong[OF _ _ refl], goal_cases) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2605 |
case 2 |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2606 |
have "monom lcr (length rr) = monom (lcr div lc) (length rr - length dd) * monom lc (length dd)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2607 |
by (simp add: mult_monom len True) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2608 |
thus ?case unfolding r d Poly_append n ring_distribs |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2609 |
by (auto simp: Poly_map smult_monom smult_monom_mult) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2610 |
qed (auto simp: len monom_Suc smult_monom) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2611 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2612 |
qed simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2613 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2614 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2615 |
lemma divide_poly_list[code]: "f div g = divide_poly_list f g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2616 |
proof - |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2617 |
note d = divide_poly_def divide_poly_list_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2618 |
show ?thesis |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2619 |
proof (cases "g = 0") |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2620 |
case True |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2621 |
show ?thesis unfolding d True by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2622 |
next |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2623 |
case False |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2624 |
then obtain cg lcg where cg: "coeffs g = cg @ [lcg]" by (cases "coeffs g" rule: rev_cases, auto) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2625 |
with False have id: "(g = 0) = False" "(cg @ [lcg] = []) = False" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2626 |
from cg False have lcg: "coeff g (degree g) = lcg" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2627 |
using last_coeffs_eq_coeff_degree last_snoc by force |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2628 |
with False have lcg0: "lcg \<noteq> 0" by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2629 |
from cg have ltp: "Poly (cg @ [lcg]) = g" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2630 |
using Poly_coeffs [of g] by auto |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2631 |
show ?thesis unfolding d cg Let_def id if_False poly_of_list_def |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2632 |
by (subst divide_poly_main_list, insert False cg lcg0, auto simp: lcg ltp, |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2633 |
simp add: degree_eq_length_coeffs) |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2634 |
qed |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2635 |
qed |
29980 | 2636 |
|
60500 | 2637 |
subsection \<open>Order of polynomial roots\<close> |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2638 |
|
52380 | 2639 |
definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat" |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2640 |
where |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2641 |
"order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2642 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2643 |
lemma coeff_linear_power: |
29979 | 2644 |
fixes a :: "'a::comm_semiring_1" |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2645 |
shows "coeff ([:a, 1:] ^ n) n = 1" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2646 |
apply (induct n, simp_all) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2647 |
apply (subst coeff_eq_0) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2648 |
apply (auto intro: le_less_trans degree_power_le) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2649 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2650 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2651 |
lemma degree_linear_power: |
29979 | 2652 |
fixes a :: "'a::comm_semiring_1" |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2653 |
shows "degree ([:a, 1:] ^ n) = n" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2654 |
apply (rule order_antisym) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2655 |
apply (rule ord_le_eq_trans [OF degree_power_le], simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2656 |
apply (rule le_degree, simp add: coeff_linear_power) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2657 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2658 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2659 |
lemma order_1: "[:-a, 1:] ^ order a p dvd p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2660 |
apply (cases "p = 0", simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2661 |
apply (cases "order a p", simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2662 |
apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)") |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2663 |
apply (drule not_less_Least, simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2664 |
apply (fold order_def, simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2665 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2666 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2667 |
lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2668 |
unfolding order_def |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2669 |
apply (rule LeastI_ex) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2670 |
apply (rule_tac x="degree p" in exI) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2671 |
apply (rule notI) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2672 |
apply (drule (1) dvd_imp_degree_le) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2673 |
apply (simp only: degree_linear_power) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2674 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2675 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2676 |
lemma order: |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2677 |
"p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2678 |
by (rule conjI [OF order_1 order_2]) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2679 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2680 |
lemma order_degree: |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2681 |
assumes p: "p \<noteq> 0" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2682 |
shows "order a p \<le> degree p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2683 |
proof - |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2684 |
have "order a p = degree ([:-a, 1:] ^ order a p)" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2685 |
by (simp only: degree_linear_power) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2686 |
also have "\<dots> \<le> degree p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2687 |
using order_1 p by (rule dvd_imp_degree_le) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2688 |
finally show ?thesis . |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2689 |
qed |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2690 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2691 |
lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2692 |
apply (cases "p = 0", simp_all) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2693 |
apply (rule iffI) |
56383 | 2694 |
apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right) |
2695 |
unfolding poly_eq_0_iff_dvd |
|
2696 |
apply (metis dvd_power dvd_trans order_1) |
|
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2697 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2698 |
|
62065 | 2699 |
lemma order_0I: "poly p a \<noteq> 0 \<Longrightarrow> order a p = 0" |
2700 |
by (subst (asm) order_root) auto |
|
2701 |
||
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
2702 |
|
62065 | 2703 |
subsection \<open>Additional induction rules on polynomials\<close> |
2704 |
||
2705 |
text \<open> |
|
2706 |
An induction rule for induction over the roots of a polynomial with a certain property. |
|
2707 |
(e.g. all positive roots) |
|
2708 |
\<close> |
|
2709 |
lemma poly_root_induct [case_names 0 no_roots root]: |
|
2710 |
fixes p :: "'a :: idom poly" |
|
2711 |
assumes "Q 0" |
|
2712 |
assumes "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> 0) \<Longrightarrow> Q p" |
|
2713 |
assumes "\<And>a p. P a \<Longrightarrow> Q p \<Longrightarrow> Q ([:a, -1:] * p)" |
|
2714 |
shows "Q p" |
|
2715 |
proof (induction "degree p" arbitrary: p rule: less_induct) |
|
2716 |
case (less p) |
|
2717 |
show ?case |
|
2718 |
proof (cases "p = 0") |
|
2719 |
assume nz: "p \<noteq> 0" |
|
2720 |
show ?case |
|
2721 |
proof (cases "\<exists>a. P a \<and> poly p a = 0") |
|
2722 |
case False |
|
2723 |
thus ?thesis by (intro assms(2)) blast |
|
2724 |
next |
|
2725 |
case True |
|
2726 |
then obtain a where a: "P a" "poly p a = 0" |
|
2727 |
by blast |
|
2728 |
hence "-[:-a, 1:] dvd p" |
|
2729 |
by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd) |
|
2730 |
then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp |
|
2731 |
with nz have q_nz: "q \<noteq> 0" by auto |
|
2732 |
have "degree p = Suc (degree q)" |
|
2733 |
by (subst q, subst degree_mult_eq) (simp_all add: q_nz) |
|
2734 |
hence "Q q" by (intro less) simp |
|
2735 |
from a(1) and this have "Q ([:a, -1:] * q)" |
|
2736 |
by (rule assms(3)) |
|
2737 |
with q show ?thesis by simp |
|
2738 |
qed |
|
2739 |
qed (simp add: assms(1)) |
|
2740 |
qed |
|
2741 |
||
2742 |
lemma dropWhile_replicate_append: |
|
2743 |
"dropWhile (op= a) (replicate n a @ ys) = dropWhile (op= a) ys" |
|
2744 |
by (induction n) simp_all |
|
2745 |
||
2746 |
lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs" |
|
2747 |
by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append) |
|
2748 |
||
2749 |
text \<open> |
|
2750 |
An induction rule for simultaneous induction over two polynomials, |
|
2751 |
prepending one coefficient in each step. |
|
2752 |
\<close> |
|
2753 |
lemma poly_induct2 [case_names 0 pCons]: |
|
2754 |
assumes "P 0 0" "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)" |
|
2755 |
shows "P p q" |
|
2756 |
proof - |
|
63040 | 2757 |
define n where "n = max (length (coeffs p)) (length (coeffs q))" |
2758 |
define xs where "xs = coeffs p @ (replicate (n - length (coeffs p)) 0)" |
|
2759 |
define ys where "ys = coeffs q @ (replicate (n - length (coeffs q)) 0)" |
|
62065 | 2760 |
have "length xs = length ys" |
2761 |
by (simp add: xs_def ys_def n_def) |
|
2762 |
hence "P (Poly xs) (Poly ys)" |
|
2763 |
by (induction rule: list_induct2) (simp_all add: assms) |
|
2764 |
also have "Poly xs = p" |
|
2765 |
by (simp add: xs_def Poly_append_replicate_0) |
|
2766 |
also have "Poly ys = q" |
|
2767 |
by (simp add: ys_def Poly_append_replicate_0) |
|
2768 |
finally show ?thesis . |
|
2769 |
qed |
|
2770 |
||
2771 |
||
60500 | 2772 |
subsection \<open>Composition of polynomials\<close> |
29478 | 2773 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2774 |
(* Several lemmas contributed by René Thiemann and Akihisa Yamada *) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2775 |
|
52380 | 2776 |
definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
2777 |
where |
|
2778 |
"pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0" |
|
2779 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2780 |
notation pcompose (infixl "\<circ>\<^sub>p" 71) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2781 |
|
52380 | 2782 |
lemma pcompose_0 [simp]: |
2783 |
"pcompose 0 q = 0" |
|
2784 |
by (simp add: pcompose_def) |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2785 |
|
52380 | 2786 |
lemma pcompose_pCons: |
2787 |
"pcompose (pCons a p) q = [:a:] + q * pcompose p q" |
|
2788 |
by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def) |
|
2789 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2790 |
lemma pcompose_1: |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2791 |
fixes p :: "'a :: comm_semiring_1 poly" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2792 |
shows "pcompose 1 p = 1" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2793 |
unfolding one_poly_def by (auto simp: pcompose_pCons) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2794 |
|
52380 | 2795 |
lemma poly_pcompose: |
2796 |
"poly (pcompose p q) x = poly p (poly q x)" |
|
2797 |
by (induct p) (simp_all add: pcompose_pCons) |
|
2798 |
||
2799 |
lemma degree_pcompose_le: |
|
2800 |
"degree (pcompose p q) \<le> degree p * degree q" |
|
2801 |
apply (induct p, simp) |
|
2802 |
apply (simp add: pcompose_pCons, clarify) |
|
2803 |
apply (rule degree_add_le, simp) |
|
2804 |
apply (rule order_trans [OF degree_mult_le], simp) |
|
29478 | 2805 |
done |
2806 |
||
62065 | 2807 |
lemma pcompose_add: |
2808 |
fixes p q r :: "'a :: {comm_semiring_0, ab_semigroup_add} poly" |
|
2809 |
shows "pcompose (p + q) r = pcompose p r + pcompose q r" |
|
2810 |
proof (induction p q rule: poly_induct2) |
|
2811 |
case (pCons a p b q) |
|
2812 |
have "pcompose (pCons a p + pCons b q) r = |
|
2813 |
[:a + b:] + r * pcompose p r + r * pcompose q r" |
|
2814 |
by (simp_all add: pcompose_pCons pCons.IH algebra_simps) |
|
2815 |
also have "[:a + b:] = [:a:] + [:b:]" by simp |
|
2816 |
also have "\<dots> + r * pcompose p r + r * pcompose q r = |
|
2817 |
pcompose (pCons a p) r + pcompose (pCons b q) r" |
|
2818 |
by (simp only: pcompose_pCons add_ac) |
|
2819 |
finally show ?case . |
|
2820 |
qed simp |
|
2821 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2822 |
lemma pcompose_uminus: |
62065 | 2823 |
fixes p r :: "'a :: comm_ring poly" |
2824 |
shows "pcompose (-p) r = -pcompose p r" |
|
2825 |
by (induction p) (simp_all add: pcompose_pCons) |
|
2826 |
||
2827 |
lemma pcompose_diff: |
|
2828 |
fixes p q r :: "'a :: comm_ring poly" |
|
2829 |
shows "pcompose (p - q) r = pcompose p r - pcompose q r" |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2830 |
using pcompose_add[of p "-q"] by (simp add: pcompose_uminus) |
62065 | 2831 |
|
2832 |
lemma pcompose_smult: |
|
2833 |
fixes p r :: "'a :: comm_semiring_0 poly" |
|
2834 |
shows "pcompose (smult a p) r = smult a (pcompose p r)" |
|
2835 |
by (induction p) |
|
2836 |
(simp_all add: pcompose_pCons pcompose_add smult_add_right) |
|
2837 |
||
2838 |
lemma pcompose_mult: |
|
2839 |
fixes p q r :: "'a :: comm_semiring_0 poly" |
|
2840 |
shows "pcompose (p * q) r = pcompose p r * pcompose q r" |
|
2841 |
by (induction p arbitrary: q) |
|
2842 |
(simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps) |
|
2843 |
||
2844 |
lemma pcompose_assoc: |
|
2845 |
"pcompose p (pcompose q r :: 'a :: comm_semiring_0 poly ) = |
|
2846 |
pcompose (pcompose p q) r" |
|
2847 |
by (induction p arbitrary: q) |
|
2848 |
(simp_all add: pcompose_pCons pcompose_add pcompose_mult) |
|
2849 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2850 |
lemma pcompose_idR[simp]: |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2851 |
fixes p :: "'a :: comm_semiring_1 poly" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2852 |
shows "pcompose p [: 0, 1 :] = p" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2853 |
by (induct p; simp add: pcompose_pCons) |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2854 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2855 |
lemma pcompose_setsum: "pcompose (setsum f A) p = setsum (\<lambda>i. pcompose (f i) p) A" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2856 |
by (cases "finite A", induction rule: finite_induct) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2857 |
(simp_all add: pcompose_1 pcompose_add) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2858 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2859 |
lemma pcompose_setprod: "pcompose (setprod f A) p = setprod (\<lambda>i. pcompose (f i) p) A" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2860 |
by (cases "finite A", induction rule: finite_induct) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2861 |
(simp_all add: pcompose_1 pcompose_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2862 |
|
62065 | 2863 |
|
2864 |
(* The remainder of this section and the next were contributed by Wenda Li *) |
|
2865 |
||
2866 |
lemma degree_mult_eq_0: |
|
63498 | 2867 |
fixes p q:: "'a :: {comm_semiring_0,semiring_no_zero_divisors} poly" |
62065 | 2868 |
shows "degree (p*q) = 0 \<longleftrightarrow> p=0 \<or> q=0 \<or> (p\<noteq>0 \<and> q\<noteq>0 \<and> degree p =0 \<and> degree q =0)" |
2869 |
by (auto simp add:degree_mult_eq) |
|
2870 |
||
2871 |
lemma pcompose_const[simp]:"pcompose [:a:] q = [:a:]" by (subst pcompose_pCons,simp) |
|
2872 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2873 |
lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2874 |
by (induct p) (auto simp add:pcompose_pCons) |
62065 | 2875 |
|
2876 |
lemma degree_pcompose: |
|
63498 | 2877 |
fixes p q:: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2878 |
shows "degree (pcompose p q) = degree p * degree q" |
62065 | 2879 |
proof (induct p) |
2880 |
case 0 |
|
2881 |
thus ?case by auto |
|
2882 |
next |
|
2883 |
case (pCons a p) |
|
2884 |
have "degree (q * pcompose p q) = 0 \<Longrightarrow> ?case" |
|
2885 |
proof (cases "p=0") |
|
2886 |
case True |
|
2887 |
thus ?thesis by auto |
|
2888 |
next |
|
2889 |
case False assume "degree (q * pcompose p q) = 0" |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2890 |
hence "degree q=0 \<or> pcompose p q=0" by (auto simp add: degree_mult_eq_0) |
62072 | 2891 |
moreover have "\<lbrakk>pcompose p q=0;degree q\<noteq>0\<rbrakk> \<Longrightarrow> False" using pCons.hyps(2) \<open>p\<noteq>0\<close> |
62065 | 2892 |
proof - |
2893 |
assume "pcompose p q=0" "degree q\<noteq>0" |
|
2894 |
hence "degree p=0" using pCons.hyps(2) by auto |
|
2895 |
then obtain a1 where "p=[:a1:]" |
|
2896 |
by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) |
|
62072 | 2897 |
thus False using \<open>pcompose p q=0\<close> \<open>p\<noteq>0\<close> by auto |
62065 | 2898 |
qed |
2899 |
ultimately have "degree (pCons a p) * degree q=0" by auto |
|
2900 |
moreover have "degree (pcompose (pCons a p) q) = 0" |
|
2901 |
proof - |
|
2902 |
have" 0 = max (degree [:a:]) (degree (q*pcompose p q))" |
|
62072 | 2903 |
using \<open>degree (q * pcompose p q) = 0\<close> by simp |
62065 | 2904 |
also have "... \<ge> degree ([:a:] + q * pcompose p q)" |
2905 |
by (rule degree_add_le_max) |
|
2906 |
finally show ?thesis by (auto simp add:pcompose_pCons) |
|
2907 |
qed |
|
2908 |
ultimately show ?thesis by simp |
|
2909 |
qed |
|
2910 |
moreover have "degree (q * pcompose p q)>0 \<Longrightarrow> ?case" |
|
2911 |
proof - |
|
2912 |
assume asm:"0 < degree (q * pcompose p q)" |
|
2913 |
hence "p\<noteq>0" "q\<noteq>0" "pcompose p q\<noteq>0" by auto |
|
2914 |
have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)" |
|
2915 |
unfolding pcompose_pCons |
|
2916 |
using degree_add_eq_right[of "[:a:]" ] asm by auto |
|
2917 |
thus ?thesis |
|
62072 | 2918 |
using pCons.hyps(2) degree_mult_eq[OF \<open>q\<noteq>0\<close> \<open>pcompose p q\<noteq>0\<close>] by auto |
62065 | 2919 |
qed |
2920 |
ultimately show ?case by blast |
|
2921 |
qed |
|
2922 |
||
2923 |
lemma pcompose_eq_0: |
|
63498 | 2924 |
fixes p q:: "'a :: {comm_semiring_0,semiring_no_zero_divisors} poly" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2925 |
assumes "pcompose p q = 0" "degree q > 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2926 |
shows "p = 0" |
62065 | 2927 |
proof - |
2928 |
have "degree p=0" using assms degree_pcompose[of p q] by auto |
|
2929 |
then obtain a where "p=[:a:]" |
|
2930 |
by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases) |
|
2931 |
hence "a=0" using assms(1) by auto |
|
62072 | 2932 |
thus ?thesis using \<open>p=[:a:]\<close> by simp |
62065 | 2933 |
qed |
2934 |
||
2935 |
||
62072 | 2936 |
subsection \<open>Leading coefficient\<close> |
62065 | 2937 |
|
2938 |
definition lead_coeff:: "'a::zero poly \<Rightarrow> 'a" where |
|
2939 |
"lead_coeff p= coeff p (degree p)" |
|
2940 |
||
2941 |
lemma lead_coeff_pCons[simp]: |
|
2942 |
"p\<noteq>0 \<Longrightarrow>lead_coeff (pCons a p) = lead_coeff p" |
|
2943 |
"p=0 \<Longrightarrow> lead_coeff (pCons a p) = a" |
|
2944 |
unfolding lead_coeff_def by auto |
|
2945 |
||
2946 |
lemma lead_coeff_0[simp]:"lead_coeff 0 =0" |
|
2947 |
unfolding lead_coeff_def by auto |
|
2948 |
||
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2949 |
lemma coeff_0_listprod: "coeff (listprod xs) 0 = listprod (map (\<lambda>p. coeff p 0) xs)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2950 |
by (induction xs) (simp_all add: coeff_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2951 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2952 |
lemma coeff_0_power: "coeff (p ^ n) 0 = coeff p 0 ^ n" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2953 |
by (induction n) (simp_all add: coeff_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2954 |
|
62065 | 2955 |
lemma lead_coeff_mult: |
63498 | 2956 |
fixes p q::"'a :: {comm_semiring_0,semiring_no_zero_divisors} poly" |
62065 | 2957 |
shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q" |
2958 |
by (unfold lead_coeff_def,cases "p=0 \<or> q=0",auto simp add:coeff_mult_degree_sum degree_mult_eq) |
|
2959 |
||
2960 |
lemma lead_coeff_add_le: |
|
2961 |
assumes "degree p < degree q" |
|
2962 |
shows "lead_coeff (p+q) = lead_coeff q" |
|
2963 |
using assms unfolding lead_coeff_def |
|
2964 |
by (metis coeff_add coeff_eq_0 monoid_add_class.add.left_neutral degree_add_eq_right) |
|
2965 |
||
2966 |
lemma lead_coeff_minus: |
|
2967 |
"lead_coeff (-p) = - lead_coeff p" |
|
2968 |
by (metis coeff_minus degree_minus lead_coeff_def) |
|
2969 |
||
63498 | 2970 |
lemma lead_coeff_smult: |
2971 |
"lead_coeff (smult c p :: 'a :: {comm_semiring_0,semiring_no_zero_divisors} poly) = c * lead_coeff p" |
|
2972 |
proof - |
|
2973 |
have "smult c p = [:c:] * p" by simp |
|
2974 |
also have "lead_coeff \<dots> = c * lead_coeff p" |
|
2975 |
by (subst lead_coeff_mult) simp_all |
|
2976 |
finally show ?thesis . |
|
2977 |
qed |
|
2978 |
||
2979 |
lemma lead_coeff_eq_zero_iff [simp]: "lead_coeff p = 0 \<longleftrightarrow> p = 0" |
|
2980 |
by (simp add: lead_coeff_def) |
|
62065 | 2981 |
|
2982 |
lemma lead_coeff_comp: |
|
63498 | 2983 |
fixes p q:: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" |
62065 | 2984 |
assumes "degree q > 0" |
2985 |
shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)" |
|
2986 |
proof (induct p) |
|
2987 |
case 0 |
|
2988 |
thus ?case unfolding lead_coeff_def by auto |
|
2989 |
next |
|
2990 |
case (pCons a p) |
|
2991 |
have "degree ( q * pcompose p q) = 0 \<Longrightarrow> ?case" |
|
2992 |
proof - |
|
2993 |
assume "degree ( q * pcompose p q) = 0" |
|
2994 |
hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv) |
|
62072 | 2995 |
hence "p=0" using pcompose_eq_0[OF _ \<open>degree q > 0\<close>] by simp |
62065 | 2996 |
thus ?thesis by auto |
2997 |
qed |
|
2998 |
moreover have "degree ( q * pcompose p q) > 0 \<Longrightarrow> ?case" |
|
2999 |
proof - |
|
3000 |
assume "degree ( q * pcompose p q) > 0" |
|
3001 |
hence "lead_coeff (pcompose (pCons a p) q) =lead_coeff ( q * pcompose p q)" |
|
3002 |
by (auto simp add:pcompose_pCons lead_coeff_add_le) |
|
3003 |
also have "... = lead_coeff q * (lead_coeff p * lead_coeff q ^ degree p)" |
|
3004 |
using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp |
|
3005 |
also have "... = lead_coeff p * lead_coeff q ^ (degree p + 1)" |
|
63498 | 3006 |
by (auto simp: mult_ac) |
62065 | 3007 |
finally show ?thesis by auto |
3008 |
qed |
|
3009 |
ultimately show ?case by blast |
|
3010 |
qed |
|
3011 |
||
3012 |
lemma lead_coeff_1 [simp]: "lead_coeff 1 = 1" |
|
3013 |
by (simp add: lead_coeff_def) |
|
3014 |
||
3015 |
lemma lead_coeff_of_nat [simp]: |
|
3016 |
"lead_coeff (of_nat n) = (of_nat n :: 'a :: {comm_semiring_1,semiring_char_0})" |
|
3017 |
by (induction n) (simp_all add: lead_coeff_def of_nat_poly) |
|
3018 |
||
3019 |
lemma lead_coeff_numeral [simp]: |
|
3020 |
"lead_coeff (numeral n) = numeral n" |
|
3021 |
unfolding lead_coeff_def |
|
3022 |
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp |
|
3023 |
||
3024 |
lemma lead_coeff_power: |
|
63498 | 3025 |
"lead_coeff (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) = lead_coeff p ^ n" |
62065 | 3026 |
by (induction n) (simp_all add: lead_coeff_mult) |
3027 |
||
3028 |
lemma lead_coeff_nonzero: "p \<noteq> 0 \<Longrightarrow> lead_coeff p \<noteq> 0" |
|
3029 |
by (simp add: lead_coeff_def) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3030 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3031 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3032 |
subsection \<open>Shifting polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3033 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3034 |
definition poly_shift :: "nat \<Rightarrow> ('a::zero) poly \<Rightarrow> 'a poly" where |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3035 |
"poly_shift n p = Abs_poly (\<lambda>i. coeff p (i + n))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3036 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3037 |
lemma nth_default_drop: "nth_default x (drop n xs) m = nth_default x xs (m + n)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3038 |
by (auto simp add: nth_default_def add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3039 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3040 |
lemma nth_default_take: "nth_default x (take n xs) m = (if m < n then nth_default x xs m else x)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3041 |
by (auto simp add: nth_default_def add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3042 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3043 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3044 |
lemma coeff_poly_shift: "coeff (poly_shift n p) i = coeff p (i + n)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3045 |
proof - |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3046 |
from MOST_coeff_eq_0[of p] obtain m where "\<forall>k>m. coeff p k = 0" by (auto simp: MOST_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3047 |
hence "\<forall>k>m. coeff p (k + n) = 0" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3048 |
hence "\<forall>\<^sub>\<infinity>k. coeff p (k + n) = 0" by (auto simp: MOST_nat) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3049 |
thus ?thesis by (simp add: poly_shift_def poly.Abs_poly_inverse) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3050 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3051 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3052 |
lemma poly_shift_id [simp]: "poly_shift 0 = (\<lambda>x. x)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3053 |
by (simp add: poly_eq_iff fun_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3054 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3055 |
lemma poly_shift_0 [simp]: "poly_shift n 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3056 |
by (simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3057 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3058 |
lemma poly_shift_1: "poly_shift n 1 = (if n = 0 then 1 else 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3059 |
by (simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3060 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3061 |
lemma poly_shift_monom: "poly_shift n (monom c m) = (if m \<ge> n then monom c (m - n) else 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3062 |
by (auto simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3063 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3064 |
lemma coeffs_shift_poly [code abstract]: "coeffs (poly_shift n p) = drop n (coeffs p)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3065 |
proof (cases "p = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3066 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3067 |
thus ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3068 |
by (intro coeffs_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3069 |
(simp_all add: coeff_poly_shift nth_default_drop last_coeffs_not_0 nth_default_coeffs_eq) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3070 |
qed simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3071 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3072 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3073 |
subsection \<open>Truncating polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3074 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3075 |
definition poly_cutoff where |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3076 |
"poly_cutoff n p = Abs_poly (\<lambda>k. if k < n then coeff p k else 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3077 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3078 |
lemma coeff_poly_cutoff: "coeff (poly_cutoff n p) k = (if k < n then coeff p k else 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3079 |
unfolding poly_cutoff_def |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3080 |
by (subst poly.Abs_poly_inverse) (auto simp: MOST_nat intro: exI[of _ n]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3081 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3082 |
lemma poly_cutoff_0 [simp]: "poly_cutoff n 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3083 |
by (simp add: poly_eq_iff coeff_poly_cutoff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3084 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3085 |
lemma poly_cutoff_1 [simp]: "poly_cutoff n 1 = (if n = 0 then 0 else 1)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3086 |
by (simp add: poly_eq_iff coeff_poly_cutoff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3087 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3088 |
lemma coeffs_poly_cutoff [code abstract]: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3089 |
"coeffs (poly_cutoff n p) = strip_while (op = 0) (take n (coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3090 |
proof (cases "strip_while (op = 0) (take n (coeffs p)) = []") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3091 |
case True |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3092 |
hence "coeff (poly_cutoff n p) k = 0" for k |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3093 |
unfolding coeff_poly_cutoff |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3094 |
by (auto simp: nth_default_coeffs_eq [symmetric] nth_default_def set_conv_nth) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3095 |
hence "poly_cutoff n p = 0" by (simp add: poly_eq_iff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3096 |
thus ?thesis by (subst True) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3097 |
next |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3098 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3099 |
have "no_trailing (op = 0) (strip_while (op = 0) (take n (coeffs p)))" by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3100 |
with False have "last (strip_while (op = 0) (take n (coeffs p))) \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3101 |
unfolding no_trailing_unfold by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3102 |
thus ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3103 |
by (intro coeffs_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3104 |
(simp_all add: coeff_poly_cutoff last_coeffs_not_0 nth_default_take nth_default_coeffs_eq) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3105 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3106 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3107 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3108 |
subsection \<open>Reflecting polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3109 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3110 |
definition reflect_poly where |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3111 |
"reflect_poly p = Poly (rev (coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3112 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3113 |
lemma coeffs_reflect_poly [code abstract]: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3114 |
"coeffs (reflect_poly p) = rev (dropWhile (op = 0) (coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3115 |
unfolding reflect_poly_def by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3116 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3117 |
lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3118 |
by (simp add: reflect_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3119 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3120 |
lemma reflect_poly_1 [simp]: "reflect_poly 1 = 1" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3121 |
by (simp add: reflect_poly_def one_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3122 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3123 |
lemma coeff_reflect_poly: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3124 |
"coeff (reflect_poly p) n = (if n > degree p then 0 else coeff p (degree p - n))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3125 |
by (cases "p = 0") (auto simp add: reflect_poly_def coeff_Poly_eq nth_default_def |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3126 |
rev_nth degree_eq_length_coeffs coeffs_nth not_less |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3127 |
dest: le_imp_less_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3128 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3129 |
lemma coeff_0_reflect_poly_0_iff [simp]: "coeff (reflect_poly p) 0 = 0 \<longleftrightarrow> p = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3130 |
by (simp add: coeff_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3131 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3132 |
lemma reflect_poly_at_0_eq_0_iff [simp]: "poly (reflect_poly p) 0 = 0 \<longleftrightarrow> p = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3133 |
by (simp add: coeff_reflect_poly poly_0_coeff_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3134 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3135 |
lemma reflect_poly_pCons': |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3136 |
"p \<noteq> 0 \<Longrightarrow> reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3137 |
by (intro poly_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3138 |
(auto simp: coeff_reflect_poly coeff_pCons not_less Suc_diff_le split: nat.split) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3139 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3140 |
lemma reflect_poly_const [simp]: "reflect_poly [:a:] = [:a:]" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3141 |
by (cases "a = 0") (simp_all add: reflect_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3142 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3143 |
lemma poly_reflect_poly_nz: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3144 |
"(x :: 'a :: field) \<noteq> 0 \<Longrightarrow> poly (reflect_poly p) x = x ^ degree p * poly p (inverse x)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3145 |
by (induction rule: pCons_induct) (simp_all add: field_simps reflect_poly_pCons' poly_monom) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3146 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3147 |
lemma coeff_0_reflect_poly [simp]: "coeff (reflect_poly p) 0 = lead_coeff p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3148 |
by (simp add: coeff_reflect_poly lead_coeff_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3149 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3150 |
lemma poly_reflect_poly_0 [simp]: "poly (reflect_poly p) 0 = lead_coeff p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3151 |
by (simp add: poly_0_coeff_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3152 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3153 |
lemma reflect_poly_reflect_poly [simp]: "coeff p 0 \<noteq> 0 \<Longrightarrow> reflect_poly (reflect_poly p) = p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3154 |
by (cases p rule: pCons_cases) (simp add: reflect_poly_def ) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3155 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3156 |
lemma degree_reflect_poly_le: "degree (reflect_poly p) \<le> degree p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3157 |
by (simp add: degree_eq_length_coeffs coeffs_reflect_poly length_dropWhile_le diff_le_mono) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3158 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3159 |
lemma reflect_poly_pCons: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3160 |
"a \<noteq> 0 \<Longrightarrow> reflect_poly (pCons a p) = Poly (rev (a # coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3161 |
by (subst coeffs_eq_iff) (simp add: coeffs_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3162 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3163 |
lemma degree_reflect_poly_eq [simp]: "coeff p 0 \<noteq> 0 \<Longrightarrow> degree (reflect_poly p) = degree p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3164 |
by (cases p rule: pCons_cases) (simp add: reflect_poly_pCons degree_eq_length_coeffs) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3165 |
|
63498 | 3166 |
(* TODO: does this work with zero divisors as well? Probably not. *) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3167 |
lemma reflect_poly_mult: |
63498 | 3168 |
"reflect_poly (p * q) = |
3169 |
reflect_poly p * reflect_poly (q :: _ :: {comm_semiring_0,semiring_no_zero_divisors} poly)" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3170 |
proof (cases "p = 0 \<or> q = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3171 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3172 |
hence [simp]: "p \<noteq> 0" "q \<noteq> 0" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3173 |
show ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3174 |
proof (rule poly_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3175 |
fix i :: nat |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3176 |
show "coeff (reflect_poly (p * q)) i = coeff (reflect_poly p * reflect_poly q) i" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3177 |
proof (cases "i \<le> degree (p * q)") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3178 |
case True |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3179 |
def A \<equiv> "{..i} \<inter> {i - degree q..degree p}" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3180 |
def B \<equiv> "{..degree p} \<inter> {degree p - i..degree (p*q) - i}" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3181 |
let ?f = "\<lambda>j. degree p - j" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3182 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3183 |
from True have "coeff (reflect_poly (p * q)) i = coeff (p * q) (degree (p * q) - i)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3184 |
by (simp add: coeff_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3185 |
also have "\<dots> = (\<Sum>j\<le>degree (p * q) - i. coeff p j * coeff q (degree (p * q) - i - j))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3186 |
unfolding coeff_mult by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3187 |
also have "\<dots> = (\<Sum>j\<in>B. coeff p j * coeff q (degree (p * q) - i - j))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3188 |
by (intro setsum.mono_neutral_right) (auto simp: B_def degree_mult_eq not_le coeff_eq_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3189 |
also from True have "\<dots> = (\<Sum>j\<in>A. coeff p (degree p - j) * coeff q (degree q - (i - j)))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3190 |
by (intro setsum.reindex_bij_witness[of _ ?f ?f]) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3191 |
(auto simp: A_def B_def degree_mult_eq add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3192 |
also have "\<dots> = (\<Sum>j\<le>i. if j \<in> {i - degree q..degree p} then |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3193 |
coeff p (degree p - j) * coeff q (degree q - (i - j)) else 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3194 |
by (subst setsum.inter_restrict [symmetric]) (simp_all add: A_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3195 |
also have "\<dots> = coeff (reflect_poly p * reflect_poly q) i" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3196 |
by (fastforce simp: coeff_mult coeff_reflect_poly intro!: setsum.cong) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3197 |
finally show ?thesis . |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3198 |
qed (auto simp: coeff_mult coeff_reflect_poly coeff_eq_0 degree_mult_eq intro!: setsum.neutral) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3199 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3200 |
qed auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3201 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3202 |
lemma reflect_poly_smult: |
63498 | 3203 |
"reflect_poly (Polynomial.smult (c::'a::{comm_semiring_0,semiring_no_zero_divisors}) p) = |
3204 |
Polynomial.smult c (reflect_poly p)" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3205 |
using reflect_poly_mult[of "[:c:]" p] by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3206 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3207 |
lemma reflect_poly_power: |
63498 | 3208 |
"reflect_poly (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) = |
3209 |
reflect_poly p ^ n" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3210 |
by (induction n) (simp_all add: reflect_poly_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3211 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3212 |
lemma reflect_poly_setprod: |
63498 | 3213 |
"reflect_poly (setprod (f :: _ \<Rightarrow> _ :: {comm_semiring_0,semiring_no_zero_divisors} poly) A) = |
3214 |
setprod (\<lambda>x. reflect_poly (f x)) A" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3215 |
by (cases "finite A", induction rule: finite_induct) (simp_all add: reflect_poly_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3216 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3217 |
lemma reflect_poly_listprod: |
63498 | 3218 |
"reflect_poly (listprod (xs :: _ :: {comm_semiring_0,semiring_no_zero_divisors} poly list)) = |
3219 |
listprod (map reflect_poly xs)" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3220 |
by (induction xs) (simp_all add: reflect_poly_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3221 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3222 |
lemma reflect_poly_Poly_nz: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3223 |
"xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0 \<Longrightarrow> reflect_poly (Poly xs) = Poly (rev xs)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3224 |
unfolding reflect_poly_def coeffs_Poly by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3225 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3226 |
lemmas reflect_poly_simps = |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3227 |
reflect_poly_0 reflect_poly_1 reflect_poly_const reflect_poly_smult reflect_poly_mult |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3228 |
reflect_poly_power reflect_poly_setprod reflect_poly_listprod |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3229 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3230 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3231 |
subsection \<open>Derivatives of univariate polynomials\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3232 |
|
63498 | 3233 |
function pderiv :: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) poly \<Rightarrow> 'a poly" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3234 |
where |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
3235 |
"pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3236 |
by (auto intro: pCons_cases) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3237 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3238 |
termination pderiv |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3239 |
by (relation "measure degree") simp_all |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3240 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
3241 |
declare pderiv.simps[simp del] |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
3242 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3243 |
lemma pderiv_0 [simp]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3244 |
"pderiv 0 = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3245 |
using pderiv.simps [of 0 0] by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3246 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3247 |
lemma pderiv_pCons: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3248 |
"pderiv (pCons a p) = p + pCons 0 (pderiv p)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3249 |
by (simp add: pderiv.simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3250 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3251 |
lemma pderiv_1 [simp]: "pderiv 1 = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3252 |
unfolding one_poly_def by (simp add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3253 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3254 |
lemma pderiv_of_nat [simp]: "pderiv (of_nat n) = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3255 |
and pderiv_numeral [simp]: "pderiv (numeral m) = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3256 |
by (simp_all add: of_nat_poly numeral_poly pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3257 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3258 |
lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3259 |
by (induct p arbitrary: n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3260 |
(auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3261 |
|
63498 | 3262 |
fun pderiv_coeffs_code |
3263 |
:: "('a :: {comm_semiring_1,semiring_no_zero_divisors}) \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3264 |
"pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3265 |
| "pderiv_coeffs_code f [] = []" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3266 |
|
63498 | 3267 |
definition pderiv_coeffs :: |
3268 |
"'a :: {comm_semiring_1,semiring_no_zero_divisors} list \<Rightarrow> 'a list" where |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3269 |
"pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3270 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3271 |
(* Efficient code for pderiv contributed by René Thiemann and Akihisa Yamada *) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3272 |
lemma pderiv_coeffs_code: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3273 |
"nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * (nth_default 0 xs n)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3274 |
proof (induct xs arbitrary: f n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3275 |
case (Cons x xs f n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3276 |
show ?case |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3277 |
proof (cases n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3278 |
case 0 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3279 |
thus ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0", auto simp: cCons_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3280 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3281 |
case (Suc m) note n = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3282 |
show ?thesis |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3283 |
proof (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3284 |
case False |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3285 |
hence "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3286 |
nth_default 0 (pderiv_coeffs_code (f + 1) xs) m" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3287 |
by (auto simp: cCons_def n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3288 |
also have "\<dots> = (f + of_nat n) * (nth_default 0 xs m)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3289 |
unfolding Cons by (simp add: n add_ac) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3290 |
finally show ?thesis by (simp add: n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3291 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3292 |
case True |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3293 |
{ |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3294 |
fix g |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3295 |
have "pderiv_coeffs_code g xs = [] \<Longrightarrow> g + of_nat m = 0 \<or> nth_default 0 xs m = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3296 |
proof (induct xs arbitrary: g m) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3297 |
case (Cons x xs g) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3298 |
from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3299 |
and g: "(g = 0 \<or> x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3300 |
by (auto simp: cCons_def split: if_splits) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3301 |
note IH = Cons(1)[OF empty] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3302 |
from IH[of m] IH[of "m - 1"] g |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3303 |
show ?case by (cases m, auto simp: field_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3304 |
qed simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3305 |
} note empty = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3306 |
from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3307 |
by (auto simp: cCons_def n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3308 |
moreover have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" using True |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3309 |
by (simp add: n, insert empty[of "f+1"], auto simp: field_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3310 |
ultimately show ?thesis by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3311 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3312 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3313 |
qed simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3314 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3315 |
lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (\<lambda> i. f (Suc i)) [0 ..< n]" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3316 |
by (induct n arbitrary: f, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3317 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3318 |
lemma coeffs_pderiv_code [code abstract]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3319 |
"coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3320 |
proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3321 |
case (1 n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3322 |
have id: "coeff p (Suc n) = nth_default 0 (map (\<lambda>i. coeff p (Suc i)) [0..<degree p]) n" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3323 |
by (cases "n < degree p", auto simp: nth_default_def coeff_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3324 |
show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3325 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3326 |
case 2 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3327 |
obtain n xs where id: "tl (coeffs p) = xs" "(1 :: 'a) = n" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3328 |
from 2 show ?case |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3329 |
unfolding id by (induct xs arbitrary: n, auto simp: cCons_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3330 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3331 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3332 |
context |
63498 | 3333 |
assumes "SORT_CONSTRAINT('a::{comm_semiring_1,semiring_no_zero_divisors, semiring_char_0})" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3334 |
begin |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3335 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3336 |
lemma pderiv_eq_0_iff: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3337 |
"pderiv (p :: 'a poly) = 0 \<longleftrightarrow> degree p = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3338 |
apply (rule iffI) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3339 |
apply (cases p, simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3340 |
apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3341 |
apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3342 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3343 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3344 |
lemma degree_pderiv: "degree (pderiv (p :: 'a poly)) = degree p - 1" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3345 |
apply (rule order_antisym [OF degree_le]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3346 |
apply (simp add: coeff_pderiv coeff_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3347 |
apply (cases "degree p", simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3348 |
apply (rule le_degree) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3349 |
apply (simp add: coeff_pderiv del: of_nat_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3350 |
apply (metis degree_0 leading_coeff_0_iff nat.distinct(1)) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3351 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3352 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3353 |
lemma not_dvd_pderiv: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3354 |
assumes "degree (p :: 'a poly) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3355 |
shows "\<not> p dvd pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3356 |
proof |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3357 |
assume dvd: "p dvd pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3358 |
then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3359 |
from dvd have le: "degree p \<le> degree (pderiv p)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3360 |
by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3361 |
from this[unfolded degree_pderiv] assms show False by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3362 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3363 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3364 |
lemma dvd_pderiv_iff [simp]: "(p :: 'a poly) dvd pderiv p \<longleftrightarrow> degree p = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3365 |
using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3366 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3367 |
end |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3368 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3369 |
lemma pderiv_singleton [simp]: "pderiv [:a:] = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3370 |
by (simp add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3371 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3372 |
lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3373 |
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3374 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3375 |
lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3376 |
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3377 |
|
63498 | 3378 |
lemma pderiv_diff: "pderiv ((p :: _ :: idom poly) - q) = pderiv p - pderiv q" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3379 |
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3380 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3381 |
lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3382 |
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3383 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3384 |
lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3385 |
by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3386 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3387 |
lemma pderiv_power_Suc: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3388 |
"pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3389 |
apply (induct n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3390 |
apply simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3391 |
apply (subst power_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3392 |
apply (subst pderiv_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3393 |
apply (erule ssubst) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3394 |
apply (simp only: of_nat_Suc smult_add_left smult_1_left) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3395 |
apply (simp add: algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3396 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3397 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3398 |
lemma pderiv_setprod: "pderiv (setprod f (as)) = |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3399 |
(\<Sum>a \<in> as. setprod f (as - {a}) * pderiv (f a))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3400 |
proof (induct as rule: infinite_finite_induct) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3401 |
case (insert a as) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3402 |
hence id: "setprod f (insert a as) = f a * setprod f as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3403 |
"\<And> g. setsum g (insert a as) = g a + setsum g as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3404 |
"insert a as - {a} = as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3405 |
by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3406 |
{ |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3407 |
fix b |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3408 |
assume "b \<in> as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3409 |
hence id2: "insert a as - {b} = insert a (as - {b})" using \<open>a \<notin> as\<close> by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3410 |
have "setprod f (insert a as - {b}) = f a * setprod f (as - {b})" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3411 |
unfolding id2 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3412 |
by (subst setprod.insert, insert insert, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3413 |
} note id2 = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3414 |
show ?case |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3415 |
unfolding id pderiv_mult insert(3) setsum_right_distrib |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3416 |
by (auto simp add: ac_simps id2 intro!: setsum.cong) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3417 |
qed auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3418 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3419 |
lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3420 |
by (rule DERIV_cong, rule DERIV_pow, simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3421 |
declare DERIV_pow2 [simp] DERIV_pow [simp] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3422 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3423 |
lemma DERIV_add_const: "DERIV f x :> D ==> DERIV (%x. a + f x :: 'a::real_normed_field) x :> D" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3424 |
by (rule DERIV_cong, rule DERIV_add, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3425 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3426 |
lemma poly_DERIV [simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3427 |
by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3428 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3429 |
lemma continuous_on_poly [continuous_intros]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3430 |
fixes p :: "'a :: {real_normed_field} poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3431 |
assumes "continuous_on A f" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3432 |
shows "continuous_on A (\<lambda>x. poly p (f x))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3433 |
proof - |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3434 |
have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3435 |
by (intro continuous_intros assms) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3436 |
also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3437 |
finally show ?thesis . |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3438 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3439 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3440 |
text\<open>Consequences of the derivative theorem above\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3441 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3442 |
lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3443 |
apply (simp add: real_differentiable_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3444 |
apply (blast intro: poly_DERIV) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3445 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3446 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3447 |
lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3448 |
by (rule poly_DERIV [THEN DERIV_isCont]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3449 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3450 |
lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3451 |
==> \<exists>x. a < x & x < b & (poly p x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3452 |
using IVT_objl [of "poly p" a 0 b] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3453 |
by (auto simp add: order_le_less) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3454 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3455 |
lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3456 |
==> \<exists>x. a < x & x < b & (poly p x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3457 |
by (insert poly_IVT_pos [where p = "- p" ]) simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3458 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3459 |
lemma poly_IVT: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3460 |
fixes p::"real poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3461 |
assumes "a<b" and "poly p a * poly p b < 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3462 |
shows "\<exists>x>a. x < b \<and> poly p x = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3463 |
by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3464 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3465 |
lemma poly_MVT: "(a::real) < b ==> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3466 |
\<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3467 |
using MVT [of a b "poly p"] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3468 |
apply auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3469 |
apply (rule_tac x = z in exI) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3470 |
apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3471 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3472 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3473 |
lemma poly_MVT': |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3474 |
assumes "{min a b..max a b} \<subseteq> A" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3475 |
shows "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3476 |
proof (cases a b rule: linorder_cases) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3477 |
case less |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3478 |
from poly_MVT[OF less, of p] guess x by (elim exE conjE) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3479 |
thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3480 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3481 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3482 |
case greater |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3483 |
from poly_MVT[OF greater, of p] guess x by (elim exE conjE) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3484 |
thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3485 |
qed (insert assms, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3486 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3487 |
lemma poly_pinfty_gt_lc: |
63649 | 3488 |
fixes p :: "real poly" |
3489 |
assumes "lead_coeff p > 0" |
|
3490 |
shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" |
|
3491 |
using assms |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3492 |
proof (induct p) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3493 |
case 0 |
63649 | 3494 |
then show ?case by auto |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3495 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3496 |
case (pCons a p) |
63649 | 3497 |
from this(1) consider "a \<noteq> 0" "p = 0" | "p \<noteq> 0" by auto |
3498 |
then show ?case |
|
3499 |
proof cases |
|
3500 |
case 1 |
|
3501 |
then show ?thesis by auto |
|
3502 |
next |
|
3503 |
case 2 |
|
3504 |
with pCons obtain n1 where gte_lcoeff: "\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" |
|
3505 |
by auto |
|
3506 |
from pCons(3) \<open>p \<noteq> 0\<close> have gt_0: "lead_coeff p > 0" by auto |
|
3507 |
define n where "n = max n1 (1 + \<bar>a\<bar> / lead_coeff p)" |
|
3508 |
have "lead_coeff (pCons a p) \<le> poly (pCons a p) x" if "n \<le> x" for x |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3509 |
proof - |
63649 | 3510 |
from gte_lcoeff that have "lead_coeff p \<le> poly p x" |
3511 |
by (auto simp: n_def) |
|
3512 |
with gt_0 have "\<bar>a\<bar> / lead_coeff p \<ge> \<bar>a\<bar> / poly p x" and "poly p x > 0" |
|
3513 |
by (auto intro: frac_le) |
|
3514 |
with \<open>n\<le>x\<close>[unfolded n_def] have "x \<ge> 1 + \<bar>a\<bar> / poly p x" |
|
3515 |
by auto |
|
3516 |
with \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x > 0\<close> \<open>p \<noteq> 0\<close> |
|
3517 |
show "lead_coeff (pCons a p) \<le> poly (pCons a p) x" |
|
3518 |
by (auto simp: field_simps) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3519 |
qed |
63649 | 3520 |
then show ?thesis by blast |
3521 |
qed |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3522 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3523 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3524 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3525 |
subsection \<open>Algebraic numbers\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3526 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3527 |
text \<open> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3528 |
Algebraic numbers can be defined in two equivalent ways: all real numbers that are |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3529 |
roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3530 |
uses the rational definition, but we need the integer definition. |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3531 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3532 |
The equivalence is obvious since any rational polynomial can be multiplied with the |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3533 |
LCM of its coefficients, yielding an integer polynomial with the same roots. |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3534 |
\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3535 |
subsection \<open>Algebraic numbers\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3536 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3537 |
definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool" where |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3538 |
"algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3539 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3540 |
lemma algebraicI: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3541 |
assumes "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3542 |
shows "algebraic x" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3543 |
using assms unfolding algebraic_def by blast |
62065 | 3544 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3545 |
lemma algebraicE: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3546 |
assumes "algebraic x" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3547 |
obtains p where "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3548 |
using assms unfolding algebraic_def by blast |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3549 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3550 |
lemma quotient_of_denom_pos': "snd (quotient_of x) > 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3551 |
using quotient_of_denom_pos[OF surjective_pairing] . |
62065 | 3552 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3553 |
lemma of_int_div_in_Ints: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3554 |
"b dvd a \<Longrightarrow> of_int a div of_int b \<in> (\<int> :: 'a :: ring_div set)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3555 |
proof (cases "of_int b = (0 :: 'a)") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3556 |
assume "b dvd a" "of_int b \<noteq> (0::'a)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3557 |
then obtain c where "a = b * c" by (elim dvdE) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3558 |
with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3559 |
qed auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3560 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3561 |
lemma of_int_divide_in_Ints: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3562 |
"b dvd a \<Longrightarrow> of_int a / of_int b \<in> (\<int> :: 'a :: field set)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3563 |
proof (cases "of_int b = (0 :: 'a)") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3564 |
assume "b dvd a" "of_int b \<noteq> (0::'a)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3565 |
then obtain c where "a = b * c" by (elim dvdE) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3566 |
with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3567 |
qed auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3568 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3569 |
lemma algebraic_altdef: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3570 |
fixes p :: "'a :: field_char_0 poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3571 |
shows "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3572 |
proof safe |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3573 |
fix p assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0" |
63040 | 3574 |
define cs where "cs = coeffs p" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3575 |
from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'" unfolding Rats_def by blast |
63060 | 3576 |
then obtain f where f: "coeff p i = of_rat (f (coeff p i))" for i |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3577 |
by (subst (asm) bchoice_iff) blast |
63040 | 3578 |
define cs' where "cs' = map (quotient_of \<circ> f) (coeffs p)" |
3579 |
define d where "d = Lcm (set (map snd cs'))" |
|
3580 |
define p' where "p' = smult (of_int d) p" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3581 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3582 |
have "\<forall>n. coeff p' n \<in> \<int>" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3583 |
proof |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3584 |
fix n :: nat |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3585 |
show "coeff p' n \<in> \<int>" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3586 |
proof (cases "n \<le> degree p") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3587 |
case True |
63040 | 3588 |
define c where "c = coeff p n" |
3589 |
define a where "a = fst (quotient_of (f (coeff p n)))" |
|
3590 |
define b where "b = snd (quotient_of (f (coeff p n)))" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3591 |
have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3592 |
have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3593 |
also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3594 |
by (subst quotient_of_div [of "f (coeff p n)", symmetric]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3595 |
(simp_all add: f [symmetric]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3596 |
also have "of_int d * \<dots> = of_rat (of_int (a*d) / of_int b)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3597 |
by (simp add: of_rat_mult of_rat_divide) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3598 |
also from nz True have "b \<in> snd ` set cs'" unfolding cs'_def |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3599 |
by (force simp: o_def b_def coeffs_def simp del: upt_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3600 |
hence "b dvd (a * d)" unfolding d_def by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3601 |
hence "of_int (a * d) / of_int b \<in> (\<int> :: rat set)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3602 |
by (rule of_int_divide_in_Ints) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3603 |
hence "of_rat (of_int (a * d) / of_int b) \<in> \<int>" by (elim Ints_cases) auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3604 |
finally show ?thesis . |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3605 |
qed (auto simp: p'_def not_le coeff_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3606 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3607 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3608 |
moreover have "set (map snd cs') \<subseteq> {0<..}" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3609 |
unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3610 |
hence "d \<noteq> 0" unfolding d_def by (induction cs') simp_all |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3611 |
with nz have "p' \<noteq> 0" by (simp add: p'_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3612 |
moreover from root have "poly p' x = 0" by (simp add: p'_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3613 |
ultimately show "algebraic x" unfolding algebraic_def by blast |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3614 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3615 |
assume "algebraic x" |
63060 | 3616 |
then obtain p where p: "coeff p i \<in> \<int>" "poly p x = 0" "p \<noteq> 0" for i |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3617 |
by (force simp: algebraic_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3618 |
moreover have "coeff p i \<in> \<int> \<Longrightarrow> coeff p i \<in> \<rat>" for i by (elim Ints_cases) simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3619 |
ultimately show "(\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3620 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3621 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3622 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3623 |
text\<open>Lemmas for Derivatives\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3624 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3625 |
lemma order_unique_lemma: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3626 |
fixes p :: "'a::idom poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3627 |
assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3628 |
shows "n = order a p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3629 |
unfolding Polynomial.order_def |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3630 |
apply (rule Least_equality [symmetric]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3631 |
apply (fact assms) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3632 |
apply (rule classical) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3633 |
apply (erule notE) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3634 |
unfolding not_less_eq_eq |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3635 |
using assms(1) apply (rule power_le_dvd) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3636 |
apply assumption |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3637 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3638 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3639 |
lemma lemma_order_pderiv1: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3640 |
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q + |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3641 |
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3642 |
apply (simp only: pderiv_mult pderiv_power_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3643 |
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3644 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3645 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3646 |
lemma lemma_order_pderiv: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3647 |
fixes p :: "'a :: field_char_0 poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3648 |
assumes n: "0 < n" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3649 |
and pd: "pderiv p \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3650 |
and pe: "p = [:- a, 1:] ^ n * q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3651 |
and nd: "~ [:- a, 1:] dvd q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3652 |
shows "n = Suc (order a (pderiv p))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3653 |
using n |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3654 |
proof - |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3655 |
have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3656 |
using assms by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3657 |
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3658 |
using assms by (cases n) auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3659 |
have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3660 |
by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3661 |
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3662 |
proof (rule order_unique_lemma) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3663 |
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3664 |
apply (subst lemma_order_pderiv1) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3665 |
apply (rule dvd_add) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3666 |
apply (metis dvdI dvd_mult2 power_Suc2) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3667 |
apply (metis dvd_smult dvd_triv_right) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3668 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3669 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3670 |
show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3671 |
apply (subst lemma_order_pderiv1) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3672 |
by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3673 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3674 |
then show ?thesis |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3675 |
by (metis \<open>n = Suc n'\<close> pe) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3676 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3677 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3678 |
lemma order_decomp: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3679 |
assumes "p \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3680 |
shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3681 |
proof - |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3682 |
from assms have A: "[:- a, 1:] ^ order a p dvd p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3683 |
and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3684 |
from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" .. |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3685 |
with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3686 |
by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3687 |
then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3688 |
by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3689 |
then have D: "\<not> [:- a, 1:] dvd q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3690 |
using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3691 |
by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3692 |
from C D show ?thesis by blast |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3693 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3694 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3695 |
lemma order_pderiv: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3696 |
"\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> \<Longrightarrow> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3697 |
(order a p = Suc (order a (pderiv p)))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3698 |
apply (case_tac "p = 0", simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3699 |
apply (drule_tac a = a and p = p in order_decomp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3700 |
using neq0_conv |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3701 |
apply (blast intro: lemma_order_pderiv) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3702 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3703 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3704 |
lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3705 |
proof - |
63040 | 3706 |
define i where "i = order a p" |
3707 |
define j where "j = order a q" |
|
3708 |
define t where "t = [:-a, 1:]" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3709 |
have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3710 |
unfolding t_def by (simp add: dvd_iff_poly_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3711 |
assume "p * q \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3712 |
then show "order a (p * q) = i + j" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3713 |
apply clarsimp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3714 |
apply (drule order [where a=a and p=p, folded i_def t_def]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3715 |
apply (drule order [where a=a and p=q, folded j_def t_def]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3716 |
apply clarify |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3717 |
apply (erule dvdE)+ |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3718 |
apply (rule order_unique_lemma [symmetric], fold t_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3719 |
apply (simp_all add: power_add t_dvd_iff) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3720 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3721 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3722 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3723 |
lemma order_smult: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3724 |
assumes "c \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3725 |
shows "order x (smult c p) = order x p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3726 |
proof (cases "p = 0") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3727 |
case False |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3728 |
have "smult c p = [:c:] * p" by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3729 |
also from assms False have "order x \<dots> = order x [:c:] + order x p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3730 |
by (subst order_mult) simp_all |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3731 |
also from assms have "order x [:c:] = 0" by (intro order_0I) auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3732 |
finally show ?thesis by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3733 |
qed simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3734 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3735 |
(* Next two lemmas contributed by Wenda Li *) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3736 |
lemma order_1_eq_0 [simp]:"order x 1 = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3737 |
by (metis order_root poly_1 zero_neq_one) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3738 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3739 |
lemma order_power_n_n: "order a ([:-a,1:]^n)=n" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3740 |
proof (induct n) (*might be proved more concisely using nat_less_induct*) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3741 |
case 0 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3742 |
thus ?case by (metis order_root poly_1 power_0 zero_neq_one) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3743 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3744 |
case (Suc n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3745 |
have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3746 |
by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3747 |
one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3748 |
moreover have "order a [:-a,1:]=1" unfolding order_def |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3749 |
proof (rule Least_equality,rule ccontr) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3750 |
assume "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3751 |
hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3752 |
hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3753 |
by (rule dvd_imp_degree_le,auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3754 |
thus False by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3755 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3756 |
fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3757 |
show "1 \<le> y" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3758 |
proof (rule ccontr) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3759 |
assume "\<not> 1 \<le> y" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3760 |
hence "y=0" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3761 |
hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3762 |
thus False using asm by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3763 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3764 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3765 |
ultimately show ?case using Suc by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3766 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3767 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3768 |
lemma order_0_monom [simp]: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3769 |
assumes "c \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3770 |
shows "order 0 (monom c n) = n" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3771 |
using assms order_power_n_n[of 0 n] by (simp add: monom_altdef order_smult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3772 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3773 |
lemma dvd_imp_order_le: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3774 |
"q \<noteq> 0 \<Longrightarrow> p dvd q \<Longrightarrow> Polynomial.order a p \<le> Polynomial.order a q" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3775 |
by (auto simp: order_mult elim: dvdE) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3776 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3777 |
text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3778 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3779 |
lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3780 |
apply (cases "p = 0", auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3781 |
apply (drule order_2 [where a=a and p=p]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3782 |
apply (metis not_less_eq_eq power_le_dvd) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3783 |
apply (erule power_le_dvd [OF order_1]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3784 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3785 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3786 |
lemma monom_1_dvd_iff: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3787 |
assumes "p \<noteq> 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3788 |
shows "monom 1 n dvd p \<longleftrightarrow> n \<le> Polynomial.order 0 p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3789 |
using assms order_divides[of 0 n p] by (simp add: monom_altdef) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3790 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3791 |
lemma poly_squarefree_decomp_order: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3792 |
assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3793 |
and p: "p = q * d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3794 |
and p': "pderiv p = e * d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3795 |
and d: "d = r * p + s * pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3796 |
shows "order a q = (if order a p = 0 then 0 else 1)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3797 |
proof (rule classical) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3798 |
assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3799 |
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3800 |
with p have "order a p = order a q + order a d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3801 |
by (simp add: order_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3802 |
with 1 have "order a p \<noteq> 0" by (auto split: if_splits) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3803 |
have "order a (pderiv p) = order a e + order a d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3804 |
using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3805 |
have "order a p = Suc (order a (pderiv p))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3806 |
using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3807 |
have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3808 |
have "([:-a, 1:] ^ (order a (pderiv p))) dvd d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3809 |
apply (simp add: d) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3810 |
apply (rule dvd_add) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3811 |
apply (rule dvd_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3812 |
apply (simp add: order_divides \<open>p \<noteq> 0\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3813 |
\<open>order a p = Suc (order a (pderiv p))\<close>) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3814 |
apply (rule dvd_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3815 |
apply (simp add: order_divides) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3816 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3817 |
then have "order a (pderiv p) \<le> order a d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3818 |
using \<open>d \<noteq> 0\<close> by (simp add: order_divides) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3819 |
show ?thesis |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3820 |
using \<open>order a p = order a q + order a d\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3821 |
using \<open>order a (pderiv p) = order a e + order a d\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3822 |
using \<open>order a p = Suc (order a (pderiv p))\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3823 |
using \<open>order a (pderiv p) \<le> order a d\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3824 |
by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3825 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3826 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3827 |
lemma poly_squarefree_decomp_order2: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3828 |
"\<lbrakk>pderiv p \<noteq> (0 :: 'a :: field_char_0 poly); |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3829 |
p = q * d; |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3830 |
pderiv p = e * d; |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3831 |
d = r * p + s * pderiv p |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3832 |
\<rbrakk> \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3833 |
by (blast intro: poly_squarefree_decomp_order) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3834 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3835 |
lemma order_pderiv2: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3836 |
"\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3837 |
\<Longrightarrow> (order a (pderiv p) = n) = (order a p = Suc n)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3838 |
by (auto dest: order_pderiv) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3839 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3840 |
definition |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3841 |
rsquarefree :: "'a::idom poly => bool" where |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3842 |
"rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3843 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3844 |
lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h :: 'a :: {semidom,semiring_char_0}:]" |
63649 | 3845 |
by (cases p) (auto simp: pderiv_eq_0_iff split: if_splits) |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3846 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3847 |
lemma rsquarefree_roots: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3848 |
fixes p :: "'a :: field_char_0 poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3849 |
shows "rsquarefree p = (\<forall>a. \<not>(poly p a = 0 \<and> poly (pderiv p) a = 0))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3850 |
apply (simp add: rsquarefree_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3851 |
apply (case_tac "p = 0", simp, simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3852 |
apply (case_tac "pderiv p = 0") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3853 |
apply simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3854 |
apply (drule pderiv_iszero, clarsimp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3855 |
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3856 |
apply (force simp add: order_root order_pderiv2) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3857 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3858 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3859 |
lemma poly_squarefree_decomp: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3860 |
assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3861 |
and "p = q * d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3862 |
and "pderiv p = e * d" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3863 |
and "d = r * p + s * pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3864 |
shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3865 |
proof - |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3866 |
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3867 |
with \<open>p = q * d\<close> have "q \<noteq> 0" by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3868 |
have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3869 |
using assms by (rule poly_squarefree_decomp_order2) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3870 |
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3871 |
by (simp add: rsquarefree_def order_root) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3872 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3873 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3874 |
lemma coeff_monom_mult: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3875 |
"coeff (monom c n * p) k = (if k < n then 0 else c * coeff p (k - n))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3876 |
proof - |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3877 |
have "coeff (monom c n * p) k = (\<Sum>i\<le>k. (if n = i then c else 0) * coeff p (k - i))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3878 |
by (simp add: coeff_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3879 |
also have "\<dots> = (\<Sum>i\<le>k. (if n = i then c * coeff p (k - i) else 0))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3880 |
by (intro setsum.cong) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3881 |
also have "\<dots> = (if k < n then 0 else c * coeff p (k - n))" by (simp add: setsum.delta') |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3882 |
finally show ?thesis . |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3883 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3884 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3885 |
lemma monom_1_dvd_iff': |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3886 |
"monom 1 n dvd p \<longleftrightarrow> (\<forall>k<n. coeff p k = 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3887 |
proof |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3888 |
assume "monom 1 n dvd p" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3889 |
then obtain r where r: "p = monom 1 n * r" by (elim dvdE) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3890 |
thus "\<forall>k<n. coeff p k = 0" by (simp add: coeff_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3891 |
next |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3892 |
assume zero: "(\<forall>k<n. coeff p k = 0)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3893 |
define r where "r = Abs_poly (\<lambda>k. coeff p (k + n))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3894 |
have "\<forall>\<^sub>\<infinity>k. coeff p (k + n) = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3895 |
by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg, |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3896 |
subst cofinite_eq_sequentially [symmetric]) transfer |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3897 |
hence coeff_r [simp]: "coeff r k = coeff p (k + n)" for k unfolding r_def |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3898 |
by (subst poly.Abs_poly_inverse) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3899 |
have "p = monom 1 n * r" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3900 |
by (intro poly_eqI, subst coeff_monom_mult) (insert zero, simp_all) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3901 |
thus "monom 1 n dvd p" by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3902 |
qed |
52380 | 3903 |
|
3904 |
no_notation cCons (infixr "##" 65) |
|
31663 | 3905 |
|
29478 | 3906 |
end |