author | haftmann |
Mon, 06 Feb 2017 20:56:34 +0100 | |
changeset 64990 | c6a7de505796 |
parent 64861 | 9e8de30fd859 |
child 65346 | 673a7b3379ec |
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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||
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section \<open>Polynomials as type over a ring structure\<close> |
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|
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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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||
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subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close> |
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|
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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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||
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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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24 |
||
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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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28 |
||
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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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32 |
||
33 |
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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36 |
||
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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" |
|
39 |
proof (cases "x = 0") |
|
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case False then show ?thesis by simp |
|
41 |
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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||
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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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||
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subsection \<open>Definition of type \<open>poly\<close>\<close> |
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|
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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 |
59 |
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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|
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||
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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" |
76 |
where |
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"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)" |
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||
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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 |
89 |
||
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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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||
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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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|
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instantiation poly :: (zero) zero |
|
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begin |
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||
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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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|
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instance .. |
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|
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end |
111 |
||
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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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|
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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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||
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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") |
124 |
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 |
131 |
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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||
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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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lemma eq_zero_or_degree_less: |
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assumes "degree p \<le> n" and "coeff p n = 0" |
|
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shows "p = 0 \<or> degree p < n" |
|
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proof (cases n) |
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case 0 |
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with \<open>degree p \<le> n\<close> and \<open>coeff p n = 0\<close> |
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have "coeff p (degree p) = 0" by simp |
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then have "p = 0" by simp |
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then show ?thesis .. |
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next |
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case (Suc m) |
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have "\<forall>i>n. coeff p i = 0" |
|
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using \<open>degree p \<le> n\<close> by (simp add: coeff_eq_0) |
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then have "\<forall>i\<ge>n. coeff p i = 0" |
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using \<open>coeff p n = 0\<close> by (simp add: le_less) |
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then have "\<forall>i>m. coeff p i = 0" |
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using \<open>n = Suc m\<close> by (simp add: less_eq_Suc_le) |
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then have "degree p \<le> m" |
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by (rule degree_le) |
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then have "degree p < n" |
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using \<open>n = Suc m\<close> by (simp add: less_Suc_eq_le) |
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then show ?thesis .. |
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qed |
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lemma coeff_0_degree_minus_1: "coeff rrr dr = 0 \<Longrightarrow> degree rrr \<le> dr \<Longrightarrow> degree rrr \<le> dr - 1" |
|
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using eq_zero_or_degree_less by fastforce |
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||
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subsection \<open>List-style constructor for polynomials\<close> |
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|
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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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|
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lemmas coeff_pCons = pCons.rep_eq |
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|
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lemma coeff_pCons_0 [simp]: |
183 |
"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: |
191 |
"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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|
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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) |
209 |
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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|
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lemma pCons_0_0 [simp]: |
216 |
"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" |
223 |
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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||
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lemma pCons_eq_0_iff [simp]: |
233 |
"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 |
235 |
||
236 |
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 |
244 |
||
245 |
lemma pCons_induct [case_names 0 pCons, induct type: poly]: |
|
246 |
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" |
249 |
proof (induct p rule: measure_induct_rule [where f=degree]) |
|
250 |
case (less p) |
|
251 |
obtain a q where "p = pCons a q" by (rule pCons_cases) |
|
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have "P q" |
|
253 |
proof (cases "q = 0") |
|
254 |
case True |
|
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then show "P q" by (simp add: zero) |
|
256 |
next |
|
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case False |
|
258 |
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" |
263 |
by (rule less.hyps) |
|
264 |
qed |
|
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have "P (pCons a q)" |
266 |
proof (cases "a \<noteq> 0 \<or> q \<noteq> 0") |
|
267 |
case True |
|
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with \<open>P q\<close> show ?thesis by (auto intro: pCons) |
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next |
270 |
case False |
|
271 |
with zero show ?thesis by simp |
|
272 |
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 |
276 |
||
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lemma degree_eq_zeroE: |
278 |
fixes p :: "'a::zero poly" |
|
279 |
assumes "degree p = 0" |
|
280 |
obtains a where "p = pCons a 0" |
|
281 |
proof - |
|
282 |
obtain a q where p: "p = pCons a q" by (cases p) |
|
283 |
with assms have "q = 0" by (cases "q = 0") simp_all |
|
284 |
with p have "p = pCons a 0" by simp |
|
285 |
with that show thesis . |
|
286 |
qed |
|
287 |
||
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|
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subsection \<open>Quickcheck generator for polynomials\<close> |
290 |
||
291 |
quickcheck_generator poly constructors: "0 :: _ poly", pCons |
|
292 |
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293 |
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subsection \<open>List-style syntax for polynomials\<close> |
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|
296 |
syntax |
|
297 |
"_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]") |
|
298 |
||
299 |
translations |
|
300 |
"[:x, xs:]" == "CONST pCons x [:xs:]" |
|
301 |
"[:x:]" == "CONST pCons x 0" |
|
302 |
"[:x:]" <= "CONST pCons x (_constrain 0 t)" |
|
303 |
||
304 |
||
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subsection \<open>Representation of polynomials by lists of coefficients\<close> |
52380 | 306 |
|
307 |
primrec Poly :: "'a::zero list \<Rightarrow> 'a poly" |
|
308 |
where |
|
54855 | 309 |
[code_post]: "Poly [] = 0" |
310 |
| [code_post]: "Poly (a # as) = pCons a (Poly as)" |
|
52380 | 311 |
|
312 |
lemma Poly_replicate_0 [simp]: |
|
313 |
"Poly (replicate n 0) = 0" |
|
314 |
by (induct n) simp_all |
|
315 |
||
316 |
lemma Poly_eq_0: |
|
317 |
"Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)" |
|
318 |
by (induct as) (auto simp add: Cons_replicate_eq) |
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319 |
|
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320 |
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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|
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324 |
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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331 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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changeset
|
332 |
lemma Poly_on_rev_starting_with_0 [simp]: |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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changeset
|
333 |
assumes "hd as = 0" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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parents:
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changeset
|
334 |
shows "Poly (rev (tl as)) = Poly (rev as)" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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changeset
|
335 |
using assms by (cases as) simp_all |
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parents:
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diff
changeset
|
336 |
|
62065 | 337 |
lemma degree_Poly: "degree (Poly xs) \<le> length xs" |
338 |
by (induction xs) simp_all |
|
63027
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several updates on polynomial long division and pseudo division
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changeset
|
339 |
|
8de0ebee3f1c
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changeset
|
340 |
lemma coeff_Poly_eq [simp]: |
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|
341 |
"coeff (Poly xs) = nth_default 0 xs" |
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|
342 |
by (induct xs) (simp_all add: fun_eq_iff coeff_pCons split: nat.splits) |
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|
343 |
|
52380 | 344 |
definition coeffs :: "'a poly \<Rightarrow> 'a::zero list" |
345 |
where |
|
346 |
"coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])" |
|
347 |
||
348 |
lemma coeffs_eq_Nil [simp]: |
|
349 |
"coeffs p = [] \<longleftrightarrow> p = 0" |
|
350 |
by (simp add: coeffs_def) |
|
351 |
||
352 |
lemma not_0_coeffs_not_Nil: |
|
353 |
"p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []" |
|
354 |
by simp |
|
355 |
||
356 |
lemma coeffs_0_eq_Nil [simp]: |
|
357 |
"coeffs 0 = []" |
|
358 |
by simp |
|
29454
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|
359 |
|
52380 | 360 |
lemma coeffs_pCons_eq_cCons [simp]: |
361 |
"coeffs (pCons a p) = a ## coeffs p" |
|
362 |
proof - |
|
363 |
{ fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a" |
|
364 |
assume "\<forall>m\<in>set ms. m > 0" |
|
55415 | 365 |
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
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57862
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|
366 |
by (induct ms) (auto split: nat.split) |
5fbe474b5da8
explicit theory with additional, less commonly used list operations
haftmann
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diff
changeset
|
367 |
} |
52380 | 368 |
note * = this |
369 |
show ?thesis |
|
60570 | 370 |
by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc) |
52380 | 371 |
qed |
372 |
||
62065 | 373 |
lemma length_coeffs: "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = degree p + 1" |
374 |
by (simp add: coeffs_def) |
|
64860 | 375 |
|
62065 | 376 |
lemma coeffs_nth: |
377 |
assumes "p \<noteq> 0" "n \<le> degree p" |
|
378 |
shows "coeffs p ! n = coeff p n" |
|
379 |
using assms unfolding coeffs_def by (auto simp del: upt_Suc) |
|
380 |
||
64860 | 381 |
lemma coeff_in_coeffs: |
382 |
"p \<noteq> 0 \<Longrightarrow> n \<le> degree p \<Longrightarrow> coeff p n \<in> set (coeffs p)" |
|
383 |
using coeffs_nth [of p n, symmetric] |
|
384 |
by (simp add: length_coeffs) |
|
385 |
||
52380 | 386 |
lemma not_0_cCons_eq [simp]: |
387 |
"p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p" |
|
388 |
by (simp add: cCons_def) |
|
389 |
||
390 |
lemma Poly_coeffs [simp, code abstype]: |
|
391 |
"Poly (coeffs p) = p" |
|
54856 | 392 |
by (induct p) auto |
52380 | 393 |
|
394 |
lemma coeffs_Poly [simp]: |
|
395 |
"coeffs (Poly as) = strip_while (HOL.eq 0) as" |
|
396 |
proof (induct as) |
|
397 |
case Nil then show ?case by simp |
|
398 |
next |
|
399 |
case (Cons a as) |
|
400 |
have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)" |
|
401 |
using replicate_length_same [of as 0] by (auto dest: sym [of _ as]) |
|
402 |
with Cons show ?case by auto |
|
403 |
qed |
|
404 |
||
405 |
lemma last_coeffs_not_0: |
|
406 |
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0" |
|
407 |
by (induct p) (auto simp add: cCons_def) |
|
408 |
||
409 |
lemma strip_while_coeffs [simp]: |
|
410 |
"strip_while (HOL.eq 0) (coeffs p) = coeffs p" |
|
411 |
by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last) |
|
412 |
||
413 |
lemma coeffs_eq_iff: |
|
414 |
"p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q") |
|
415 |
proof |
|
416 |
assume ?P then show ?Q by simp |
|
417 |
next |
|
418 |
assume ?Q |
|
419 |
then have "Poly (coeffs p) = Poly (coeffs q)" by simp |
|
420 |
then show ?P by simp |
|
421 |
qed |
|
422 |
||
423 |
lemma nth_default_coeffs_eq: |
|
424 |
"nth_default 0 (coeffs p) = coeff p" |
|
425 |
by (simp add: fun_eq_iff coeff_Poly_eq [symmetric]) |
|
426 |
||
427 |
lemma [code]: |
|
428 |
"coeff p = nth_default 0 (coeffs p)" |
|
429 |
by (simp add: nth_default_coeffs_eq) |
|
430 |
||
431 |
lemma coeffs_eqI: |
|
432 |
assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n" |
|
433 |
assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" |
|
434 |
shows "coeffs p = xs" |
|
435 |
proof - |
|
63027
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Rene Thiemann <rene.thiemann@uibk.ac.at>
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62422
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changeset
|
436 |
from coeff have "p = Poly xs" by (simp add: poly_eq_iff) |
52380 | 437 |
with zero show ?thesis by simp (cases xs, simp_all) |
438 |
qed |
|
439 |
||
440 |
lemma degree_eq_length_coeffs [code]: |
|
441 |
"degree p = length (coeffs p) - 1" |
|
442 |
by (simp add: coeffs_def) |
|
443 |
||
444 |
lemma length_coeffs_degree: |
|
445 |
"p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)" |
|
446 |
by (induct p) (auto simp add: cCons_def) |
|
447 |
||
448 |
lemma [code abstract]: |
|
449 |
"coeffs 0 = []" |
|
450 |
by (fact coeffs_0_eq_Nil) |
|
451 |
||
452 |
lemma [code abstract]: |
|
453 |
"coeffs (pCons a p) = a ## coeffs p" |
|
454 |
by (fact coeffs_pCons_eq_cCons) |
|
455 |
||
456 |
instantiation poly :: ("{zero, equal}") equal |
|
457 |
begin |
|
458 |
||
459 |
definition |
|
460 |
[code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)" |
|
461 |
||
60679 | 462 |
instance |
463 |
by standard (simp add: equal equal_poly_def coeffs_eq_iff) |
|
52380 | 464 |
|
465 |
end |
|
466 |
||
60679 | 467 |
lemma [code nbe]: "HOL.equal (p :: _ poly) p \<longleftrightarrow> True" |
52380 | 468 |
by (fact equal_refl) |
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
469 |
|
52380 | 470 |
definition is_zero :: "'a::zero poly \<Rightarrow> bool" |
471 |
where |
|
472 |
[code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)" |
|
473 |
||
474 |
lemma is_zero_null [code_abbrev]: |
|
475 |
"is_zero p \<longleftrightarrow> p = 0" |
|
476 |
by (simp add: is_zero_def null_def) |
|
477 |
||
63027
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several updates on polynomial long division and pseudo division
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62422
diff
changeset
|
478 |
subsubsection \<open>Reconstructing the polynomial from the list\<close> |
63145 | 479 |
\<comment> \<open>contributed by Sebastiaan J.C. Joosten and René Thiemann\<close> |
63027
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several updates on polynomial long division and pseudo division
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parents:
62422
diff
changeset
|
480 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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parents:
62422
diff
changeset
|
481 |
definition poly_of_list :: "'a::comm_monoid_add list \<Rightarrow> 'a poly" |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
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parents:
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changeset
|
482 |
where |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
483 |
[simp]: "poly_of_list = Poly" |
8de0ebee3f1c
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parents:
62422
diff
changeset
|
484 |
|
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
485 |
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
|
486 |
"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
|
487 |
by simp |
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
488 |
|
52380 | 489 |
|
60500 | 490 |
subsection \<open>Fold combinator for polynomials\<close> |
52380 | 491 |
|
492 |
definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b" |
|
493 |
where |
|
494 |
"fold_coeffs f p = foldr f (coeffs p)" |
|
495 |
||
496 |
lemma fold_coeffs_0_eq [simp]: |
|
497 |
"fold_coeffs f 0 = id" |
|
498 |
by (simp add: fold_coeffs_def) |
|
499 |
||
500 |
lemma fold_coeffs_pCons_eq [simp]: |
|
501 |
"f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
502 |
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
|
503 |
|
52380 | 504 |
lemma fold_coeffs_pCons_0_0_eq [simp]: |
505 |
"fold_coeffs f (pCons 0 0) = id" |
|
506 |
by (simp add: fold_coeffs_def) |
|
507 |
||
508 |
lemma fold_coeffs_pCons_coeff_not_0_eq [simp]: |
|
509 |
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
510 |
by (simp add: fold_coeffs_def) |
|
511 |
||
512 |
lemma fold_coeffs_pCons_not_0_0_eq [simp]: |
|
513 |
"p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p" |
|
514 |
by (simp add: fold_coeffs_def) |
|
515 |
||
64795 | 516 |
|
60500 | 517 |
subsection \<open>Canonical morphism on polynomials -- evaluation\<close> |
52380 | 518 |
|
519 |
definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" |
|
520 |
where |
|
61585 | 521 |
"poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" \<comment> \<open>The Horner Schema\<close> |
52380 | 522 |
|
523 |
lemma poly_0 [simp]: |
|
524 |
"poly 0 x = 0" |
|
525 |
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
|
526 |
|
52380 | 527 |
lemma poly_pCons [simp]: |
528 |
"poly (pCons a p) x = a + x * poly p x" |
|
529 |
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
|
530 |
|
62065 | 531 |
lemma poly_altdef: |
532 |
"poly p (x :: 'a :: {comm_semiring_0, semiring_1}) = (\<Sum>i\<le>degree p. coeff p i * x ^ i)" |
|
533 |
proof (induction p rule: pCons_induct) |
|
534 |
case (pCons a p) |
|
535 |
show ?case |
|
536 |
proof (cases "p = 0") |
|
537 |
case False |
|
538 |
let ?p' = "pCons a p" |
|
539 |
note poly_pCons[of a p x] |
|
540 |
also note pCons.IH |
|
541 |
also have "a + x * (\<Sum>i\<le>degree p. coeff p i * x ^ i) = |
|
542 |
coeff ?p' 0 * x^0 + (\<Sum>i\<le>degree p. coeff ?p' (Suc i) * x^Suc i)" |
|
64267 | 543 |
by (simp add: field_simps sum_distrib_left coeff_pCons) |
544 |
also note sum_atMost_Suc_shift[symmetric] |
|
62072 | 545 |
also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric] |
62065 | 546 |
finally show ?thesis . |
547 |
qed simp |
|
548 |
qed simp |
|
549 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
550 |
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
|
551 |
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
|
552 |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
553 |
|
60500 | 554 |
subsection \<open>Monomials\<close> |
29451 | 555 |
|
52380 | 556 |
lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" |
557 |
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
|
558 |
by (simp add: MOST_iff_cofinite) |
52380 | 559 |
|
560 |
lemma coeff_monom [simp]: |
|
561 |
"coeff (monom a m) n = (if m = n then a else 0)" |
|
562 |
by transfer rule |
|
29451 | 563 |
|
52380 | 564 |
lemma monom_0: |
565 |
"monom a 0 = pCons a 0" |
|
566 |
by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) |
|
29451 | 567 |
|
52380 | 568 |
lemma monom_Suc: |
569 |
"monom a (Suc n) = pCons 0 (monom a n)" |
|
570 |
by (rule poly_eqI) (simp add: coeff_pCons split: nat.split) |
|
29451 | 571 |
|
572 |
lemma monom_eq_0 [simp]: "monom 0 n = 0" |
|
52380 | 573 |
by (rule poly_eqI) simp |
29451 | 574 |
|
575 |
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0" |
|
52380 | 576 |
by (simp add: poly_eq_iff) |
29451 | 577 |
|
578 |
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b" |
|
52380 | 579 |
by (simp add: poly_eq_iff) |
29451 | 580 |
|
581 |
lemma degree_monom_le: "degree (monom a n) \<le> n" |
|
582 |
by (rule degree_le, simp) |
|
583 |
||
584 |
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n" |
|
585 |
apply (rule order_antisym [OF degree_monom_le]) |
|
586 |
apply (rule le_degree, simp) |
|
587 |
done |
|
588 |
||
52380 | 589 |
lemma coeffs_monom [code abstract]: |
590 |
"coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])" |
|
591 |
by (induct n) (simp_all add: monom_0 monom_Suc) |
|
592 |
||
593 |
lemma fold_coeffs_monom [simp]: |
|
594 |
"a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a" |
|
595 |
by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff) |
|
596 |
||
597 |
lemma poly_monom: |
|
598 |
fixes a x :: "'a::{comm_semiring_1}" |
|
599 |
shows "poly (monom a n) x = a * x ^ n" |
|
600 |
by (cases "a = 0", simp_all) |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
601 |
(induct n, simp_all add: mult.left_commute poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
602 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
603 |
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
|
604 |
by (auto simp: poly_eq_iff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
605 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
606 |
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
|
607 |
using monom_eq_iff'[of c n d 0] by (simp add: monom_0) |
64795 | 608 |
|
609 |
||
610 |
subsection \<open>Leading coefficient\<close> |
|
611 |
||
612 |
abbreviation lead_coeff:: "'a::zero poly \<Rightarrow> 'a" |
|
613 |
where "lead_coeff p \<equiv> coeff p (degree p)" |
|
614 |
||
615 |
lemma lead_coeff_pCons[simp]: |
|
616 |
"p \<noteq> 0 \<Longrightarrow> lead_coeff (pCons a p) = lead_coeff p" |
|
617 |
"p = 0 \<Longrightarrow> lead_coeff (pCons a p) = a" |
|
618 |
by auto |
|
619 |
||
620 |
lemma lead_coeff_monom [simp]: "lead_coeff (monom c n) = c" |
|
621 |
by (cases "c = 0") (simp_all add: degree_monom_eq) |
|
622 |
||
623 |
||
60500 | 624 |
subsection \<open>Addition and subtraction\<close> |
29451 | 625 |
|
626 |
instantiation poly :: (comm_monoid_add) comm_monoid_add |
|
627 |
begin |
|
628 |
||
52380 | 629 |
lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
630 |
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
|
631 |
proof - |
60679 | 632 |
fix q p :: "'a poly" |
633 |
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
|
634 |
using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp |
52380 | 635 |
qed |
29451 | 636 |
|
60679 | 637 |
lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n" |
52380 | 638 |
by (simp add: plus_poly.rep_eq) |
29451 | 639 |
|
60679 | 640 |
instance |
641 |
proof |
|
29451 | 642 |
fix p q r :: "'a poly" |
643 |
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
|
644 |
by (simp add: poly_eq_iff add.assoc) |
29451 | 645 |
show "p + q = q + p" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57482
diff
changeset
|
646 |
by (simp add: poly_eq_iff add.commute) |
29451 | 647 |
show "0 + p = p" |
52380 | 648 |
by (simp add: poly_eq_iff) |
29451 | 649 |
qed |
650 |
||
651 |
end |
|
652 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
653 |
instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
654 |
begin |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
655 |
|
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
656 |
lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
657 |
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
|
658 |
proof - |
60679 | 659 |
fix q p :: "'a poly" |
660 |
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
|
661 |
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
|
662 |
qed |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
663 |
|
60679 | 664 |
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
|
665 |
by (simp add: minus_poly.rep_eq) |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
666 |
|
60679 | 667 |
instance |
668 |
proof |
|
29540 | 669 |
fix p q r :: "'a poly" |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
670 |
show "p + q - p = q" |
52380 | 671 |
by (simp add: poly_eq_iff) |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
672 |
show "p - q - r = p - (q + r)" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
673 |
by (simp add: poly_eq_iff diff_diff_eq) |
29540 | 674 |
qed |
675 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
676 |
end |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
677 |
|
29451 | 678 |
instantiation poly :: (ab_group_add) ab_group_add |
679 |
begin |
|
680 |
||
52380 | 681 |
lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly" |
682 |
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
|
683 |
proof - |
60679 | 684 |
fix p :: "'a poly" |
685 |
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
|
686 |
using MOST_coeff_eq_0 by simp |
52380 | 687 |
qed |
29451 | 688 |
|
689 |
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" |
|
52380 | 690 |
by (simp add: uminus_poly.rep_eq) |
29451 | 691 |
|
60679 | 692 |
instance |
693 |
proof |
|
29451 | 694 |
fix p q :: "'a poly" |
695 |
show "- p + p = 0" |
|
52380 | 696 |
by (simp add: poly_eq_iff) |
29451 | 697 |
show "p - q = p + - q" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
52380
diff
changeset
|
698 |
by (simp add: poly_eq_iff) |
29451 | 699 |
qed |
700 |
||
701 |
end |
|
702 |
||
703 |
lemma add_pCons [simp]: |
|
704 |
"pCons a p + pCons b q = pCons (a + b) (p + q)" |
|
52380 | 705 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 706 |
|
707 |
lemma minus_pCons [simp]: |
|
708 |
"- pCons a p = pCons (- a) (- p)" |
|
52380 | 709 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 710 |
|
711 |
lemma diff_pCons [simp]: |
|
712 |
"pCons a p - pCons b q = pCons (a - b) (p - q)" |
|
52380 | 713 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 714 |
|
29539 | 715 |
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)" |
29451 | 716 |
by (rule degree_le, auto simp add: coeff_eq_0) |
717 |
||
29539 | 718 |
lemma degree_add_le: |
719 |
"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n" |
|
720 |
by (auto intro: order_trans degree_add_le_max) |
|
721 |
||
29453 | 722 |
lemma degree_add_less: |
723 |
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n" |
|
29539 | 724 |
by (auto intro: le_less_trans degree_add_le_max) |
29453 | 725 |
|
29451 | 726 |
lemma degree_add_eq_right: |
727 |
"degree p < degree q \<Longrightarrow> degree (p + q) = degree q" |
|
728 |
apply (cases "q = 0", simp) |
|
729 |
apply (rule order_antisym) |
|
29539 | 730 |
apply (simp add: degree_add_le) |
29451 | 731 |
apply (rule le_degree) |
732 |
apply (simp add: coeff_eq_0) |
|
733 |
done |
|
734 |
||
735 |
lemma degree_add_eq_left: |
|
736 |
"degree q < degree p \<Longrightarrow> degree (p + q) = degree p" |
|
737 |
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
|
738 |
by (simp add: add.commute) |
29451 | 739 |
|
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
740 |
lemma degree_minus [simp]: |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
741 |
"degree (- p) = degree p" |
29451 | 742 |
unfolding degree_def by simp |
743 |
||
64795 | 744 |
lemma lead_coeff_add_le: |
745 |
assumes "degree p < degree q" |
|
746 |
shows "lead_coeff (p + q) = lead_coeff q" |
|
747 |
using assms |
|
748 |
by (metis coeff_add coeff_eq_0 monoid_add_class.add.left_neutral degree_add_eq_right) |
|
749 |
||
750 |
lemma lead_coeff_minus: |
|
751 |
"lead_coeff (- p) = - lead_coeff p" |
|
752 |
by (metis coeff_minus degree_minus) |
|
753 |
||
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
754 |
lemma degree_diff_le_max: |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
755 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
756 |
shows "degree (p - q) \<le> max (degree p) (degree q)" |
29451 | 757 |
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
|
758 |
by simp |
29451 | 759 |
|
29539 | 760 |
lemma degree_diff_le: |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
761 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
762 |
assumes "degree p \<le> n" and "degree q \<le> n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
763 |
shows "degree (p - q) \<le> n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
764 |
using assms degree_add_le [of p n "- q"] by simp |
29539 | 765 |
|
29453 | 766 |
lemma degree_diff_less: |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
767 |
fixes p q :: "'a :: ab_group_add poly" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
768 |
assumes "degree p < n" and "degree q < n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
769 |
shows "degree (p - q) < n" |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59557
diff
changeset
|
770 |
using assms degree_add_less [of p n "- q"] by simp |
29453 | 771 |
|
29451 | 772 |
lemma add_monom: "monom a n + monom b n = monom (a + b) n" |
52380 | 773 |
by (rule poly_eqI) simp |
29451 | 774 |
|
775 |
lemma diff_monom: "monom a n - monom b n = monom (a - b) n" |
|
52380 | 776 |
by (rule poly_eqI) simp |
29451 | 777 |
|
778 |
lemma minus_monom: "- monom a n = monom (-a) n" |
|
52380 | 779 |
by (rule poly_eqI) simp |
29451 | 780 |
|
64267 | 781 |
lemma coeff_sum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)" |
29451 | 782 |
by (cases "finite A", induct set: finite, simp_all) |
783 |
||
64267 | 784 |
lemma monom_sum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)" |
785 |
by (rule poly_eqI) (simp add: coeff_sum) |
|
52380 | 786 |
|
787 |
fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
|
788 |
where |
|
789 |
"plus_coeffs xs [] = xs" |
|
790 |
| "plus_coeffs [] ys = ys" |
|
791 |
| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys" |
|
792 |
||
793 |
lemma coeffs_plus_eq_plus_coeffs [code abstract]: |
|
794 |
"coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)" |
|
795 |
proof - |
|
796 |
{ fix xs ys :: "'a list" and n |
|
797 |
have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n" |
|
798 |
proof (induct xs ys arbitrary: n rule: plus_coeffs.induct) |
|
60679 | 799 |
case (3 x xs y ys n) |
800 |
then show ?case by (cases n) (auto simp add: cCons_def) |
|
52380 | 801 |
qed simp_all } |
802 |
note * = this |
|
803 |
{ fix xs ys :: "'a list" |
|
804 |
assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0" |
|
805 |
moreover assume "plus_coeffs xs ys \<noteq> []" |
|
806 |
ultimately have "last (plus_coeffs xs ys) \<noteq> 0" |
|
807 |
proof (induct xs ys rule: plus_coeffs.induct) |
|
808 |
case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis |
|
809 |
qed simp_all } |
|
810 |
note ** = this |
|
811 |
show ?thesis |
|
812 |
apply (rule coeffs_eqI) |
|
813 |
apply (simp add: * nth_default_coeffs_eq) |
|
814 |
apply (rule **) |
|
815 |
apply (auto dest: last_coeffs_not_0) |
|
816 |
done |
|
817 |
qed |
|
818 |
||
819 |
lemma coeffs_uminus [code abstract]: |
|
820 |
"coeffs (- p) = map (\<lambda>a. - a) (coeffs p)" |
|
821 |
by (rule coeffs_eqI) |
|
822 |
(simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) |
|
823 |
||
824 |
lemma [code]: |
|
825 |
fixes p q :: "'a::ab_group_add poly" |
|
826 |
shows "p - q = p + - q" |
|
59557 | 827 |
by (fact diff_conv_add_uminus) |
52380 | 828 |
|
829 |
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" |
|
830 |
apply (induct p arbitrary: q, simp) |
|
831 |
apply (case_tac q, simp, simp add: algebra_simps) |
|
832 |
done |
|
833 |
||
834 |
lemma poly_minus [simp]: |
|
835 |
fixes x :: "'a::comm_ring" |
|
836 |
shows "poly (- p) x = - poly p x" |
|
837 |
by (induct p) simp_all |
|
838 |
||
839 |
lemma poly_diff [simp]: |
|
840 |
fixes x :: "'a::comm_ring" |
|
841 |
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
|
842 |
using poly_add [of p "- q" x] by simp |
52380 | 843 |
|
64267 | 844 |
lemma poly_sum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" |
52380 | 845 |
by (induct A rule: infinite_finite_induct) simp_all |
29451 | 846 |
|
64267 | 847 |
lemma degree_sum_le: "finite S \<Longrightarrow> (\<And> p . p \<in> S \<Longrightarrow> degree (f p) \<le> n) |
848 |
\<Longrightarrow> degree (sum f S) \<le> n" |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
849 |
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
|
850 |
case (insert p S) |
64267 | 851 |
hence "degree (sum f S) \<le> n" "degree (f p) \<le> n" by auto |
852 |
thus ?case unfolding sum.insert[OF insert(1-2)] by (metis degree_add_le) |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
853 |
qed simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
854 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
855 |
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
|
856 |
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
|
857 |
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
|
858 |
proof - |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
859 |
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
|
860 |
by auto |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
861 |
show ?thesis |
64267 | 862 |
using n by (simp add: poly_eq_iff coeff_sum coeff_eq_0 sum.If_cases eq |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
863 |
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
|
864 |
qed |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
865 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
866 |
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
|
867 |
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
|
868 |
|
62065 | 869 |
lemma Poly_snoc: "Poly (xs @ [x]) = Poly xs + monom x (length xs)" |
870 |
by (induction xs) (simp_all add: monom_0 monom_Suc) |
|
871 |
||
29451 | 872 |
|
60500 | 873 |
subsection \<open>Multiplication by a constant, polynomial multiplication and the unit polynomial\<close> |
29451 | 874 |
|
52380 | 875 |
lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
876 |
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
|
877 |
proof - |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
59983
diff
changeset
|
878 |
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
|
879 |
using MOST_coeff_eq_0[of p] by eventually_elim simp |
52380 | 880 |
qed |
29451 | 881 |
|
52380 | 882 |
lemma coeff_smult [simp]: |
883 |
"coeff (smult a p) n = a * coeff p n" |
|
884 |
by (simp add: smult.rep_eq) |
|
29451 | 885 |
|
886 |
lemma degree_smult_le: "degree (smult a p) \<le> degree p" |
|
887 |
by (rule degree_le, simp add: coeff_eq_0) |
|
888 |
||
29472 | 889 |
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
|
890 |
by (rule poly_eqI, simp add: mult.assoc) |
29451 | 891 |
|
892 |
lemma smult_0_right [simp]: "smult a 0 = 0" |
|
52380 | 893 |
by (rule poly_eqI, simp) |
29451 | 894 |
|
895 |
lemma smult_0_left [simp]: "smult 0 p = 0" |
|
52380 | 896 |
by (rule poly_eqI, simp) |
29451 | 897 |
|
898 |
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" |
|
52380 | 899 |
by (rule poly_eqI, simp) |
29451 | 900 |
|
901 |
lemma smult_add_right: |
|
902 |
"smult a (p + q) = smult a p + smult a q" |
|
52380 | 903 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 904 |
|
905 |
lemma smult_add_left: |
|
906 |
"smult (a + b) p = smult a p + smult b p" |
|
52380 | 907 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 908 |
|
29457 | 909 |
lemma smult_minus_right [simp]: |
29451 | 910 |
"smult (a::'a::comm_ring) (- p) = - smult a p" |
52380 | 911 |
by (rule poly_eqI, simp) |
29451 | 912 |
|
29457 | 913 |
lemma smult_minus_left [simp]: |
29451 | 914 |
"smult (- a::'a::comm_ring) p = - smult a p" |
52380 | 915 |
by (rule poly_eqI, simp) |
29451 | 916 |
|
917 |
lemma smult_diff_right: |
|
918 |
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" |
|
52380 | 919 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 920 |
|
921 |
lemma smult_diff_left: |
|
922 |
"smult (a - b::'a::comm_ring) p = smult a p - smult b p" |
|
52380 | 923 |
by (rule poly_eqI, simp add: algebra_simps) |
29451 | 924 |
|
29472 | 925 |
lemmas smult_distribs = |
926 |
smult_add_left smult_add_right |
|
927 |
smult_diff_left smult_diff_right |
|
928 |
||
29451 | 929 |
lemma smult_pCons [simp]: |
930 |
"smult a (pCons b p) = pCons (a * b) (smult a p)" |
|
52380 | 931 |
by (rule poly_eqI, simp add: coeff_pCons split: nat.split) |
29451 | 932 |
|
933 |
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" |
|
934 |
by (induct n, simp add: monom_0, simp add: monom_Suc) |
|
935 |
||
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
936 |
lemma smult_Poly: "smult c (Poly xs) = Poly (map (op * c) xs)" |
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
937 |
by (auto simp add: poly_eq_iff nth_default_def) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
938 |
|
29659 | 939 |
lemma degree_smult_eq [simp]: |
63498 | 940 |
fixes a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" |
29659 | 941 |
shows "degree (smult a p) = (if a = 0 then 0 else degree p)" |
942 |
by (cases "a = 0", simp, simp add: degree_def) |
|
943 |
||
944 |
lemma smult_eq_0_iff [simp]: |
|
63498 | 945 |
fixes a :: "'a::{comm_semiring_0,semiring_no_zero_divisors}" |
29659 | 946 |
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0" |
52380 | 947 |
by (simp add: poly_eq_iff) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
948 |
|
52380 | 949 |
lemma coeffs_smult [code abstract]: |
63498 | 950 |
fixes p :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
52380 | 951 |
shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))" |
952 |
by (rule coeffs_eqI) |
|
953 |
(auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq) |
|
64795 | 954 |
|
955 |
lemma smult_eq_iff: |
|
956 |
assumes "(b :: 'a :: field) \<noteq> 0" |
|
957 |
shows "smult a p = smult b q \<longleftrightarrow> smult (a / b) p = q" |
|
958 |
proof |
|
959 |
assume "smult a p = smult b q" |
|
960 |
also from assms have "smult (inverse b) \<dots> = q" by simp |
|
961 |
finally show "smult (a / b) p = q" by (simp add: field_simps) |
|
962 |
qed (insert assms, auto) |
|
963 |
||
29451 | 964 |
instantiation poly :: (comm_semiring_0) comm_semiring_0 |
965 |
begin |
|
966 |
||
967 |
definition |
|
52380 | 968 |
"p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0" |
29474 | 969 |
|
970 |
lemma mult_poly_0_left: "(0::'a poly) * q = 0" |
|
52380 | 971 |
by (simp add: times_poly_def) |
29474 | 972 |
|
973 |
lemma mult_pCons_left [simp]: |
|
974 |
"pCons a p * q = smult a q + pCons 0 (p * q)" |
|
52380 | 975 |
by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def) |
29474 | 976 |
|
977 |
lemma mult_poly_0_right: "p * (0::'a poly) = 0" |
|
52380 | 978 |
by (induct p) (simp add: mult_poly_0_left, simp) |
29451 | 979 |
|
29474 | 980 |
lemma mult_pCons_right [simp]: |
981 |
"p * pCons a q = smult a p + pCons 0 (p * q)" |
|
52380 | 982 |
by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps) |
29474 | 983 |
|
984 |
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right |
|
985 |
||
52380 | 986 |
lemma mult_smult_left [simp]: |
987 |
"smult a p * q = smult a (p * q)" |
|
988 |
by (induct p) (simp add: mult_poly_0, simp add: smult_add_right) |
|
29474 | 989 |
|
52380 | 990 |
lemma mult_smult_right [simp]: |
991 |
"p * smult a q = smult a (p * q)" |
|
992 |
by (induct q) (simp add: mult_poly_0, simp add: smult_add_right) |
|
29474 | 993 |
|
994 |
lemma mult_poly_add_left: |
|
995 |
fixes p q r :: "'a poly" |
|
996 |
shows "(p + q) * r = p * r + q * r" |
|
52380 | 997 |
by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps) |
29451 | 998 |
|
60679 | 999 |
instance |
1000 |
proof |
|
29451 | 1001 |
fix p q r :: "'a poly" |
1002 |
show 0: "0 * p = 0" |
|
29474 | 1003 |
by (rule mult_poly_0_left) |
29451 | 1004 |
show "p * 0 = 0" |
29474 | 1005 |
by (rule mult_poly_0_right) |
29451 | 1006 |
show "(p + q) * r = p * r + q * r" |
29474 | 1007 |
by (rule mult_poly_add_left) |
29451 | 1008 |
show "(p * q) * r = p * (q * r)" |
29474 | 1009 |
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) |
29451 | 1010 |
show "p * q = q * p" |
29474 | 1011 |
by (induct p, simp add: mult_poly_0, simp) |
29451 | 1012 |
qed |
1013 |
||
1014 |
end |
|
1015 |
||
63498 | 1016 |
lemma coeff_mult_degree_sum: |
1017 |
"coeff (p * q) (degree p + degree q) = |
|
1018 |
coeff p (degree p) * coeff q (degree q)" |
|
1019 |
by (induct p, simp, simp add: coeff_eq_0) |
|
1020 |
||
1021 |
instance poly :: ("{comm_semiring_0,semiring_no_zero_divisors}") semiring_no_zero_divisors |
|
1022 |
proof |
|
1023 |
fix p q :: "'a poly" |
|
1024 |
assume "p \<noteq> 0" and "q \<noteq> 0" |
|
1025 |
have "coeff (p * q) (degree p + degree q) = |
|
1026 |
coeff p (degree p) * coeff q (degree q)" |
|
1027 |
by (rule coeff_mult_degree_sum) |
|
1028 |
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" |
|
1029 |
using \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> by simp |
|
1030 |
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. |
|
1031 |
thus "p * q \<noteq> 0" by (simp add: poly_eq_iff) |
|
1032 |
qed |
|
1033 |
||
29540 | 1034 |
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
1035 |
||
29474 | 1036 |
lemma coeff_mult: |
1037 |
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))" |
|
1038 |
proof (induct p arbitrary: n) |
|
1039 |
case 0 show ?case by simp |
|
1040 |
next |
|
1041 |
case (pCons a p n) thus ?case |
|
64267 | 1042 |
by (cases n, simp, simp add: sum_atMost_Suc_shift |
1043 |
del: sum_atMost_Suc) |
|
29474 | 1044 |
qed |
29451 | 1045 |
|
29474 | 1046 |
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" |
1047 |
apply (rule degree_le) |
|
1048 |
apply (induct p) |
|
1049 |
apply simp |
|
1050 |
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) |
|
29451 | 1051 |
done |
1052 |
||
1053 |
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" |
|
60679 | 1054 |
by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc) |
29451 | 1055 |
|
1056 |
instantiation poly :: (comm_semiring_1) comm_semiring_1 |
|
1057 |
begin |
|
1058 |
||
60679 | 1059 |
definition one_poly_def: "1 = pCons 1 0" |
29451 | 1060 |
|
60679 | 1061 |
instance |
1062 |
proof |
|
1063 |
show "1 * p = p" for p :: "'a poly" |
|
52380 | 1064 |
unfolding one_poly_def by simp |
29451 | 1065 |
show "0 \<noteq> (1::'a poly)" |
1066 |
unfolding one_poly_def by simp |
|
1067 |
qed |
|
1068 |
||
1069 |
end |
|
1070 |
||
63498 | 1071 |
instance poly :: ("{comm_semiring_1,semiring_1_no_zero_divisors}") semiring_1_no_zero_divisors .. |
1072 |
||
52380 | 1073 |
instance poly :: (comm_ring) comm_ring .. |
1074 |
||
1075 |
instance poly :: (comm_ring_1) comm_ring_1 .. |
|
1076 |
||
63498 | 1077 |
instance poly :: (comm_ring_1) comm_semiring_1_cancel .. |
1078 |
||
29451 | 1079 |
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" |
1080 |
unfolding one_poly_def |
|
1081 |
by (simp add: coeff_pCons split: nat.split) |
|
1082 |
||
60570 | 1083 |
lemma monom_eq_1 [simp]: |
1084 |
"monom 1 0 = 1" |
|
1085 |
by (simp add: monom_0 one_poly_def) |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1086 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1087 |
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
|
1088 |
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
|
1089 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1090 |
lemma monom_altdef: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1091 |
"monom c n = smult c ([:0, 1:]^n)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
1092 |
by (induction n) (simp_all add: monom_0 monom_Suc one_poly_def) |
60570 | 1093 |
|
29451 | 1094 |
lemma degree_1 [simp]: "degree 1 = 0" |
1095 |
unfolding one_poly_def |
|
1096 |
by (rule degree_pCons_0) |
|
1097 |
||
52380 | 1098 |
lemma coeffs_1_eq [simp, code abstract]: |
1099 |
"coeffs 1 = [1]" |
|
1100 |
by (simp add: one_poly_def) |
|
1101 |
||
1102 |
lemma degree_power_le: |
|
1103 |
"degree (p ^ n) \<le> degree p * n" |
|
1104 |
by (induct n) (auto intro: order_trans degree_mult_le) |
|
1105 |
||
64795 | 1106 |
lemma coeff_0_power: |
1107 |
"coeff (p ^ n) 0 = coeff p 0 ^ n" |
|
1108 |
by (induction n) (simp_all add: coeff_mult) |
|
1109 |
||
52380 | 1110 |
lemma poly_smult [simp]: |
1111 |
"poly (smult a p) x = a * poly p x" |
|
1112 |
by (induct p, simp, simp add: algebra_simps) |
|
1113 |
||
1114 |
lemma poly_mult [simp]: |
|
1115 |
"poly (p * q) x = poly p x * poly q x" |
|
1116 |
by (induct p, simp_all, simp add: algebra_simps) |
|
1117 |
||
1118 |
lemma poly_1 [simp]: |
|
1119 |
"poly 1 x = 1" |
|
1120 |
by (simp add: one_poly_def) |
|
1121 |
||
1122 |
lemma poly_power [simp]: |
|
1123 |
fixes p :: "'a::{comm_semiring_1} poly" |
|
1124 |
shows "poly (p ^ n) x = poly p x ^ n" |
|
1125 |
by (induct n) simp_all |
|
1126 |
||
64272 | 1127 |
lemma poly_prod: "poly (\<Prod>k\<in>A. p k) x = (\<Prod>k\<in>A. poly (p k) x)" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1128 |
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
|
1129 |
|
64272 | 1130 |
lemma degree_prod_sum_le: "finite S \<Longrightarrow> degree (prod f S) \<le> sum (degree o f) S" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1131 |
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
|
1132 |
case (insert a S) |
64272 | 1133 |
show ?case unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)] |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1134 |
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
|
1135 |
qed simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1136 |
|
64795 | 1137 |
lemma coeff_0_prod_list: |
1138 |
"coeff (prod_list xs) 0 = prod_list (map (\<lambda>p. coeff p 0) xs)" |
|
1139 |
by (induction xs) (simp_all add: coeff_mult) |
|
1140 |
||
1141 |
lemma coeff_monom_mult: |
|
1142 |
"coeff (monom c n * p) k = (if k < n then 0 else c * coeff p (k - n))" |
|
1143 |
proof - |
|
1144 |
have "coeff (monom c n * p) k = (\<Sum>i\<le>k. (if n = i then c else 0) * coeff p (k - i))" |
|
1145 |
by (simp add: coeff_mult) |
|
1146 |
also have "\<dots> = (\<Sum>i\<le>k. (if n = i then c * coeff p (k - i) else 0))" |
|
1147 |
by (intro sum.cong) simp_all |
|
1148 |
also have "\<dots> = (if k < n then 0 else c * coeff p (k - n))" by (simp add: sum.delta') |
|
1149 |
finally show ?thesis . |
|
1150 |
qed |
|
1151 |
||
1152 |
lemma monom_1_dvd_iff': |
|
1153 |
"monom 1 n dvd p \<longleftrightarrow> (\<forall>k<n. coeff p k = 0)" |
|
1154 |
proof |
|
1155 |
assume "monom 1 n dvd p" |
|
1156 |
then obtain r where r: "p = monom 1 n * r" by (elim dvdE) |
|
1157 |
thus "\<forall>k<n. coeff p k = 0" by (simp add: coeff_mult) |
|
1158 |
next |
|
1159 |
assume zero: "(\<forall>k<n. coeff p k = 0)" |
|
1160 |
define r where "r = Abs_poly (\<lambda>k. coeff p (k + n))" |
|
1161 |
have "\<forall>\<^sub>\<infinity>k. coeff p (k + n) = 0" |
|
1162 |
by (subst cofinite_eq_sequentially, subst eventually_sequentially_seg, |
|
1163 |
subst cofinite_eq_sequentially [symmetric]) transfer |
|
1164 |
hence coeff_r [simp]: "coeff r k = coeff p (k + n)" for k unfolding r_def |
|
1165 |
by (subst poly.Abs_poly_inverse) simp_all |
|
1166 |
have "p = monom 1 n * r" |
|
1167 |
by (intro poly_eqI, subst coeff_monom_mult) (insert zero, simp_all) |
|
1168 |
thus "monom 1 n dvd p" by simp |
|
1169 |
qed |
|
1170 |
||
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1171 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1172 |
subsection \<open>Mapping polynomials\<close> |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1173 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1174 |
definition map_poly |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1175 |
:: "('a :: zero \<Rightarrow> 'b :: zero) \<Rightarrow> 'a poly \<Rightarrow> 'b poly" where |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1176 |
"map_poly f p = Poly (map f (coeffs p))" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1177 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1178 |
lemma map_poly_0 [simp]: "map_poly f 0 = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1179 |
by (simp add: map_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1180 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1181 |
lemma map_poly_1: "map_poly f 1 = [:f 1:]" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1182 |
by (simp add: map_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1183 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1184 |
lemma map_poly_1' [simp]: "f 1 = 1 \<Longrightarrow> map_poly f 1 = 1" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1185 |
by (simp add: map_poly_def one_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1186 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1187 |
lemma coeff_map_poly: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1188 |
assumes "f 0 = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1189 |
shows "coeff (map_poly f p) n = f (coeff p n)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1190 |
by (auto simp: map_poly_def nth_default_def coeffs_def assms |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1191 |
not_less Suc_le_eq coeff_eq_0 simp del: upt_Suc) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1192 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1193 |
lemma coeffs_map_poly [code abstract]: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1194 |
"coeffs (map_poly f p) = strip_while (op = 0) (map f (coeffs p))" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1195 |
by (simp add: map_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1196 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1197 |
lemma set_coeffs_map_poly: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1198 |
"(\<And>x. f x = 0 \<longleftrightarrow> x = 0) \<Longrightarrow> set (coeffs (map_poly f p)) = f ` set (coeffs p)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1199 |
by (cases "p = 0") (auto simp: coeffs_map_poly last_map last_coeffs_not_0) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1200 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1201 |
lemma coeffs_map_poly': |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1202 |
assumes "(\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1203 |
shows "coeffs (map_poly f p) = map f (coeffs p)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1204 |
by (cases "p = 0") (auto simp: coeffs_map_poly last_map last_coeffs_not_0 assms |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1205 |
intro!: strip_while_not_last split: if_splits) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1206 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1207 |
lemma degree_map_poly: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1208 |
assumes "\<And>x. x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1209 |
shows "degree (map_poly f p) = degree p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1210 |
by (simp add: degree_eq_length_coeffs coeffs_map_poly' assms) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1211 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1212 |
lemma map_poly_eq_0_iff: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1213 |
assumes "f 0 = 0" "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> f x \<noteq> 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1214 |
shows "map_poly f p = 0 \<longleftrightarrow> p = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1215 |
proof - |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1216 |
{ |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1217 |
fix n :: nat |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1218 |
have "coeff (map_poly f p) n = f (coeff p n)" by (simp add: coeff_map_poly assms) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1219 |
also have "\<dots> = 0 \<longleftrightarrow> coeff p n = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1220 |
proof (cases "n < length (coeffs p)") |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1221 |
case True |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1222 |
hence "coeff p n \<in> set (coeffs p)" by (auto simp: coeffs_def simp del: upt_Suc) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1223 |
with assms show "f (coeff p n) = 0 \<longleftrightarrow> coeff p n = 0" by auto |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1224 |
qed (auto simp: assms length_coeffs nth_default_coeffs_eq [symmetric] nth_default_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1225 |
finally have "(coeff (map_poly f p) n = 0) = (coeff p n = 0)" . |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1226 |
} |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1227 |
thus ?thesis by (auto simp: poly_eq_iff) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1228 |
qed |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1229 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1230 |
lemma map_poly_smult: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1231 |
assumes "f 0 = 0""\<And>c x. f (c * x) = f c * f x" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1232 |
shows "map_poly f (smult c p) = smult (f c) (map_poly f p)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1233 |
by (intro poly_eqI) (simp_all add: assms coeff_map_poly) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1234 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1235 |
lemma map_poly_pCons: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1236 |
assumes "f 0 = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1237 |
shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1238 |
by (intro poly_eqI) (simp_all add: assms coeff_map_poly coeff_pCons split: nat.splits) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1239 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1240 |
lemma map_poly_map_poly: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1241 |
assumes "f 0 = 0" "g 0 = 0" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1242 |
shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1243 |
by (intro poly_eqI) (simp add: coeff_map_poly assms) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1244 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1245 |
lemma map_poly_id [simp]: "map_poly id p = p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1246 |
by (simp add: map_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1247 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1248 |
lemma map_poly_id' [simp]: "map_poly (\<lambda>x. x) p = p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1249 |
by (simp add: map_poly_def) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1250 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1251 |
lemma map_poly_cong: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1252 |
assumes "(\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = g x)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1253 |
shows "map_poly f p = map_poly g p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1254 |
proof - |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1255 |
from assms have "map f (coeffs p) = map g (coeffs p)" by (intro map_cong) simp_all |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1256 |
thus ?thesis by (simp only: coeffs_eq_iff coeffs_map_poly) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1257 |
qed |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1258 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1259 |
lemma map_poly_monom: "f 0 = 0 \<Longrightarrow> map_poly f (monom c n) = monom (f c) n" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1260 |
by (intro poly_eqI) (simp_all add: coeff_map_poly) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1261 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1262 |
lemma map_poly_idI: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1263 |
assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1264 |
shows "map_poly f p = p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1265 |
using map_poly_cong[OF assms, of _ id] by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1266 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1267 |
lemma map_poly_idI': |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1268 |
assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1269 |
shows "p = map_poly f p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1270 |
using map_poly_cong[OF assms, of _ id] by simp |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1271 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1272 |
lemma smult_conv_map_poly: "smult c p = map_poly (\<lambda>x. c * x) p" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1273 |
by (intro poly_eqI) (simp_all add: coeff_map_poly) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1274 |
|
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1275 |
|
64793 | 1276 |
subsection \<open>Conversions from @{typ nat} and @{typ int} numbers\<close> |
62065 | 1277 |
|
1278 |
lemma of_nat_poly: "of_nat n = [:of_nat n :: 'a :: comm_semiring_1:]" |
|
1279 |
proof (induction n) |
|
1280 |
case (Suc n) |
|
1281 |
hence "of_nat (Suc n) = 1 + (of_nat n :: 'a poly)" |
|
1282 |
by simp |
|
1283 |
also have "(of_nat n :: 'a poly) = [: of_nat n :]" |
|
1284 |
by (subst Suc) (rule refl) |
|
1285 |
also have "1 = [:1:]" by (simp add: one_poly_def) |
|
1286 |
finally show ?case by (subst (asm) add_pCons) simp |
|
1287 |
qed simp |
|
1288 |
||
1289 |
lemma degree_of_nat [simp]: "degree (of_nat n) = 0" |
|
1290 |
by (simp add: of_nat_poly) |
|
1291 |
||
64795 | 1292 |
lemma lead_coeff_of_nat [simp]: |
1293 |
"lead_coeff (of_nat n) = (of_nat n :: 'a :: {comm_semiring_1,semiring_char_0})" |
|
1294 |
by (simp add: of_nat_poly) |
|
1295 |
||
1296 |
lemma of_int_poly: "of_int k = [:of_int k :: 'a :: comm_ring_1:]" |
|
64793 | 1297 |
by (simp only: of_int_of_nat of_nat_poly) simp |
1298 |
||
64795 | 1299 |
lemma degree_of_int [simp]: "degree (of_int k) = 0" |
1300 |
by (simp add: of_int_poly) |
|
1301 |
||
1302 |
lemma lead_coeff_of_int [simp]: |
|
1303 |
"lead_coeff (of_int k) = (of_int k :: 'a :: {comm_ring_1,ring_char_0})" |
|
64793 | 1304 |
by (simp add: of_int_poly) |
62065 | 1305 |
|
1306 |
lemma numeral_poly: "numeral n = [:numeral n:]" |
|
1307 |
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp |
|
64793 | 1308 |
|
1309 |
lemma degree_numeral [simp]: "degree (numeral n) = 0" |
|
1310 |
by (subst of_nat_numeral [symmetric], subst of_nat_poly) simp |
|
52380 | 1311 |
|
64795 | 1312 |
lemma lead_coeff_numeral [simp]: |
1313 |
"lead_coeff (numeral n) = numeral n" |
|
1314 |
by (simp add: numeral_poly) |
|
1315 |
||
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1316 |
|
60500 | 1317 |
subsection \<open>Lemmas about divisibility\<close> |
29979 | 1318 |
|
1319 |
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q" |
|
1320 |
proof - |
|
1321 |
assume "p dvd q" |
|
1322 |
then obtain k where "q = p * k" .. |
|
1323 |
then have "smult a q = p * smult a k" by simp |
|
1324 |
then show "p dvd smult a q" .. |
|
1325 |
qed |
|
1326 |
||
1327 |
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
|
1328 |
fixes a :: "'a :: field" |
29979 | 1329 |
shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q" |
1330 |
by (drule dvd_smult [where a="inverse a"]) simp |
|
1331 |
||
1332 |
lemma dvd_smult_iff: |
|
1333 |
fixes a :: "'a::field" |
|
1334 |
shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q" |
|
1335 |
by (safe elim!: dvd_smult dvd_smult_cancel) |
|
1336 |
||
31663 | 1337 |
lemma smult_dvd_cancel: |
1338 |
"smult a p dvd q \<Longrightarrow> p dvd q" |
|
1339 |
proof - |
|
1340 |
assume "smult a p dvd q" |
|
1341 |
then obtain k where "q = smult a p * k" .. |
|
1342 |
then have "q = p * smult a k" by simp |
|
1343 |
then show "p dvd q" .. |
|
1344 |
qed |
|
1345 |
||
1346 |
lemma smult_dvd: |
|
1347 |
fixes a :: "'a::field" |
|
1348 |
shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q" |
|
1349 |
by (rule smult_dvd_cancel [where a="inverse a"]) simp |
|
1350 |
||
1351 |
lemma smult_dvd_iff: |
|
1352 |
fixes a :: "'a::field" |
|
1353 |
shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)" |
|
1354 |
by (auto elim: smult_dvd smult_dvd_cancel) |
|
1355 |
||
64795 | 1356 |
lemma is_unit_smult_iff: "smult c p dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1" |
1357 |
proof - |
|
1358 |
have "smult c p = [:c:] * p" by simp |
|
1359 |
also have "\<dots> dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1" |
|
1360 |
proof safe |
|
1361 |
assume A: "[:c:] * p dvd 1" |
|
1362 |
thus "p dvd 1" by (rule dvd_mult_right) |
|
1363 |
from A obtain q where B: "1 = [:c:] * p * q" by (erule dvdE) |
|
1364 |
have "c dvd c * (coeff p 0 * coeff q 0)" by simp |
|
1365 |
also have "\<dots> = coeff ([:c:] * p * q) 0" by (simp add: mult.assoc coeff_mult) |
|
1366 |
also note B [symmetric] |
|
1367 |
finally show "c dvd 1" by simp |
|
1368 |
next |
|
1369 |
assume "c dvd 1" "p dvd 1" |
|
1370 |
from \<open>c dvd 1\<close> obtain d where "1 = c * d" by (erule dvdE) |
|
1371 |
hence "1 = [:c:] * [:d:]" by (simp add: one_poly_def mult_ac) |
|
1372 |
hence "[:c:] dvd 1" by (rule dvdI) |
|
1373 |
from mult_dvd_mono[OF this \<open>p dvd 1\<close>] show "[:c:] * p dvd 1" by simp |
|
1374 |
qed |
|
1375 |
finally show ?thesis . |
|
1376 |
qed |
|
1377 |
||
29451 | 1378 |
|
60500 | 1379 |
subsection \<open>Polynomials form an integral domain\<close> |
29451 | 1380 |
|
63498 | 1381 |
instance poly :: (idom) idom .. |
29451 | 1382 |
|
1383 |
lemma degree_mult_eq: |
|
63498 | 1384 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
29451 | 1385 |
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" |
1386 |
apply (rule order_antisym [OF degree_mult_le le_degree]) |
|
1387 |
apply (simp add: coeff_mult_degree_sum) |
|
1388 |
done |
|
1389 |
||
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1390 |
lemma degree_mult_eq_0: |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1391 |
fixes p q:: "'a :: {comm_semiring_0,semiring_no_zero_divisors} poly" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1392 |
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)" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1393 |
by (auto simp add: degree_mult_eq) |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
1394 |
|
60570 | 1395 |
lemma degree_mult_right_le: |
63498 | 1396 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
60570 | 1397 |
assumes "q \<noteq> 0" |
1398 |
shows "degree p \<le> degree (p * q)" |
|
1399 |
using assms by (cases "p = 0") (simp_all add: degree_mult_eq) |
|
1400 |
||
1401 |
lemma coeff_degree_mult: |
|
63498 | 1402 |
fixes p q :: "'a::{comm_semiring_0,semiring_no_zero_divisors} poly" |
60570 | 1403 |
shows "coeff (p * q) (degree (p * q)) = |
1404 |
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
|
1405 |
by (cases "p = 0 \<or> q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum mult_ac) |
60570 | 1406 |
|
29451 | 1407 |
lemma dvd_imp_degree_le: |
63498 | 1408 |
fixes p q :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" |
29451 | 1409 |
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
|
1410 |
by (erule dvdE, hypsubst, subst degree_mult_eq) auto |
29451 | 1411 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1412 |
lemma divides_degree: |
63498 | 1413 |
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
|
1414 |
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
|
1415 |
by (metis dvd_imp_degree_le pq) |
63498 | 1416 |
|
1417 |
lemma const_poly_dvd_iff: |
|
1418 |
fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}" |
|
1419 |
shows "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)" |
|
1420 |
proof (cases "c = 0 \<or> p = 0") |
|
1421 |
case False |
|
1422 |
show ?thesis |
|
1423 |
proof |
|
1424 |
assume "[:c:] dvd p" |
|
1425 |
thus "\<forall>n. c dvd coeff p n" by (auto elim!: dvdE simp: coeffs_def) |
|
1426 |
next |
|
1427 |
assume *: "\<forall>n. c dvd coeff p n" |
|
1428 |
define mydiv where "mydiv = (\<lambda>x y :: 'a. SOME z. x = y * z)" |
|
1429 |
have mydiv: "x = y * mydiv x y" if "y dvd x" for x y |
|
1430 |
using that unfolding mydiv_def dvd_def by (rule someI_ex) |
|
1431 |
define q where "q = Poly (map (\<lambda>a. mydiv a c) (coeffs p))" |
|
1432 |
from False * have "p = q * [:c:]" |
|
1433 |
by (intro poly_eqI) (auto simp: q_def nth_default_def not_less length_coeffs_degree |
|
1434 |
coeffs_nth intro!: coeff_eq_0 mydiv) |
|
1435 |
thus "[:c:] dvd p" by (simp only: dvd_triv_right) |
|
1436 |
qed |
|
1437 |
qed (auto intro!: poly_eqI) |
|
1438 |
||
1439 |
lemma const_poly_dvd_const_poly_iff [simp]: |
|
1440 |
"[:a::'a::{comm_semiring_1,semiring_no_zero_divisors}:] dvd [:b:] \<longleftrightarrow> a dvd b" |
|
1441 |
by (subst const_poly_dvd_iff) (auto simp: coeff_pCons split: nat.splits) |
|
1442 |
||
64795 | 1443 |
lemma lead_coeff_mult: |
1444 |
fixes p q :: "'a :: {comm_semiring_0, semiring_no_zero_divisors} poly" |
|
1445 |
shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q" |
|
1446 |
by (cases "p=0 \<or> q=0", auto simp add:coeff_mult_degree_sum degree_mult_eq) |
|
1447 |
||
1448 |
lemma lead_coeff_smult: |
|
1449 |
"lead_coeff (smult c p :: 'a :: {comm_semiring_0,semiring_no_zero_divisors} poly) = c * lead_coeff p" |
|
1450 |
proof - |
|
1451 |
have "smult c p = [:c:] * p" by simp |
|
1452 |
also have "lead_coeff \<dots> = c * lead_coeff p" |
|
1453 |
by (subst lead_coeff_mult) simp_all |
|
1454 |
finally show ?thesis . |
|
1455 |
qed |
|
1456 |
||
1457 |
lemma lead_coeff_1 [simp]: "lead_coeff 1 = 1" |
|
1458 |
by simp |
|
1459 |
||
1460 |
lemma lead_coeff_power: |
|
1461 |
"lead_coeff (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) = lead_coeff p ^ n" |
|
1462 |
by (induction n) (simp_all add: lead_coeff_mult) |
|
1463 |
||
29451 | 1464 |
|
60500 | 1465 |
subsection \<open>Polynomials form an ordered integral domain\<close> |
29878 | 1466 |
|
63498 | 1467 |
definition pos_poly :: "'a::linordered_semidom poly \<Rightarrow> bool" |
29878 | 1468 |
where |
1469 |
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)" |
|
1470 |
||
1471 |
lemma pos_poly_pCons: |
|
1472 |
"pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)" |
|
1473 |
unfolding pos_poly_def by simp |
|
1474 |
||
1475 |
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0" |
|
1476 |
unfolding pos_poly_def by simp |
|
1477 |
||
1478 |
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)" |
|
1479 |
apply (induct p arbitrary: q, simp) |
|
1480 |
apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos) |
|
1481 |
done |
|
1482 |
||
1483 |
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)" |
|
1484 |
unfolding pos_poly_def |
|
1485 |
apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0") |
|
56544 | 1486 |
apply (simp add: degree_mult_eq coeff_mult_degree_sum) |
29878 | 1487 |
apply auto |
1488 |
done |
|
1489 |
||
63498 | 1490 |
lemma pos_poly_total: "(p :: 'a :: linordered_idom poly) = 0 \<or> pos_poly p \<or> pos_poly (- p)" |
29878 | 1491 |
by (induct p) (auto simp add: pos_poly_pCons) |
1492 |
||
52380 | 1493 |
lemma last_coeffs_eq_coeff_degree: |
1494 |
"p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)" |
|
1495 |
by (simp add: coeffs_def) |
|
1496 |
||
1497 |
lemma pos_poly_coeffs [code]: |
|
1498 |
"pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q") |
|
1499 |
proof |
|
1500 |
assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree) |
|
1501 |
next |
|
1502 |
assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def) |
|
1503 |
then have "p \<noteq> 0" by auto |
|
1504 |
with * show ?Q by (simp add: last_coeffs_eq_coeff_degree) |
|
1505 |
qed |
|
1506 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34973
diff
changeset
|
1507 |
instantiation poly :: (linordered_idom) linordered_idom |
29878 | 1508 |
begin |
1509 |
||
1510 |
definition |
|
37765 | 1511 |
"x < y \<longleftrightarrow> pos_poly (y - x)" |
29878 | 1512 |
|
1513 |
definition |
|
37765 | 1514 |
"x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)" |
29878 | 1515 |
|
1516 |
definition |
|
61945 | 1517 |
"\<bar>x::'a poly\<bar> = (if x < 0 then - x else x)" |
29878 | 1518 |
|
1519 |
definition |
|
37765 | 1520 |
"sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
29878 | 1521 |
|
60679 | 1522 |
instance |
1523 |
proof |
|
1524 |
fix x y z :: "'a poly" |
|
29878 | 1525 |
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
1526 |
unfolding less_eq_poly_def less_poly_def |
|
1527 |
apply safe |
|
1528 |
apply simp |
|
1529 |
apply (drule (1) pos_poly_add) |
|
1530 |
apply simp |
|
1531 |
done |
|
60679 | 1532 |
show "x \<le> x" |
29878 | 1533 |
unfolding less_eq_poly_def by simp |
60679 | 1534 |
show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
29878 | 1535 |
unfolding less_eq_poly_def |
1536 |
apply safe |
|
1537 |
apply (drule (1) pos_poly_add) |
|
1538 |
apply (simp add: algebra_simps) |
|
1539 |
done |
|
60679 | 1540 |
show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
29878 | 1541 |
unfolding less_eq_poly_def |
1542 |
apply safe |
|
1543 |
apply (drule (1) pos_poly_add) |
|
1544 |
apply simp |
|
1545 |
done |
|
60679 | 1546 |
show "x \<le> y \<Longrightarrow> z + x \<le> z + y" |
29878 | 1547 |
unfolding less_eq_poly_def |
1548 |
apply safe |
|
1549 |
apply (simp add: algebra_simps) |
|
1550 |
done |
|
1551 |
show "x \<le> y \<or> y \<le> x" |
|
1552 |
unfolding less_eq_poly_def |
|
1553 |
using pos_poly_total [of "x - y"] |
|
1554 |
by auto |
|
60679 | 1555 |
show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y" |
29878 | 1556 |
unfolding less_poly_def |
1557 |
by (simp add: right_diff_distrib [symmetric] pos_poly_mult) |
|
1558 |
show "\<bar>x\<bar> = (if x < 0 then - x else x)" |
|
1559 |
by (rule abs_poly_def) |
|
1560 |
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
|
1561 |
by (rule sgn_poly_def) |
|
1562 |
qed |
|
1563 |
||
1564 |
end |
|
1565 |
||
60500 | 1566 |
text \<open>TODO: Simplification rules for comparisons\<close> |
29878 | 1567 |
|
1568 |
||
60500 | 1569 |
subsection \<open>Synthetic division and polynomial roots\<close> |
52380 | 1570 |
|
64795 | 1571 |
subsubsection \<open>Synthetic division\<close> |
1572 |
||
60500 | 1573 |
text \<open> |
52380 | 1574 |
Synthetic division is simply division by the linear polynomial @{term "x - c"}. |
60500 | 1575 |
\<close> |
52380 | 1576 |
|
1577 |
definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" |
|
1578 |
where |
|
1579 |
"synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)" |
|
1580 |
||
1581 |
definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" |
|
1582 |
where |
|
1583 |
"synthetic_div p c = fst (synthetic_divmod p c)" |
|
1584 |
||
1585 |
lemma synthetic_divmod_0 [simp]: |
|
1586 |
"synthetic_divmod 0 c = (0, 0)" |
|
1587 |
by (simp add: synthetic_divmod_def) |
|
1588 |
||
1589 |
lemma synthetic_divmod_pCons [simp]: |
|
1590 |
"synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" |
|
1591 |
by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def) |
|
1592 |
||
1593 |
lemma synthetic_div_0 [simp]: |
|
1594 |
"synthetic_div 0 c = 0" |
|
1595 |
unfolding synthetic_div_def by simp |
|
1596 |
||
1597 |
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" |
|
1598 |
by (induct p arbitrary: a) simp_all |
|
1599 |
||
1600 |
lemma snd_synthetic_divmod: |
|
1601 |
"snd (synthetic_divmod p c) = poly p c" |
|
1602 |
by (induct p, simp, simp add: split_def) |
|
1603 |
||
1604 |
lemma synthetic_div_pCons [simp]: |
|
1605 |
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" |
|
1606 |
unfolding synthetic_div_def |
|
1607 |
by (simp add: split_def snd_synthetic_divmod) |
|
1608 |
||
1609 |
lemma synthetic_div_eq_0_iff: |
|
1610 |
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0" |
|
63649 | 1611 |
proof (induct p) |
1612 |
case 0 |
|
1613 |
then show ?case by simp |
|
1614 |
next |
|
1615 |
case (pCons a p) |
|
1616 |
then show ?case by (cases p) simp |
|
1617 |
qed |
|
52380 | 1618 |
|
1619 |
lemma degree_synthetic_div: |
|
1620 |
"degree (synthetic_div p c) = degree p - 1" |
|
63649 | 1621 |
by (induct p) (simp_all add: synthetic_div_eq_0_iff) |
52380 | 1622 |
|
1623 |
lemma synthetic_div_correct: |
|
1624 |
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" |
|
1625 |
by (induct p) simp_all |
|
1626 |
||
1627 |
lemma synthetic_div_unique: |
|
1628 |
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" |
|
1629 |
apply (induct p arbitrary: q r) |
|
1630 |
apply (simp, frule synthetic_div_unique_lemma, simp) |
|
1631 |
apply (case_tac q, force) |
|
1632 |
done |
|
1633 |
||
1634 |
lemma synthetic_div_correct': |
|
1635 |
fixes c :: "'a::comm_ring_1" |
|
1636 |
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" |
|
1637 |
using synthetic_div_correct [of p c] |
|
1638 |
by (simp add: algebra_simps) |
|
1639 |
||
64795 | 1640 |
|
1641 |
subsubsection \<open>Polynomial roots\<close> |
|
1642 |
||
52380 | 1643 |
lemma poly_eq_0_iff_dvd: |
63498 | 1644 |
fixes c :: "'a::{comm_ring_1}" |
64795 | 1645 |
shows "poly p c = 0 \<longleftrightarrow> [:- c, 1:] dvd p" |
52380 | 1646 |
proof |
1647 |
assume "poly p c = 0" |
|
1648 |
with synthetic_div_correct' [of c p] |
|
1649 |
have "p = [:-c, 1:] * synthetic_div p c" by simp |
|
1650 |
then show "[:-c, 1:] dvd p" .. |
|
1651 |
next |
|
1652 |
assume "[:-c, 1:] dvd p" |
|
1653 |
then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) |
|
1654 |
then show "poly p c = 0" by simp |
|
1655 |
qed |
|
1656 |
||
1657 |
lemma dvd_iff_poly_eq_0: |
|
63498 | 1658 |
fixes c :: "'a::{comm_ring_1}" |
64795 | 1659 |
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (- c) = 0" |
52380 | 1660 |
by (simp add: poly_eq_0_iff_dvd) |
1661 |
||
1662 |
lemma poly_roots_finite: |
|
63498 | 1663 |
fixes p :: "'a::{comm_ring_1,ring_no_zero_divisors} poly" |
52380 | 1664 |
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" |
1665 |
proof (induct n \<equiv> "degree p" arbitrary: p) |
|
1666 |
case (0 p) |
|
1667 |
then obtain a where "a \<noteq> 0" and "p = [:a:]" |
|
1668 |
by (cases p, simp split: if_splits) |
|
1669 |
then show "finite {x. poly p x = 0}" by simp |
|
1670 |
next |
|
1671 |
case (Suc n p) |
|
1672 |
show "finite {x. poly p x = 0}" |
|
1673 |
proof (cases "\<exists>x. poly p x = 0") |
|
1674 |
case False |
|
1675 |
then show "finite {x. poly p x = 0}" by simp |
|
1676 |
next |
|
1677 |
case True |
|
1678 |
then obtain a where "poly p a = 0" .. |
|
1679 |
then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) |
|
1680 |
then obtain k where k: "p = [:-a, 1:] * k" .. |
|
60500 | 1681 |
with \<open>p \<noteq> 0\<close> have "k \<noteq> 0" by auto |
52380 | 1682 |
with k have "degree p = Suc (degree k)" |
1683 |
by (simp add: degree_mult_eq del: mult_pCons_left) |
|
60500 | 1684 |
with \<open>Suc n = degree p\<close> have "n = degree k" by simp |
1685 |
then have "finite {x. poly k x = 0}" using \<open>k \<noteq> 0\<close> by (rule Suc.hyps) |
|
52380 | 1686 |
then have "finite (insert a {x. poly k x = 0})" by simp |
1687 |
then show "finite {x. poly p x = 0}" |
|
57862 | 1688 |
by (simp add: k Collect_disj_eq del: mult_pCons_left) |
52380 | 1689 |
qed |
1690 |
qed |
|
1691 |
||
1692 |
lemma poly_eq_poly_eq_iff: |
|
63498 | 1693 |
fixes p q :: "'a::{comm_ring_1,ring_no_zero_divisors,ring_char_0} poly" |
52380 | 1694 |
shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q") |
1695 |
proof |
|
1696 |
assume ?Q then show ?P by simp |
|
1697 |
next |
|
63498 | 1698 |
{ fix p :: "'a poly" |
52380 | 1699 |
have "poly p = poly 0 \<longleftrightarrow> p = 0" |
1700 |
apply (cases "p = 0", simp_all) |
|
1701 |
apply (drule poly_roots_finite) |
|
1702 |
apply (auto simp add: infinite_UNIV_char_0) |
|
1703 |
done |
|
1704 |
} note this [of "p - q"] |
|
1705 |
moreover assume ?P |
|
1706 |
ultimately show ?Q by auto |
|
1707 |
qed |
|
1708 |
||
1709 |
lemma poly_all_0_iff_0: |
|
63498 | 1710 |
fixes p :: "'a::{ring_char_0, comm_ring_1,ring_no_zero_divisors} poly" |
52380 | 1711 |
shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0" |
1712 |
by (auto simp add: poly_eq_poly_eq_iff [symmetric]) |
|
1713 |
||
64795 | 1714 |
|
1715 |
subsubsection \<open>Order of polynomial roots\<close> |
|
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1716 |
|
52380 | 1717 |
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
|
1718 |
where |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1719 |
"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
|
1720 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1721 |
lemma coeff_linear_power: |
29979 | 1722 |
fixes a :: "'a::comm_semiring_1" |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1723 |
shows "coeff ([:a, 1:] ^ n) n = 1" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1724 |
apply (induct n, simp_all) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1725 |
apply (subst coeff_eq_0) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1726 |
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
|
1727 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1728 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1729 |
lemma degree_linear_power: |
29979 | 1730 |
fixes a :: "'a::comm_semiring_1" |
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1731 |
shows "degree ([:a, 1:] ^ n) = n" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1732 |
apply (rule order_antisym) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1733 |
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
|
1734 |
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
|
1735 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1736 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1737 |
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
|
1738 |
apply (cases "p = 0", simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1739 |
apply (cases "order a p", simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1740 |
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
|
1741 |
apply (drule not_less_Least, simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1742 |
apply (fold order_def, simp) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1743 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1744 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1745 |
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
|
1746 |
unfolding order_def |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1747 |
apply (rule LeastI_ex) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1748 |
apply (rule_tac x="degree p" in exI) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1749 |
apply (rule notI) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1750 |
apply (drule (1) dvd_imp_degree_le) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1751 |
apply (simp only: degree_linear_power) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1752 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1753 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1754 |
lemma order: |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1755 |
"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
|
1756 |
by (rule conjI [OF order_1 order_2]) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1757 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1758 |
lemma order_degree: |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1759 |
assumes p: "p \<noteq> 0" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1760 |
shows "order a p \<le> degree p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1761 |
proof - |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1762 |
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
|
1763 |
by (simp only: degree_linear_power) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1764 |
also have "\<dots> \<le> degree p" |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1765 |
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
|
1766 |
finally show ?thesis . |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1767 |
qed |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1768 |
|
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1769 |
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
|
1770 |
apply (cases "p = 0", simp_all) |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1771 |
apply (rule iffI) |
56383 | 1772 |
apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right) |
1773 |
unfolding poly_eq_0_iff_dvd |
|
1774 |
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
|
1775 |
done |
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1776 |
|
62065 | 1777 |
lemma order_0I: "poly p a \<noteq> 0 \<Longrightarrow> order a p = 0" |
1778 |
by (subst (asm) order_root) auto |
|
1779 |
||
64795 | 1780 |
lemma order_unique_lemma: |
1781 |
fixes p :: "'a::idom poly" |
|
1782 |
assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p" |
|
1783 |
shows "n = order a p" |
|
1784 |
unfolding Polynomial.order_def |
|
1785 |
apply (rule Least_equality [symmetric]) |
|
1786 |
apply (fact assms) |
|
1787 |
apply (rule classical) |
|
1788 |
apply (erule notE) |
|
1789 |
unfolding not_less_eq_eq |
|
1790 |
using assms(1) apply (rule power_le_dvd) |
|
1791 |
apply assumption |
|
1792 |
done |
|
1793 |
||
1794 |
lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q" |
|
1795 |
proof - |
|
1796 |
define i where "i = order a p" |
|
1797 |
define j where "j = order a q" |
|
1798 |
define t where "t = [:-a, 1:]" |
|
1799 |
have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0" |
|
1800 |
unfolding t_def by (simp add: dvd_iff_poly_eq_0) |
|
1801 |
assume "p * q \<noteq> 0" |
|
1802 |
then show "order a (p * q) = i + j" |
|
1803 |
apply clarsimp |
|
1804 |
apply (drule order [where a=a and p=p, folded i_def t_def]) |
|
1805 |
apply (drule order [where a=a and p=q, folded j_def t_def]) |
|
1806 |
apply clarify |
|
1807 |
apply (erule dvdE)+ |
|
1808 |
apply (rule order_unique_lemma [symmetric], fold t_def) |
|
1809 |
apply (simp_all add: power_add t_dvd_iff) |
|
1810 |
done |
|
1811 |
qed |
|
1812 |
||
1813 |
lemma order_smult: |
|
1814 |
assumes "c \<noteq> 0" |
|
1815 |
shows "order x (smult c p) = order x p" |
|
1816 |
proof (cases "p = 0") |
|
1817 |
case False |
|
1818 |
have "smult c p = [:c:] * p" by simp |
|
1819 |
also from assms False have "order x \<dots> = order x [:c:] + order x p" |
|
1820 |
by (subst order_mult) simp_all |
|
1821 |
also from assms have "order x [:c:] = 0" by (intro order_0I) auto |
|
1822 |
finally show ?thesis by simp |
|
1823 |
qed simp |
|
1824 |
||
1825 |
(* Next two lemmas contributed by Wenda Li *) |
|
1826 |
lemma order_1_eq_0 [simp]:"order x 1 = 0" |
|
1827 |
by (metis order_root poly_1 zero_neq_one) |
|
1828 |
||
1829 |
lemma order_power_n_n: "order a ([:-a,1:]^n)=n" |
|
1830 |
proof (induct n) (*might be proved more concisely using nat_less_induct*) |
|
1831 |
case 0 |
|
1832 |
thus ?case by (metis order_root poly_1 power_0 zero_neq_one) |
|
1833 |
next |
|
1834 |
case (Suc n) |
|
1835 |
have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" |
|
1836 |
by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral |
|
1837 |
one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right) |
|
1838 |
moreover have "order a [:-a,1:]=1" unfolding order_def |
|
1839 |
proof (rule Least_equality,rule ccontr) |
|
1840 |
assume "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" |
|
1841 |
hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp |
|
1842 |
hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" |
|
1843 |
by (rule dvd_imp_degree_le,auto) |
|
1844 |
thus False by auto |
|
1845 |
next |
|
1846 |
fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]" |
|
1847 |
show "1 \<le> y" |
|
1848 |
proof (rule ccontr) |
|
1849 |
assume "\<not> 1 \<le> y" |
|
1850 |
hence "y=0" by auto |
|
1851 |
hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto |
|
1852 |
thus False using asm by auto |
|
1853 |
qed |
|
1854 |
qed |
|
1855 |
ultimately show ?case using Suc by auto |
|
1856 |
qed |
|
1857 |
||
1858 |
lemma order_0_monom [simp]: |
|
1859 |
assumes "c \<noteq> 0" |
|
1860 |
shows "order 0 (monom c n) = n" |
|
1861 |
using assms order_power_n_n[of 0 n] by (simp add: monom_altdef order_smult) |
|
1862 |
||
1863 |
lemma dvd_imp_order_le: |
|
1864 |
"q \<noteq> 0 \<Longrightarrow> p dvd q \<Longrightarrow> Polynomial.order a p \<le> Polynomial.order a q" |
|
1865 |
by (auto simp: order_mult elim: dvdE) |
|
1866 |
||
1867 |
text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close> |
|
1868 |
||
1869 |
lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p" |
|
1870 |
apply (cases "p = 0", auto) |
|
1871 |
apply (drule order_2 [where a=a and p=p]) |
|
1872 |
apply (metis not_less_eq_eq power_le_dvd) |
|
1873 |
apply (erule power_le_dvd [OF order_1]) |
|
1874 |
done |
|
1875 |
||
1876 |
lemma order_decomp: |
|
1877 |
assumes "p \<noteq> 0" |
|
1878 |
shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q" |
|
1879 |
proof - |
|
1880 |
from assms have A: "[:- a, 1:] ^ order a p dvd p" |
|
1881 |
and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order) |
|
1882 |
from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" .. |
|
1883 |
with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q" |
|
1884 |
by simp |
|
1885 |
then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q" |
|
1886 |
by simp |
|
1887 |
then have D: "\<not> [:- a, 1:] dvd q" |
|
1888 |
using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q] |
|
1889 |
by auto |
|
1890 |
from C D show ?thesis by blast |
|
1891 |
qed |
|
1892 |
||
1893 |
lemma monom_1_dvd_iff: |
|
1894 |
assumes "p \<noteq> 0" |
|
1895 |
shows "monom 1 n dvd p \<longleftrightarrow> n \<le> order 0 p" |
|
1896 |
using assms order_divides[of 0 n p] by (simp add: monom_altdef) |
|
1897 |
||
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset
|
1898 |
|
62065 | 1899 |
subsection \<open>Additional induction rules on polynomials\<close> |
1900 |
||
1901 |
text \<open> |
|
1902 |
An induction rule for induction over the roots of a polynomial with a certain property. |
|
1903 |
(e.g. all positive roots) |
|
1904 |
\<close> |
|
1905 |
lemma poly_root_induct [case_names 0 no_roots root]: |
|
1906 |
fixes p :: "'a :: idom poly" |
|
1907 |
assumes "Q 0" |
|
1908 |
assumes "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> 0) \<Longrightarrow> Q p" |
|
1909 |
assumes "\<And>a p. P a \<Longrightarrow> Q p \<Longrightarrow> Q ([:a, -1:] * p)" |
|
1910 |
shows "Q p" |
|
1911 |
proof (induction "degree p" arbitrary: p rule: less_induct) |
|
1912 |
case (less p) |
|
1913 |
show ?case |
|
1914 |
proof (cases "p = 0") |
|
1915 |
assume nz: "p \<noteq> 0" |
|
1916 |
show ?case |
|
1917 |
proof (cases "\<exists>a. P a \<and> poly p a = 0") |
|
1918 |
case False |
|
1919 |
thus ?thesis by (intro assms(2)) blast |
|
1920 |
next |
|
1921 |
case True |
|
1922 |
then obtain a where a: "P a" "poly p a = 0" |
|
1923 |
by blast |
|
1924 |
hence "-[:-a, 1:] dvd p" |
|
1925 |
by (subst minus_dvd_iff) (simp add: poly_eq_0_iff_dvd) |
|
1926 |
then obtain q where q: "p = [:a, -1:] * q" by (elim dvdE) simp |
|
1927 |
with nz have q_nz: "q \<noteq> 0" by auto |
|
1928 |
have "degree p = Suc (degree q)" |
|
1929 |
by (subst q, subst degree_mult_eq) (simp_all add: q_nz) |
|
1930 |
hence "Q q" by (intro less) simp |
|
1931 |
from a(1) and this have "Q ([:a, -1:] * q)" |
|
1932 |
by (rule assms(3)) |
|
1933 |
with q show ?thesis by simp |
|
1934 |
qed |
|
1935 |
qed (simp add: assms(1)) |
|
1936 |
qed |
|
1937 |
||
1938 |
lemma dropWhile_replicate_append: |
|
1939 |
"dropWhile (op= a) (replicate n a @ ys) = dropWhile (op= a) ys" |
|
1940 |
by (induction n) simp_all |
|
1941 |
||
1942 |
lemma Poly_append_replicate_0: "Poly (xs @ replicate n 0) = Poly xs" |
|
1943 |
by (subst coeffs_eq_iff) (simp_all add: strip_while_def dropWhile_replicate_append) |
|
1944 |
||
1945 |
text \<open> |
|
1946 |
An induction rule for simultaneous induction over two polynomials, |
|
1947 |
prepending one coefficient in each step. |
|
1948 |
\<close> |
|
1949 |
lemma poly_induct2 [case_names 0 pCons]: |
|
1950 |
assumes "P 0 0" "\<And>a p b q. P p q \<Longrightarrow> P (pCons a p) (pCons b q)" |
|
1951 |
shows "P p q" |
|
1952 |
proof - |
|
63040 | 1953 |
define n where "n = max (length (coeffs p)) (length (coeffs q))" |
1954 |
define xs where "xs = coeffs p @ (replicate (n - length (coeffs p)) 0)" |
|
1955 |
define ys where "ys = coeffs q @ (replicate (n - length (coeffs q)) 0)" |
|
62065 | 1956 |
have "length xs = length ys" |
1957 |
by (simp add: xs_def ys_def n_def) |
|
1958 |
hence "P (Poly xs) (Poly ys)" |
|
1959 |
by (induction rule: list_induct2) (simp_all add: assms) |
|
1960 |
also have "Poly xs = p" |
|
1961 |
by (simp add: xs_def Poly_append_replicate_0) |
|
1962 |
also have "Poly ys = q" |
|
1963 |
by (simp add: ys_def Poly_append_replicate_0) |
|
1964 |
finally show ?thesis . |
|
1965 |
qed |
|
1966 |
||
64795 | 1967 |
|
60500 | 1968 |
subsection \<open>Composition of polynomials\<close> |
29478 | 1969 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1970 |
(* 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
|
1971 |
|
52380 | 1972 |
definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
1973 |
where |
|
1974 |
"pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0" |
|
1975 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1976 |
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
|
1977 |
|
52380 | 1978 |
lemma pcompose_0 [simp]: |
1979 |
"pcompose 0 q = 0" |
|
1980 |
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
|
1981 |
|
52380 | 1982 |
lemma pcompose_pCons: |
1983 |
"pcompose (pCons a p) q = [:a:] + q * pcompose p q" |
|
1984 |
by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def) |
|
1985 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1986 |
lemma pcompose_1: |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
1987 |
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
|
1988 |
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
|
1989 |
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
|
1990 |
|
52380 | 1991 |
lemma poly_pcompose: |
1992 |
"poly (pcompose p q) x = poly p (poly q x)" |
|
1993 |
by (induct p) (simp_all add: pcompose_pCons) |
|
1994 |
||
1995 |
lemma degree_pcompose_le: |
|
1996 |
"degree (pcompose p q) \<le> degree p * degree q" |
|
1997 |
apply (induct p, simp) |
|
1998 |
apply (simp add: pcompose_pCons, clarify) |
|
1999 |
apply (rule degree_add_le, simp) |
|
2000 |
apply (rule order_trans [OF degree_mult_le], simp) |
|
29478 | 2001 |
done |
2002 |
||
62065 | 2003 |
lemma pcompose_add: |
2004 |
fixes p q r :: "'a :: {comm_semiring_0, ab_semigroup_add} poly" |
|
2005 |
shows "pcompose (p + q) r = pcompose p r + pcompose q r" |
|
2006 |
proof (induction p q rule: poly_induct2) |
|
2007 |
case (pCons a p b q) |
|
2008 |
have "pcompose (pCons a p + pCons b q) r = |
|
2009 |
[:a + b:] + r * pcompose p r + r * pcompose q r" |
|
2010 |
by (simp_all add: pcompose_pCons pCons.IH algebra_simps) |
|
2011 |
also have "[:a + b:] = [:a:] + [:b:]" by simp |
|
2012 |
also have "\<dots> + r * pcompose p r + r * pcompose q r = |
|
2013 |
pcompose (pCons a p) r + pcompose (pCons b q) r" |
|
2014 |
by (simp only: pcompose_pCons add_ac) |
|
2015 |
finally show ?case . |
|
2016 |
qed simp |
|
2017 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2018 |
lemma pcompose_uminus: |
62065 | 2019 |
fixes p r :: "'a :: comm_ring poly" |
2020 |
shows "pcompose (-p) r = -pcompose p r" |
|
2021 |
by (induction p) (simp_all add: pcompose_pCons) |
|
2022 |
||
2023 |
lemma pcompose_diff: |
|
2024 |
fixes p q r :: "'a :: comm_ring poly" |
|
2025 |
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
|
2026 |
using pcompose_add[of p "-q"] by (simp add: pcompose_uminus) |
62065 | 2027 |
|
2028 |
lemma pcompose_smult: |
|
2029 |
fixes p r :: "'a :: comm_semiring_0 poly" |
|
2030 |
shows "pcompose (smult a p) r = smult a (pcompose p r)" |
|
2031 |
by (induction p) |
|
2032 |
(simp_all add: pcompose_pCons pcompose_add smult_add_right) |
|
2033 |
||
2034 |
lemma pcompose_mult: |
|
2035 |
fixes p q r :: "'a :: comm_semiring_0 poly" |
|
2036 |
shows "pcompose (p * q) r = pcompose p r * pcompose q r" |
|
2037 |
by (induction p arbitrary: q) |
|
2038 |
(simp_all add: pcompose_add pcompose_smult pcompose_pCons algebra_simps) |
|
2039 |
||
2040 |
lemma pcompose_assoc: |
|
2041 |
"pcompose p (pcompose q r :: 'a :: comm_semiring_0 poly ) = |
|
2042 |
pcompose (pcompose p q) r" |
|
2043 |
by (induction p arbitrary: q) |
|
2044 |
(simp_all add: pcompose_pCons pcompose_add pcompose_mult) |
|
2045 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2046 |
lemma pcompose_idR[simp]: |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2047 |
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
|
2048 |
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
|
2049 |
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
|
2050 |
|
64267 | 2051 |
lemma pcompose_sum: "pcompose (sum f A) p = sum (\<lambda>i. pcompose (f i) p) A" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2052 |
by (cases "finite A", induction rule: finite_induct) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2053 |
(simp_all add: pcompose_1 pcompose_add) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2054 |
|
64272 | 2055 |
lemma pcompose_prod: "pcompose (prod f A) p = prod (\<lambda>i. pcompose (f i) p) A" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2056 |
by (cases "finite A", induction rule: finite_induct) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2057 |
(simp_all add: pcompose_1 pcompose_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2058 |
|
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
2059 |
lemma pcompose_const [simp]: "pcompose [:a:] q = [:a:]" |
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
2060 |
by (subst pcompose_pCons) simp |
62065 | 2061 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
62072
diff
changeset
|
2062 |
lemma pcompose_0': "pcompose p 0 = [:coeff p 0:]" |
64591
240a39af9ec4
restructured matter on polynomials and normalized fractions
haftmann
parents:
64272
diff
changeset
|
2063 |
by (induct p) (auto simp add: pcompose_pCons) |
62065 | 2064 |
|
2065 |
lemma degree_pcompose: |
|
63498 | 2066 |
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
|
2067 |
shows "degree (pcompose p q) = degree p * degree q" |
62065 | 2068 |
proof (induct p) |
2069 |
case 0 |
|
2070 |
thus ?case by auto |
|
2071 |
next |
|
2072 |
case (pCons a p) |
|
2073 |
have "degree (q * pcompose p q) = 0 \<Longrightarrow> ?case" |
|
2074 |
proof (cases "p=0") |
|
2075 |
case True |
|
2076 |
thus ?thesis by auto |
|
2077 |
next |
|
2078 |
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
|
2079 |
hence "degree q=0 \<or> pcompose p q=0" by (auto simp add: degree_mult_eq_0) |
62072 | 2080 |
moreover have "\<lbrakk>pcompose p q=0;degree q\<noteq>0\<rbrakk> \<Longrightarrow> False" using pCons.hyps(2) \<open>p\<noteq>0\<close> |
62065 | 2081 |
proof - |
2082 |
assume "pcompose p q=0" "degree q\<noteq>0" |
|
2083 |
hence "degree p=0" using pCons.hyps(2) by auto |
|
2084 |
then obtain a1 where "p=[:a1:]" |
|
2085 |
by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) |
|
62072 | 2086 |
thus False using \<open>pcompose p q=0\<close> \<open>p\<noteq>0\<close> by auto |
62065 | 2087 |
qed |
2088 |
ultimately have "degree (pCons a p) * degree q=0" by auto |
|
2089 |
moreover have "degree (pcompose (pCons a p) q) = 0" |
|
2090 |
proof - |
|
2091 |
have" 0 = max (degree [:a:]) (degree (q*pcompose p q))" |
|
62072 | 2092 |
using \<open>degree (q * pcompose p q) = 0\<close> by simp |
62065 | 2093 |
also have "... \<ge> degree ([:a:] + q * pcompose p q)" |
2094 |
by (rule degree_add_le_max) |
|
2095 |
finally show ?thesis by (auto simp add:pcompose_pCons) |
|
2096 |
qed |
|
2097 |
ultimately show ?thesis by simp |
|
2098 |
qed |
|
2099 |
moreover have "degree (q * pcompose p q)>0 \<Longrightarrow> ?case" |
|
2100 |
proof - |
|
2101 |
assume asm:"0 < degree (q * pcompose p q)" |
|
2102 |
hence "p\<noteq>0" "q\<noteq>0" "pcompose p q\<noteq>0" by auto |
|
2103 |
have "degree (pcompose (pCons a p) q) = degree ( q * pcompose p q)" |
|
2104 |
unfolding pcompose_pCons |
|
2105 |
using degree_add_eq_right[of "[:a:]" ] asm by auto |
|
2106 |
thus ?thesis |
|
62072 | 2107 |
using pCons.hyps(2) degree_mult_eq[OF \<open>q\<noteq>0\<close> \<open>pcompose p q\<noteq>0\<close>] by auto |
62065 | 2108 |
qed |
2109 |
ultimately show ?case by blast |
|
2110 |
qed |
|
2111 |
||
2112 |
lemma pcompose_eq_0: |
|
63498 | 2113 |
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
|
2114 |
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
|
2115 |
shows "p = 0" |
62065 | 2116 |
proof - |
2117 |
have "degree p=0" using assms degree_pcompose[of p q] by auto |
|
2118 |
then obtain a where "p=[:a:]" |
|
2119 |
by (metis degree_pCons_eq_if gr0_conv_Suc neq0_conv pCons_cases) |
|
2120 |
hence "a=0" using assms(1) by auto |
|
62072 | 2121 |
thus ?thesis using \<open>p=[:a:]\<close> by simp |
62065 | 2122 |
qed |
2123 |
||
2124 |
lemma lead_coeff_comp: |
|
63498 | 2125 |
fixes p q:: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly" |
62065 | 2126 |
assumes "degree q > 0" |
2127 |
shows "lead_coeff (pcompose p q) = lead_coeff p * lead_coeff q ^ (degree p)" |
|
2128 |
proof (induct p) |
|
2129 |
case 0 |
|
64794 | 2130 |
thus ?case by auto |
62065 | 2131 |
next |
2132 |
case (pCons a p) |
|
2133 |
have "degree ( q * pcompose p q) = 0 \<Longrightarrow> ?case" |
|
2134 |
proof - |
|
2135 |
assume "degree ( q * pcompose p q) = 0" |
|
2136 |
hence "pcompose p q = 0" by (metis assms degree_0 degree_mult_eq_0 neq0_conv) |
|
62072 | 2137 |
hence "p=0" using pcompose_eq_0[OF _ \<open>degree q > 0\<close>] by simp |
62065 | 2138 |
thus ?thesis by auto |
2139 |
qed |
|
2140 |
moreover have "degree ( q * pcompose p q) > 0 \<Longrightarrow> ?case" |
|
2141 |
proof - |
|
64794 | 2142 |
assume "degree (q * pcompose p q) > 0" |
2143 |
then have "degree [:a:] < degree (q * pcompose p q)" |
|
2144 |
by simp |
|
2145 |
then have "lead_coeff ([:a:] + q * p \<circ>\<^sub>p q) = lead_coeff (q * p \<circ>\<^sub>p q)" |
|
2146 |
by (rule lead_coeff_add_le) |
|
2147 |
then have "lead_coeff (pcompose (pCons a p) q) = lead_coeff (q * pcompose p q)" |
|
2148 |
by (simp add: pcompose_pCons) |
|
62065 | 2149 |
also have "... = lead_coeff q * (lead_coeff p * lead_coeff q ^ degree p)" |
2150 |
using pCons.hyps(2) lead_coeff_mult[of q "pcompose p q"] by simp |
|
2151 |
also have "... = lead_coeff p * lead_coeff q ^ (degree p + 1)" |
|
63498 | 2152 |
by (auto simp: mult_ac) |
62065 | 2153 |
finally show ?thesis by auto |
2154 |
qed |
|
2155 |
ultimately show ?case by blast |
|
2156 |
qed |
|
2157 |
||
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2158 |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2159 |
subsection \<open>Shifting polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2160 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2161 |
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
|
2162 |
"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
|
2163 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2164 |
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
|
2165 |
by (auto simp add: nth_default_def add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2166 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2167 |
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
|
2168 |
by (auto simp add: nth_default_def add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2169 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2170 |
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
|
2171 |
proof - |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2172 |
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
|
2173 |
hence "\<forall>k>m. coeff p (k + n) = 0" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2174 |
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
|
2175 |
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
|
2176 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2177 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2178 |
lemma poly_shift_id [simp]: "poly_shift 0 = (\<lambda>x. x)" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2179 |
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
|
2180 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2181 |
lemma poly_shift_0 [simp]: "poly_shift n 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2182 |
by (simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2183 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2184 |
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
|
2185 |
by (simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2186 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2187 |
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
|
2188 |
by (auto simp add: poly_eq_iff coeff_poly_shift) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2189 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2190 |
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
|
2191 |
proof (cases "p = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2192 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2193 |
thus ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2194 |
by (intro coeffs_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2195 |
(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
|
2196 |
qed simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2197 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2198 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2199 |
subsection \<open>Truncating polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2200 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2201 |
definition poly_cutoff where |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2202 |
"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
|
2203 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2204 |
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
|
2205 |
unfolding poly_cutoff_def |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2206 |
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
|
2207 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2208 |
lemma poly_cutoff_0 [simp]: "poly_cutoff n 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2209 |
by (simp add: poly_eq_iff coeff_poly_cutoff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2210 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2211 |
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
|
2212 |
by (simp add: poly_eq_iff coeff_poly_cutoff) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2213 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2214 |
lemma coeffs_poly_cutoff [code abstract]: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2215 |
"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
|
2216 |
proof (cases "strip_while (op = 0) (take n (coeffs p)) = []") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2217 |
case True |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2218 |
hence "coeff (poly_cutoff n p) k = 0" for k |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2219 |
unfolding coeff_poly_cutoff |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2220 |
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
|
2221 |
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
|
2222 |
thus ?thesis by (subst True) simp_all |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2223 |
next |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2224 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2225 |
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
|
2226 |
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
|
2227 |
unfolding no_trailing_unfold by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2228 |
thus ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2229 |
by (intro coeffs_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2230 |
(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
|
2231 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2232 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2233 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2234 |
subsection \<open>Reflecting polynomials\<close> |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2235 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2236 |
definition reflect_poly where |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2237 |
"reflect_poly p = Poly (rev (coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2238 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2239 |
lemma coeffs_reflect_poly [code abstract]: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2240 |
"coeffs (reflect_poly p) = rev (dropWhile (op = 0) (coeffs p))" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2241 |
unfolding reflect_poly_def by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2242 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2243 |
lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2244 |
by (simp add: reflect_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2245 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2246 |
lemma reflect_poly_1 [simp]: "reflect_poly 1 = 1" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2247 |
by (simp add: reflect_poly_def one_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2248 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2249 |
lemma coeff_reflect_poly: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2250 |
"coeff (reflect_poly p) n = (if n > degree p then 0 else coeff p (degree p - n))" |
64811 | 2251 |
by (cases "p = 0") (auto simp add: reflect_poly_def nth_default_def |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2252 |
rev_nth degree_eq_length_coeffs coeffs_nth not_less |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2253 |
dest: le_imp_less_Suc) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2254 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2255 |
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
|
2256 |
by (simp add: coeff_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2257 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2258 |
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
|
2259 |
by (simp add: coeff_reflect_poly poly_0_coeff_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2260 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2261 |
lemma reflect_poly_pCons': |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2262 |
"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
|
2263 |
by (intro poly_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2264 |
(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
|
2265 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2266 |
lemma reflect_poly_const [simp]: "reflect_poly [:a:] = [:a:]" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2267 |
by (cases "a = 0") (simp_all add: reflect_poly_def) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2268 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2269 |
lemma poly_reflect_poly_nz: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2270 |
"(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
|
2271 |
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
|
2272 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2273 |
lemma coeff_0_reflect_poly [simp]: "coeff (reflect_poly p) 0 = lead_coeff p" |
64794 | 2274 |
by (simp add: coeff_reflect_poly) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2275 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2276 |
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
|
2277 |
by (simp add: poly_0_coeff_0) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2278 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2279 |
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
|
2280 |
by (cases p rule: pCons_cases) (simp add: reflect_poly_def ) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2281 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2282 |
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
|
2283 |
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
|
2284 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2285 |
lemma reflect_poly_pCons: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2286 |
"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
|
2287 |
by (subst coeffs_eq_iff) (simp add: coeffs_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2288 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2289 |
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
|
2290 |
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
|
2291 |
|
63498 | 2292 |
(* 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
|
2293 |
lemma reflect_poly_mult: |
63498 | 2294 |
"reflect_poly (p * q) = |
2295 |
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
|
2296 |
proof (cases "p = 0 \<or> q = 0") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2297 |
case False |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2298 |
hence [simp]: "p \<noteq> 0" "q \<noteq> 0" by auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2299 |
show ?thesis |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2300 |
proof (rule poly_eqI) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2301 |
fix i :: nat |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2302 |
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
|
2303 |
proof (cases "i \<le> degree (p * q)") |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2304 |
case True |
64811 | 2305 |
define A where "A = {..i} \<inter> {i - degree q..degree p}" |
2306 |
define B where "B = {..degree p} \<inter> {degree p - i..degree (p*q) - i}" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2307 |
let ?f = "\<lambda>j. degree p - j" |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2308 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2309 |
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
|
2310 |
by (simp add: coeff_reflect_poly) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2311 |
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
|
2312 |
unfolding coeff_mult by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2313 |
also have "\<dots> = (\<Sum>j\<in>B. coeff p j * coeff q (degree (p * q) - i - j))" |
64267 | 2314 |
by (intro sum.mono_neutral_right) (auto simp: B_def degree_mult_eq not_le coeff_eq_0) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2315 |
also from True have "\<dots> = (\<Sum>j\<in>A. coeff p (degree p - j) * coeff q (degree q - (i - j)))" |
64267 | 2316 |
by (intro sum.reindex_bij_witness[of _ ?f ?f]) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2317 |
(auto simp: A_def B_def degree_mult_eq add_ac) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2318 |
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
|
2319 |
coeff p (degree p - j) * coeff q (degree q - (i - j)) else 0)" |
64267 | 2320 |
by (subst sum.inter_restrict [symmetric]) (simp_all add: A_def) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2321 |
also have "\<dots> = coeff (reflect_poly p * reflect_poly q) i" |
64267 | 2322 |
by (fastforce simp: coeff_mult coeff_reflect_poly intro!: sum.cong) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2323 |
finally show ?thesis . |
64267 | 2324 |
qed (auto simp: coeff_mult coeff_reflect_poly coeff_eq_0 degree_mult_eq intro!: sum.neutral) |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2325 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2326 |
qed auto |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2327 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2328 |
lemma reflect_poly_smult: |
63498 | 2329 |
"reflect_poly (Polynomial.smult (c::'a::{comm_semiring_0,semiring_no_zero_divisors}) p) = |
2330 |
Polynomial.smult c (reflect_poly p)" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2331 |
using reflect_poly_mult[of "[:c:]" p] by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2332 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2333 |
lemma reflect_poly_power: |
63498 | 2334 |
"reflect_poly (p ^ n :: 'a :: {comm_semiring_1,semiring_no_zero_divisors} poly) = |
2335 |
reflect_poly p ^ n" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2336 |
by (induction n) (simp_all add: reflect_poly_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2337 |
|
64272 | 2338 |
lemma reflect_poly_prod: |
2339 |
"reflect_poly (prod (f :: _ \<Rightarrow> _ :: {comm_semiring_0,semiring_no_zero_divisors} poly) A) = |
|
2340 |
prod (\<lambda>x. reflect_poly (f x)) A" |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2341 |
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
|
2342 |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63649
diff
changeset
|
2343 |
lemma reflect_poly_prod_list: |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63649
diff
changeset
|
2344 |
"reflect_poly (prod_list (xs :: _ :: {comm_semiring_0,semiring_no_zero_divisors} poly list)) = |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63649
diff
changeset
|
2345 |
prod_list (map reflect_poly xs)" |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2346 |
by (induction xs) (simp_all add: reflect_poly_mult) |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2347 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2348 |
lemma reflect_poly_Poly_nz: |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2349 |
"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
|
2350 |
unfolding reflect_poly_def coeffs_Poly by simp |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2351 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2352 |
lemmas reflect_poly_simps = |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2353 |
reflect_poly_0 reflect_poly_1 reflect_poly_const reflect_poly_smult reflect_poly_mult |
64272 | 2354 |
reflect_poly_power reflect_poly_prod reflect_poly_prod_list |
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2355 |
|
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
2356 |
|
64795 | 2357 |
subsection \<open>Derivatives\<close> |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2358 |
|
63498 | 2359 |
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
|
2360 |
where |
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2361 |
"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
|
2362 |
by (auto intro: pCons_cases) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2363 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2364 |
termination pderiv |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2365 |
by (relation "measure degree") simp_all |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2366 |
|
63027
8de0ebee3f1c
several updates on polynomial long division and pseudo division
Rene Thiemann <rene.thiemann@uibk.ac.at>
parents:
62422
diff
changeset
|
2367 |
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
|
2368 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2369 |
lemma pderiv_0 [simp]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2370 |
"pderiv 0 = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2371 |
using pderiv.simps [of 0 0] by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2372 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2373 |
lemma pderiv_pCons: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2374 |
"pderiv (pCons a p) = p + pCons 0 (pderiv p)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2375 |
by (simp add: pderiv.simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2376 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2377 |
lemma pderiv_1 [simp]: "pderiv 1 = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2378 |
unfolding one_poly_def by (simp add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2379 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2380 |
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
|
2381 |
and pderiv_numeral [simp]: "pderiv (numeral m) = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2382 |
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
|
2383 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2384 |
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
|
2385 |
by (induct p arbitrary: n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2386 |
(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
|
2387 |
|
63498 | 2388 |
fun pderiv_coeffs_code |
2389 |
:: "('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
|
2390 |
"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
|
2391 |
| "pderiv_coeffs_code f [] = []" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2392 |
|
63498 | 2393 |
definition pderiv_coeffs :: |
2394 |
"'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
|
2395 |
"pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2396 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2397 |
(* 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
|
2398 |
lemma pderiv_coeffs_code: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2399 |
"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
|
2400 |
proof (induct xs arbitrary: f n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2401 |
case (Cons x xs f n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2402 |
show ?case |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2403 |
proof (cases n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2404 |
case 0 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2405 |
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
|
2406 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2407 |
case (Suc m) note n = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2408 |
show ?thesis |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2409 |
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
|
2410 |
case False |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2411 |
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
|
2412 |
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
|
2413 |
by (auto simp: cCons_def n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2414 |
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
|
2415 |
unfolding Cons by (simp add: n add_ac) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2416 |
finally show ?thesis by (simp add: n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2417 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2418 |
case True |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2419 |
{ |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2420 |
fix g |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2421 |
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
|
2422 |
proof (induct xs arbitrary: g m) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2423 |
case (Cons x xs g) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2424 |
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
|
2425 |
and g: "(g = 0 \<or> x = 0)" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2426 |
by (auto simp: cCons_def split: if_splits) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2427 |
note IH = Cons(1)[OF empty] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2428 |
from IH[of m] IH[of "m - 1"] g |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2429 |
show ?case by (cases m, auto simp: field_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2430 |
qed simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2431 |
} note empty = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2432 |
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
|
2433 |
by (auto simp: cCons_def n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2434 |
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
|
2435 |
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
|
2436 |
ultimately show ?thesis by simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2437 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2438 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2439 |
qed simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2440 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2441 |
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
|
2442 |
by (induct n arbitrary: f, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2443 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2444 |
lemma coeffs_pderiv_code [code abstract]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2445 |
"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
|
2446 |
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
|
2447 |
case (1 n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2448 |
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
|
2449 |
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
|
2450 |
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
|
2451 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2452 |
case 2 |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2453 |
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
|
2454 |
from 2 show ?case |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2455 |
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
|
2456 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2457 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2458 |
context |
63498 | 2459 |
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
|
2460 |
begin |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2461 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2462 |
lemma pderiv_eq_0_iff: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2463 |
"pderiv (p :: 'a poly) = 0 \<longleftrightarrow> degree p = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2464 |
apply (rule iffI) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2465 |
apply (cases p, simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2466 |
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
|
2467 |
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
|
2468 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2469 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2470 |
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
|
2471 |
apply (rule order_antisym [OF degree_le]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2472 |
apply (simp add: coeff_pderiv coeff_eq_0) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2473 |
apply (cases "degree p", simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2474 |
apply (rule le_degree) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2475 |
apply (simp add: coeff_pderiv del: of_nat_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2476 |
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
|
2477 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2478 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2479 |
lemma not_dvd_pderiv: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2480 |
assumes "degree (p :: 'a poly) \<noteq> 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2481 |
shows "\<not> p dvd pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2482 |
proof |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2483 |
assume dvd: "p dvd pderiv p" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2484 |
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
|
2485 |
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
|
2486 |
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
|
2487 |
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
|
2488 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2489 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2490 |
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
|
2491 |
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
|
2492 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2493 |
end |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2494 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2495 |
lemma pderiv_singleton [simp]: "pderiv [:a:] = 0" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2496 |
by (simp add: pderiv_pCons) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2497 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2498 |
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
|
2499 |
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
|
2500 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2501 |
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
|
2502 |
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
|
2503 |
|
63498 | 2504 |
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
|
2505 |
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
|
2506 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2507 |
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
|
2508 |
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
|
2509 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2510 |
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
|
2511 |
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
|
2512 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2513 |
lemma pderiv_power_Suc: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2514 |
"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
|
2515 |
apply (induct n) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2516 |
apply simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2517 |
apply (subst power_Suc) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2518 |
apply (subst pderiv_mult) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2519 |
apply (erule ssubst) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2520 |
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
|
2521 |
apply (simp add: algebra_simps) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2522 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2523 |
|
64272 | 2524 |
lemma pderiv_prod: "pderiv (prod f (as)) = |
2525 |
(\<Sum>a \<in> as. prod f (as - {a}) * pderiv (f a))" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2526 |
proof (induct as rule: infinite_finite_induct) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2527 |
case (insert a as) |
64272 | 2528 |
hence id: "prod f (insert a as) = f a * prod f as" |
64267 | 2529 |
"\<And> g. sum g (insert a as) = g a + sum g as" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2530 |
"insert a as - {a} = as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2531 |
by auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2532 |
{ |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2533 |
fix b |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2534 |
assume "b \<in> as" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2535 |
hence id2: "insert a as - {b} = insert a (as - {b})" using \<open>a \<notin> as\<close> by auto |
64272 | 2536 |
have "prod f (insert a as - {b}) = f a * prod f (as - {b})" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2537 |
unfolding id2 |
64272 | 2538 |
by (subst prod.insert, insert insert, auto) |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2539 |
} note id2 = this |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2540 |
show ?case |
64267 | 2541 |
unfolding id pderiv_mult insert(3) sum_distrib_left |
2542 |
by (auto simp add: ac_simps id2 intro!: sum.cong) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2543 |
qed auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2544 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2545 |
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
|
2546 |
by (rule DERIV_cong, rule DERIV_pow, simp) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2547 |
declare DERIV_pow2 [simp] DERIV_pow [simp] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2548 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2549 |
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
|
2550 |
by (rule DERIV_cong, rule DERIV_add, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2551 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2552 |
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
|
2553 |
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
|
2554 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2555 |
lemma continuous_on_poly [continuous_intros]: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2556 |
fixes p :: "'a :: {real_normed_field} poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2557 |
assumes "continuous_on A f" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2558 |
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
|
2559 |
proof - |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2560 |
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
|
2561 |
by (intro continuous_intros assms) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2562 |
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
|
2563 |
finally show ?thesis . |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2564 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2565 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2566 |
text\<open>Consequences of the derivative theorem above\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2567 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2568 |
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
|
2569 |
apply (simp add: real_differentiable_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2570 |
apply (blast intro: poly_DERIV) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2571 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2572 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2573 |
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
|
2574 |
by (rule poly_DERIV [THEN DERIV_isCont]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2575 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2576 |
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
|
2577 |
==> \<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
|
2578 |
using IVT_objl [of "poly p" a 0 b] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2579 |
by (auto simp add: order_le_less) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2580 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2581 |
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
|
2582 |
==> \<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
|
2583 |
by (insert poly_IVT_pos [where p = "- p" ]) simp |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2584 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2585 |
lemma poly_IVT: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2586 |
fixes p::"real poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2587 |
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
|
2588 |
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
|
2589 |
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
|
2590 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2591 |
lemma poly_MVT: "(a::real) < b ==> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2592 |
\<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
|
2593 |
using MVT [of a b "poly p"] |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2594 |
apply auto |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2595 |
apply (rule_tac x = z in exI) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2596 |
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
|
2597 |
done |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2598 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2599 |
lemma poly_MVT': |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2600 |
assumes "{min a b..max a b} \<subseteq> A" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2601 |
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
|
2602 |
proof (cases a b rule: linorder_cases) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2603 |
case less |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2604 |
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
|
2605 |
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
|
2606 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2607 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2608 |
case greater |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2609 |
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
|
2610 |
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
|
2611 |
qed (insert assms, auto) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2612 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2613 |
lemma poly_pinfty_gt_lc: |
63649 | 2614 |
fixes p :: "real poly" |
2615 |
assumes "lead_coeff p > 0" |
|
2616 |
shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" |
|
2617 |
using assms |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2618 |
proof (induct p) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2619 |
case 0 |
63649 | 2620 |
then show ?case by auto |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2621 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2622 |
case (pCons a p) |
63649 | 2623 |
from this(1) consider "a \<noteq> 0" "p = 0" | "p \<noteq> 0" by auto |
2624 |
then show ?case |
|
2625 |
proof cases |
|
2626 |
case 1 |
|
2627 |
then show ?thesis by auto |
|
2628 |
next |
|
2629 |
case 2 |
|
2630 |
with pCons obtain n1 where gte_lcoeff: "\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" |
|
2631 |
by auto |
|
2632 |
from pCons(3) \<open>p \<noteq> 0\<close> have gt_0: "lead_coeff p > 0" by auto |
|
2633 |
define n where "n = max n1 (1 + \<bar>a\<bar> / lead_coeff p)" |
|
2634 |
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
|
2635 |
proof - |
63649 | 2636 |
from gte_lcoeff that have "lead_coeff p \<le> poly p x" |
2637 |
by (auto simp: n_def) |
|
2638 |
with gt_0 have "\<bar>a\<bar> / lead_coeff p \<ge> \<bar>a\<bar> / poly p x" and "poly p x > 0" |
|
2639 |
by (auto intro: frac_le) |
|
2640 |
with \<open>n\<le>x\<close>[unfolded n_def] have "x \<ge> 1 + \<bar>a\<bar> / poly p x" |
|
2641 |
by auto |
|
2642 |
with \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x > 0\<close> \<open>p \<noteq> 0\<close> |
|
2643 |
show "lead_coeff (pCons a p) \<le> poly (pCons a p) x" |
|
2644 |
by (auto simp: field_simps) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2645 |
qed |
63649 | 2646 |
then show ?thesis by blast |
2647 |
qed |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2648 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2649 |
|
64795 | 2650 |
lemma lemma_order_pderiv1: |
2651 |
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q + |
|
2652 |
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)" |
|
2653 |
apply (simp only: pderiv_mult pderiv_power_Suc) |
|
2654 |
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons) |
|
2655 |
done |
|
2656 |
||
2657 |
lemma lemma_order_pderiv: |
|
2658 |
fixes p :: "'a :: field_char_0 poly" |
|
2659 |
assumes n: "0 < n" |
|
2660 |
and pd: "pderiv p \<noteq> 0" |
|
2661 |
and pe: "p = [:- a, 1:] ^ n * q" |
|
2662 |
and nd: "~ [:- a, 1:] dvd q" |
|
2663 |
shows "n = Suc (order a (pderiv p))" |
|
2664 |
using n |
|
2665 |
proof - |
|
2666 |
have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0" |
|
2667 |
using assms by auto |
|
2668 |
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0" |
|
2669 |
using assms by (cases n) auto |
|
2670 |
have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l" |
|
2671 |
by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff) |
|
2672 |
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" |
|
2673 |
proof (rule order_unique_lemma) |
|
2674 |
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
2675 |
apply (subst lemma_order_pderiv1) |
|
2676 |
apply (rule dvd_add) |
|
2677 |
apply (metis dvdI dvd_mult2 power_Suc2) |
|
2678 |
apply (metis dvd_smult dvd_triv_right) |
|
2679 |
done |
|
2680 |
next |
|
2681 |
show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
2682 |
apply (subst lemma_order_pderiv1) |
|
2683 |
by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one) |
|
2684 |
qed |
|
2685 |
then show ?thesis |
|
2686 |
by (metis \<open>n = Suc n'\<close> pe) |
|
2687 |
qed |
|
2688 |
||
2689 |
lemma order_pderiv: |
|
2690 |
"\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> \<Longrightarrow> |
|
2691 |
(order a p = Suc (order a (pderiv p)))" |
|
2692 |
apply (case_tac "p = 0", simp) |
|
2693 |
apply (drule_tac a = a and p = p in order_decomp) |
|
2694 |
using neq0_conv |
|
2695 |
apply (blast intro: lemma_order_pderiv) |
|
2696 |
done |
|
2697 |
||
2698 |
lemma poly_squarefree_decomp_order: |
|
2699 |
assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0" |
|
2700 |
and p: "p = q * d" |
|
2701 |
and p': "pderiv p = e * d" |
|
2702 |
and d: "d = r * p + s * pderiv p" |
|
2703 |
shows "order a q = (if order a p = 0 then 0 else 1)" |
|
2704 |
proof (rule classical) |
|
2705 |
assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)" |
|
2706 |
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto |
|
2707 |
with p have "order a p = order a q + order a d" |
|
2708 |
by (simp add: order_mult) |
|
2709 |
with 1 have "order a p \<noteq> 0" by (auto split: if_splits) |
|
2710 |
have "order a (pderiv p) = order a e + order a d" |
|
2711 |
using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult) |
|
2712 |
have "order a p = Suc (order a (pderiv p))" |
|
2713 |
using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv) |
|
2714 |
have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp |
|
2715 |
have "([:-a, 1:] ^ (order a (pderiv p))) dvd d" |
|
2716 |
apply (simp add: d) |
|
2717 |
apply (rule dvd_add) |
|
2718 |
apply (rule dvd_mult) |
|
2719 |
apply (simp add: order_divides \<open>p \<noteq> 0\<close> |
|
2720 |
\<open>order a p = Suc (order a (pderiv p))\<close>) |
|
2721 |
apply (rule dvd_mult) |
|
2722 |
apply (simp add: order_divides) |
|
2723 |
done |
|
2724 |
then have "order a (pderiv p) \<le> order a d" |
|
2725 |
using \<open>d \<noteq> 0\<close> by (simp add: order_divides) |
|
2726 |
show ?thesis |
|
2727 |
using \<open>order a p = order a q + order a d\<close> |
|
2728 |
using \<open>order a (pderiv p) = order a e + order a d\<close> |
|
2729 |
using \<open>order a p = Suc (order a (pderiv p))\<close> |
|
2730 |
using \<open>order a (pderiv p) \<le> order a d\<close> |
|
2731 |
by auto |
|
2732 |
qed |
|
2733 |
||
2734 |
lemma poly_squarefree_decomp_order2: |
|
2735 |
"\<lbrakk>pderiv p \<noteq> (0 :: 'a :: field_char_0 poly); |
|
2736 |
p = q * d; |
|
2737 |
pderiv p = e * d; |
|
2738 |
d = r * p + s * pderiv p |
|
2739 |
\<rbrakk> \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
|
2740 |
by (blast intro: poly_squarefree_decomp_order) |
|
2741 |
||
2742 |
lemma order_pderiv2: |
|
2743 |
"\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> |
|
2744 |
\<Longrightarrow> (order a (pderiv p) = n) = (order a p = Suc n)" |
|
2745 |
by (auto dest: order_pderiv) |
|
2746 |
||
2747 |
definition rsquarefree :: "'a::idom poly \<Rightarrow> bool" |
|
2748 |
where "rsquarefree p \<longleftrightarrow> p \<noteq> 0 \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)" |
|
2749 |
||
2750 |
lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h :: 'a :: {semidom,semiring_char_0}:]" |
|
2751 |
by (cases p) (auto simp: pderiv_eq_0_iff split: if_splits) |
|
2752 |
||
2753 |
lemma rsquarefree_roots: |
|
2754 |
fixes p :: "'a :: field_char_0 poly" |
|
2755 |
shows "rsquarefree p = (\<forall>a. \<not>(poly p a = 0 \<and> poly (pderiv p) a = 0))" |
|
2756 |
apply (simp add: rsquarefree_def) |
|
2757 |
apply (case_tac "p = 0", simp, simp) |
|
2758 |
apply (case_tac "pderiv p = 0") |
|
2759 |
apply simp |
|
2760 |
apply (drule pderiv_iszero, clarsimp) |
|
2761 |
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree) |
|
2762 |
apply (force simp add: order_root order_pderiv2) |
|
2763 |
done |
|
2764 |
||
2765 |
lemma poly_squarefree_decomp: |
|
2766 |
assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0" |
|
2767 |
and "p = q * d" |
|
2768 |
and "pderiv p = e * d" |
|
2769 |
and "d = r * p + s * pderiv p" |
|
2770 |
shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))" |
|
2771 |
proof - |
|
2772 |
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto |
|
2773 |
with \<open>p = q * d\<close> have "q \<noteq> 0" by simp |
|
2774 |
have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
|
2775 |
using assms by (rule poly_squarefree_decomp_order2) |
|
2776 |
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis |
|
2777 |
by (simp add: rsquarefree_def order_root) |
|
2778 |
qed |
|
2779 |
||
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2780 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2781 |
subsection \<open>Algebraic numbers\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2782 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2783 |
text \<open> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2784 |
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
|
2785 |
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
|
2786 |
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
|
2787 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2788 |
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
|
2789 |
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
|
2790 |
\<close> |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2791 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2792 |
definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool" where |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2793 |
"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
|
2794 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2795 |
lemma algebraicI: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2796 |
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
|
2797 |
shows "algebraic x" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2798 |
using assms unfolding algebraic_def by blast |
62065 | 2799 |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2800 |
lemma algebraicE: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2801 |
assumes "algebraic x" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2802 |
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
|
2803 |
using assms unfolding algebraic_def by blast |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2804 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2805 |
lemma algebraic_altdef: |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2806 |
fixes p :: "'a :: field_char_0 poly" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2807 |
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
|
2808 |
proof safe |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2809 |
fix p assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0" |
63040 | 2810 |
define cs where "cs = coeffs p" |
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2811 |
from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'" unfolding Rats_def by blast |
63060 | 2812 |
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
|
2813 |
by (subst (asm) bchoice_iff) blast |
63040 | 2814 |
define cs' where "cs' = map (quotient_of \<circ> f) (coeffs p)" |
2815 |
define d where "d = Lcm (set (map snd cs'))" |
|
2816 |
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
|
2817 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2818 |
have "\<forall>n. coeff p' n \<in> \<int>" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2819 |
proof |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2820 |
fix n :: nat |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2821 |
show "coeff p' n \<in> \<int>" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2822 |
proof (cases "n \<le> degree p") |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2823 |
case True |
63040 | 2824 |
define c where "c = coeff p n" |
2825 |
define a where "a = fst (quotient_of (f (coeff p n)))" |
|
2826 |
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
|
2827 |
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
|
2828 |
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
|
2829 |
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
|
2830 |
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
|
2831 |
(simp_all add: f [symmetric]) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2832 |
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
|
2833 |
by (simp add: of_rat_mult of_rat_divide) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2834 |
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
|
2835 |
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
|
2836 |
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
|
2837 |
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
|
2838 |
by (rule of_int_divide_in_Ints) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2839 |
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
|
2840 |
finally show ?thesis . |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2841 |
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
|
2842 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2843 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2844 |
moreover have "set (map snd cs') \<subseteq> {0<..}" |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2845 |
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
|
2846 |
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
|
2847 |
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
|
2848 |
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
|
2849 |
ultimately show "algebraic x" unfolding algebraic_def by blast |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2850 |
next |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2851 |
assume "algebraic x" |
63060 | 2852 |
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
|
2853 |
by (force simp: algebraic_def) |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2854 |
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
|
2855 |
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
|
2856 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2857 |
|
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2858 |
|
64795 | 2859 |
subsection \<open>Content and primitive part of a polynomial\<close> |
2860 |
||
64860 | 2861 |
definition content :: "('a :: semiring_gcd poly) \<Rightarrow> 'a" where |
2862 |
"content p = gcd_list (coeffs p)" |
|
2863 |
||
2864 |
lemma content_eq_fold_coeffs [code]: |
|
2865 |
"content p = fold_coeffs gcd p 0" |
|
2866 |
by (simp add: content_def Gcd_fin.set_eq_fold fold_coeffs_def foldr_fold fun_eq_iff ac_simps) |
|
64795 | 2867 |
|
2868 |
lemma content_0 [simp]: "content 0 = 0" |
|
2869 |
by (simp add: content_def) |
|
2870 |
||
2871 |
lemma content_1 [simp]: "content 1 = 1" |
|
2872 |
by (simp add: content_def) |
|
2873 |
||
2874 |
lemma content_const [simp]: "content [:c:] = normalize c" |
|
2875 |
by (simp add: content_def cCons_def) |
|
2876 |
||
2877 |
lemma const_poly_dvd_iff_dvd_content: |
|
64860 | 2878 |
fixes c :: "'a :: semiring_gcd" |
64795 | 2879 |
shows "[:c:] dvd p \<longleftrightarrow> c dvd content p" |
2880 |
proof (cases "p = 0") |
|
2881 |
case [simp]: False |
|
2882 |
have "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)" by (rule const_poly_dvd_iff) |
|
2883 |
also have "\<dots> \<longleftrightarrow> (\<forall>a\<in>set (coeffs p). c dvd a)" |
|
2884 |
proof safe |
|
2885 |
fix n :: nat assume "\<forall>a\<in>set (coeffs p). c dvd a" |
|
2886 |
thus "c dvd coeff p n" |
|
2887 |
by (cases "n \<le> degree p") (auto simp: coeff_eq_0 coeffs_def split: if_splits) |
|
2888 |
qed (auto simp: coeffs_def simp del: upt_Suc split: if_splits) |
|
2889 |
also have "\<dots> \<longleftrightarrow> c dvd content p" |
|
64860 | 2890 |
by (simp add: content_def dvd_Gcd_fin_iff dvd_mult_unit_iff) |
64795 | 2891 |
finally show ?thesis . |
2892 |
qed simp_all |
|
2893 |
||
2894 |
lemma content_dvd [simp]: "[:content p:] dvd p" |
|
2895 |
by (subst const_poly_dvd_iff_dvd_content) simp_all |
|
2896 |
||
2897 |
lemma content_dvd_coeff [simp]: "content p dvd coeff p n" |
|
64860 | 2898 |
proof (cases "p = 0") |
2899 |
case True |
|
2900 |
then show ?thesis |
|
2901 |
by simp |
|
2902 |
next |
|
2903 |
case False |
|
2904 |
then show ?thesis |
|
2905 |
by (cases "n \<le> degree p") |
|
2906 |
(auto simp add: content_def not_le coeff_eq_0 coeff_in_coeffs intro: Gcd_fin_dvd) |
|
2907 |
qed |
|
2908 |
||
64795 | 2909 |
lemma content_dvd_coeffs: "c \<in> set (coeffs p) \<Longrightarrow> content p dvd c" |
64860 | 2910 |
by (simp add: content_def Gcd_fin_dvd) |
64795 | 2911 |
|
2912 |
lemma normalize_content [simp]: "normalize (content p) = content p" |
|
2913 |
by (simp add: content_def) |
|
2914 |
||
2915 |
lemma is_unit_content_iff [simp]: "is_unit (content p) \<longleftrightarrow> content p = 1" |
|
2916 |
proof |
|
2917 |
assume "is_unit (content p)" |
|
2918 |
hence "normalize (content p) = 1" by (simp add: is_unit_normalize del: normalize_content) |
|
2919 |
thus "content p = 1" by simp |
|
2920 |
qed auto |
|
2921 |
||
2922 |
lemma content_smult [simp]: "content (smult c p) = normalize c * content p" |
|
64860 | 2923 |
by (simp add: content_def coeffs_smult Gcd_fin_mult) |
64795 | 2924 |
|
2925 |
lemma content_eq_zero_iff [simp]: "content p = 0 \<longleftrightarrow> p = 0" |
|
2926 |
by (auto simp: content_def simp: poly_eq_iff coeffs_def) |
|
2927 |
||
64860 | 2928 |
definition primitive_part :: "'a :: semiring_gcd poly \<Rightarrow> 'a poly" where |
2929 |
"primitive_part p = map_poly (\<lambda>x. x div content p) p" |
|
2930 |
||
64795 | 2931 |
lemma primitive_part_0 [simp]: "primitive_part 0 = 0" |
2932 |
by (simp add: primitive_part_def) |
|
2933 |
||
2934 |
lemma content_times_primitive_part [simp]: |
|
64860 | 2935 |
fixes p :: "'a :: semiring_gcd poly" |
64795 | 2936 |
shows "smult (content p) (primitive_part p) = p" |
2937 |
proof (cases "p = 0") |
|
2938 |
case False |
|
2939 |
thus ?thesis |
|
2940 |
unfolding primitive_part_def |
|
2941 |
by (auto simp: smult_conv_map_poly map_poly_map_poly o_def content_dvd_coeffs |
|
2942 |
intro: map_poly_idI) |
|
2943 |
qed simp_all |
|
2944 |
||
2945 |
lemma primitive_part_eq_0_iff [simp]: "primitive_part p = 0 \<longleftrightarrow> p = 0" |
|
2946 |
proof (cases "p = 0") |
|
2947 |
case False |
|
2948 |
hence "primitive_part p = map_poly (\<lambda>x. x div content p) p" |
|
2949 |
by (simp add: primitive_part_def) |
|
2950 |
also from False have "\<dots> = 0 \<longleftrightarrow> p = 0" |
|
2951 |
by (intro map_poly_eq_0_iff) (auto simp: dvd_div_eq_0_iff content_dvd_coeffs) |
|
2952 |
finally show ?thesis using False by simp |
|
2953 |
qed simp |
|
2954 |
||
2955 |
lemma content_primitive_part [simp]: |
|
2956 |
assumes "p \<noteq> 0" |
|
2957 |
shows "content (primitive_part p) = 1" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
2958 |
proof - |
64795 | 2959 |
have "p = smult (content p) (primitive_part p)" by simp |
64860 | 2960 |
also have "content \<dots> = content (primitive_part p) * content p" |
2961 |
by (simp del: content_times_primitive_part add: ac_simps) |
|
2962 |
finally have "1 * content p = content (primitive_part p) * content p" |
|
2963 |
by simp |
|
2964 |
then have "1 * content p div content p = content (primitive_part p) * content p div content p" |
|
2965 |
by simp |
|
2966 |
with assms show ?thesis |
|
2967 |
by simp |
|
64795 | 2968 |
qed |
2969 |
||
2970 |
lemma content_decompose: |
|
64860 | 2971 |
fixes p :: "'a :: semiring_gcd poly" |
64795 | 2972 |
obtains p' where "p = smult (content p) p'" "content p' = 1" |
2973 |
proof (cases "p = 0") |
|
2974 |
case True |
|
2975 |
thus ?thesis by (intro that[of 1]) simp_all |
|
2976 |
next |
|
2977 |
case False |
|
2978 |
from content_dvd[of p] obtain r where r: "p = [:content p:] * r" by (erule dvdE) |
|
2979 |
have "content p * 1 = content p * content r" by (subst r) simp |
|
2980 |
with False have "content r = 1" by (subst (asm) mult_left_cancel) simp_all |
|
2981 |
with r show ?thesis by (intro that[of r]) simp_all |
|
2982 |
qed |
|
2983 |
||
2984 |
lemma content_dvd_contentI [intro]: |
|
2985 |
"p dvd q \<Longrightarrow> content p dvd content q" |
|
2986 |
using const_poly_dvd_iff_dvd_content content_dvd dvd_trans by blast |
|
2987 |
||
2988 |
lemma primitive_part_const_poly [simp]: "primitive_part [:x:] = [:unit_factor x:]" |
|
2989 |
by (simp add: primitive_part_def map_poly_pCons) |
|
2990 |
||
2991 |
lemma primitive_part_prim: "content p = 1 \<Longrightarrow> primitive_part p = p" |
|
2992 |
by (auto simp: primitive_part_def) |
|
2993 |
||
2994 |
lemma degree_primitive_part [simp]: "degree (primitive_part p) = degree p" |
|
2995 |
proof (cases "p = 0") |
|
2996 |
case False |
|
2997 |
have "p = smult (content p) (primitive_part p)" by simp |
|
2998 |
also from False have "degree \<dots> = degree (primitive_part p)" |
|
2999 |
by (subst degree_smult_eq) simp_all |
|
3000 |
finally show ?thesis .. |
|
3001 |
qed simp_all |
|
3002 |
||
3003 |
||
3004 |
subsection \<open>Division of polynomials\<close> |
|
3005 |
||
3006 |
subsubsection \<open>Division in general\<close> |
|
3007 |
||
3008 |
instantiation poly :: (idom_divide) idom_divide |
|
3009 |
begin |
|
3010 |
||
3011 |
fun divide_poly_main :: "'a \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly |
|
3012 |
\<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly" where |
|
3013 |
"divide_poly_main lc q r d dr (Suc n) = (let cr = coeff r dr; a = cr div lc; mon = monom a n in |
|
3014 |
if False \<or> a * lc = cr then (* False \<or> is only because of problem in function-package *) |
|
3015 |
divide_poly_main |
|
3016 |
lc |
|
3017 |
(q + mon) |
|
3018 |
(r - mon * d) |
|
3019 |
d (dr - 1) n else 0)" |
|
3020 |
| "divide_poly_main lc q r d dr 0 = q" |
|
3021 |
||
3022 |
definition divide_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
3023 |
"divide_poly f g = (if g = 0 then 0 else |
|
3024 |
divide_poly_main (coeff g (degree g)) 0 f g (degree f) (1 + length (coeffs f) - length (coeffs g)))" |
|
3025 |
||
3026 |
lemma divide_poly_main: |
|
3027 |
assumes d: "d \<noteq> 0" "lc = coeff d (degree d)" |
|
3028 |
and *: "degree (d * r) \<le> dr" "divide_poly_main lc q (d * r) d dr n = q'" |
|
3029 |
"n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> d * r = 0" |
|
3030 |
shows "q' = q + r" |
|
3031 |
using * |
|
3032 |
proof (induct n arbitrary: q r dr) |
|
3033 |
case (Suc n q r dr) |
|
3034 |
let ?rr = "d * r" |
|
3035 |
let ?a = "coeff ?rr dr" |
|
3036 |
let ?qq = "?a div lc" |
|
3037 |
define b where [simp]: "b = monom ?qq n" |
|
3038 |
let ?rrr = "d * (r - b)" |
|
3039 |
let ?qqq = "q + b" |
|
3040 |
note res = Suc(3) |
|
3041 |
have dr: "dr = n + degree d" using Suc(4) by auto |
|
3042 |
have lc: "lc \<noteq> 0" using d by auto |
|
3043 |
have "coeff (b * d) dr = coeff b n * coeff d (degree d)" |
|
3044 |
proof (cases "?qq = 0") |
|
3045 |
case False |
|
3046 |
hence n: "n = degree b" by (simp add: degree_monom_eq) |
|
3047 |
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) |
|
3048 |
qed simp |
|
3049 |
also have "\<dots> = lc * coeff b n" unfolding d by simp |
|
3050 |
finally have c2: "coeff (b * d) dr = lc * coeff b n" . |
|
3051 |
have rrr: "?rrr = ?rr - b * d" by (simp add: field_simps) |
|
3052 |
have c1: "coeff (d * r) dr = lc * coeff r n" |
|
3053 |
proof (cases "degree r = n") |
|
3054 |
case True |
|
3055 |
thus ?thesis using Suc(2) unfolding dr using coeff_mult_degree_sum[of d r] d by (auto simp: ac_simps) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3056 |
next |
64795 | 3057 |
case False |
3058 |
have "degree r \<le> n" using dr Suc(2) by auto |
|
3059 |
(metis add.commute add_le_cancel_left d(1) degree_0 degree_mult_eq diff_is_0_eq diff_zero le_cases) |
|
3060 |
with False have r_n: "degree r < n" by auto |
|
3061 |
hence right: "lc * coeff r n = 0" by (simp add: coeff_eq_0) |
|
3062 |
have "coeff (d * r) dr = coeff (d * r) (degree d + n)" unfolding dr by (simp add: ac_simps) |
|
3063 |
also have "\<dots> = 0" using r_n |
|
3064 |
by (metis False Suc.prems(1) add.commute add_left_imp_eq coeff_degree_mult coeff_eq_0 |
|
3065 |
coeff_mult_degree_sum degree_mult_le dr le_eq_less_or_eq) |
|
3066 |
finally show ?thesis unfolding right . |
|
3067 |
qed |
|
3068 |
have c0: "coeff ?rrr dr = 0" |
|
3069 |
and id: "lc * (coeff (d * r) dr div lc) = coeff (d * r) dr" unfolding rrr coeff_diff c2 |
|
3070 |
unfolding b_def coeff_monom coeff_smult c1 using lc by auto |
|
3071 |
from res[unfolded divide_poly_main.simps[of lc q] Let_def] id |
|
3072 |
have res: "divide_poly_main lc ?qqq ?rrr d (dr - 1) n = q'" |
|
3073 |
by (simp del: divide_poly_main.simps add: field_simps) |
|
3074 |
note IH = Suc(1)[OF _ res] |
|
3075 |
have dr: "dr = n + degree d" using Suc(4) by auto |
|
3076 |
have deg_rr: "degree ?rr \<le> dr" using Suc(2) by auto |
|
3077 |
have deg_bd: "degree (b * d) \<le> dr" |
|
3078 |
unfolding dr b_def by (rule order.trans[OF degree_mult_le], auto simp: degree_monom_le) |
|
3079 |
have "degree ?rrr \<le> dr" unfolding rrr by (rule degree_diff_le[OF deg_rr deg_bd]) |
|
3080 |
with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)" |
|
3081 |
by (rule coeff_0_degree_minus_1) |
|
3082 |
have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0" |
|
3083 |
proof (cases dr) |
|
3084 |
case 0 |
|
3085 |
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto |
|
3086 |
with deg_rrr have "degree ?rrr = 0" by simp |
|
3087 |
from degree_eq_zeroE[OF this] obtain a where rrr: "?rrr = [:a:]" by metis |
|
3088 |
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp |
|
3089 |
qed (insert Suc(4), auto) |
|
3090 |
note IH = IH[OF deg_rrr this] |
|
3091 |
show ?case using IH by simp |
|
3092 |
next |
|
3093 |
case (0 q r dr) |
|
3094 |
show ?case |
|
3095 |
proof (cases "r = 0") |
|
3096 |
case True |
|
3097 |
thus ?thesis using 0 by auto |
|
3098 |
next |
|
3099 |
case False |
|
3100 |
have "degree (d * r) = degree d + degree r" using d False |
|
3101 |
by (subst degree_mult_eq, auto) |
|
3102 |
thus ?thesis using 0 d by auto |
|
3103 |
qed |
|
3104 |
qed |
|
3105 |
||
3106 |
lemma divide_poly_main_0: "divide_poly_main 0 0 r d dr n = 0" |
|
3107 |
proof (induct n arbitrary: r d dr) |
|
3108 |
case (Suc n r d dr) |
|
3109 |
show ?case unfolding divide_poly_main.simps[of _ _ r] Let_def |
|
3110 |
by (simp add: Suc del: divide_poly_main.simps) |
|
3111 |
qed simp |
|
3112 |
||
3113 |
lemma divide_poly: |
|
3114 |
assumes g: "g \<noteq> 0" |
|
3115 |
shows "(f * g) div g = (f :: 'a poly)" |
|
3116 |
proof - |
|
3117 |
have "divide_poly_main (coeff g (degree g)) 0 (g * f) g (degree (g * f)) (1 + length (coeffs (g * f)) - length (coeffs g)) |
|
3118 |
= (f * g) div g" unfolding divide_poly_def Let_def by (simp add: ac_simps) |
|
3119 |
note main = divide_poly_main[OF g refl le_refl this] |
|
3120 |
{ |
|
3121 |
fix f :: "'a poly" |
|
3122 |
assume "f \<noteq> 0" |
|
3123 |
hence "length (coeffs f) = Suc (degree f)" unfolding degree_eq_length_coeffs by auto |
|
3124 |
} note len = this |
|
3125 |
have "(f * g) div g = 0 + f" |
|
3126 |
proof (rule main, goal_cases) |
|
3127 |
case 1 |
|
3128 |
show ?case |
|
3129 |
proof (cases "f = 0") |
|
3130 |
case True |
|
3131 |
with g show ?thesis by (auto simp: degree_eq_length_coeffs) |
|
3132 |
next |
|
3133 |
case False |
|
3134 |
with g have fg: "g * f \<noteq> 0" by auto |
|
3135 |
show ?thesis unfolding len[OF fg] len[OF g] by auto |
|
3136 |
qed |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3137 |
qed |
64795 | 3138 |
thus ?thesis by simp |
3139 |
qed |
|
3140 |
||
3141 |
lemma divide_poly_0: "f div 0 = (0 :: 'a poly)" |
|
3142 |
by (simp add: divide_poly_def Let_def divide_poly_main_0) |
|
3143 |
||
3144 |
instance |
|
3145 |
by standard (auto simp: divide_poly divide_poly_0) |
|
3146 |
||
3147 |
end |
|
3148 |
||
3149 |
instance poly :: (idom_divide) algebraic_semidom .. |
|
3150 |
||
3151 |
lemma div_const_poly_conv_map_poly: |
|
3152 |
assumes "[:c:] dvd p" |
|
3153 |
shows "p div [:c:] = map_poly (\<lambda>x. x div c) p" |
|
3154 |
proof (cases "c = 0") |
|
3155 |
case False |
|
3156 |
from assms obtain q where p: "p = [:c:] * q" by (erule dvdE) |
|
3157 |
moreover { |
|
3158 |
have "smult c q = [:c:] * q" by simp |
|
3159 |
also have "\<dots> div [:c:] = q" by (rule nonzero_mult_div_cancel_left) (insert False, auto) |
|
3160 |
finally have "smult c q div [:c:] = q" . |
|
3161 |
} |
|
3162 |
ultimately show ?thesis by (intro poly_eqI) (auto simp: coeff_map_poly False) |
|
3163 |
qed (auto intro!: poly_eqI simp: coeff_map_poly) |
|
3164 |
||
3165 |
lemma is_unit_monom_0: |
|
3166 |
fixes a :: "'a::field" |
|
3167 |
assumes "a \<noteq> 0" |
|
3168 |
shows "is_unit (monom a 0)" |
|
3169 |
proof |
|
3170 |
from assms show "1 = monom a 0 * monom (inverse a) 0" |
|
3171 |
by (simp add: mult_monom) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3172 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3173 |
|
64795 | 3174 |
lemma is_unit_triv: |
3175 |
fixes a :: "'a::field" |
|
3176 |
assumes "a \<noteq> 0" |
|
3177 |
shows "is_unit [:a:]" |
|
3178 |
using assms by (simp add: is_unit_monom_0 monom_0 [symmetric]) |
|
3179 |
||
3180 |
lemma is_unit_iff_degree: |
|
3181 |
assumes "p \<noteq> (0 :: _ :: field poly)" |
|
3182 |
shows "is_unit p \<longleftrightarrow> degree p = 0" (is "?P \<longleftrightarrow> ?Q") |
|
3183 |
proof |
|
3184 |
assume ?Q |
|
3185 |
then obtain a where "p = [:a:]" by (rule degree_eq_zeroE) |
|
3186 |
with assms show ?P by (simp add: is_unit_triv) |
|
3187 |
next |
|
3188 |
assume ?P |
|
3189 |
then obtain q where "q \<noteq> 0" "p * q = 1" .. |
|
3190 |
then have "degree (p * q) = degree 1" |
|
3191 |
by simp |
|
3192 |
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0" |
|
3193 |
by (simp add: degree_mult_eq) |
|
3194 |
then show ?Q by simp |
|
3195 |
qed |
|
3196 |
||
3197 |
lemma is_unit_pCons_iff: |
|
3198 |
"is_unit (pCons (a::_::field) p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0" |
|
3199 |
by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree) |
|
3200 |
||
3201 |
lemma is_unit_monom_trival: |
|
3202 |
fixes p :: "'a::field poly" |
|
3203 |
assumes "is_unit p" |
|
3204 |
shows "monom (coeff p (degree p)) 0 = p" |
|
3205 |
using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff) |
|
3206 |
||
3207 |
lemma is_unit_const_poly_iff: |
|
3208 |
"[:c :: 'a :: {comm_semiring_1,semiring_no_zero_divisors}:] dvd 1 \<longleftrightarrow> c dvd 1" |
|
3209 |
by (auto simp: one_poly_def) |
|
3210 |
||
3211 |
lemma is_unit_polyE: |
|
3212 |
fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly" |
|
3213 |
assumes "p dvd 1" obtains c where "p = [:c:]" "c dvd 1" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3214 |
proof - |
64795 | 3215 |
from assms obtain q where "1 = p * q" |
3216 |
by (rule dvdE) |
|
3217 |
then have "p \<noteq> 0" and "q \<noteq> 0" |
|
3218 |
by auto |
|
3219 |
from \<open>1 = p * q\<close> have "degree 1 = degree (p * q)" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3220 |
by simp |
64795 | 3221 |
also from \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> have "\<dots> = degree p + degree q" |
3222 |
by (simp add: degree_mult_eq) |
|
3223 |
finally have "degree p = 0" by simp |
|
3224 |
with degree_eq_zeroE obtain c where c: "p = [:c:]" . |
|
3225 |
moreover with \<open>p dvd 1\<close> have "c dvd 1" |
|
3226 |
by (simp add: is_unit_const_poly_iff) |
|
3227 |
ultimately show thesis |
|
3228 |
by (rule that) |
|
3229 |
qed |
|
3230 |
||
3231 |
lemma is_unit_polyE': |
|
3232 |
assumes "is_unit (p::_::field poly)" |
|
3233 |
obtains a where "p = monom a 0" and "a \<noteq> 0" |
|
3234 |
proof - |
|
3235 |
obtain a q where "p = pCons a q" by (cases p) |
|
3236 |
with assms have "p = [:a:]" and "a \<noteq> 0" |
|
3237 |
by (simp_all add: is_unit_pCons_iff) |
|
3238 |
with that show thesis by (simp add: monom_0) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3239 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3240 |
|
64795 | 3241 |
lemma is_unit_poly_iff: |
3242 |
fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly" |
|
3243 |
shows "p dvd 1 \<longleftrightarrow> (\<exists>c. p = [:c:] \<and> c dvd 1)" |
|
3244 |
by (auto elim: is_unit_polyE simp add: is_unit_const_poly_iff) |
|
3245 |
||
3246 |
||
3247 |
subsubsection \<open>Pseudo-Division\<close> |
|
3248 |
||
3249 |
text\<open>This part is by René Thiemann and Akihisa Yamada.\<close> |
|
3250 |
||
3251 |
fun pseudo_divmod_main :: "'a :: comm_ring_1 \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly |
|
3252 |
\<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a poly \<times> 'a poly" where |
|
3253 |
"pseudo_divmod_main lc q r d dr (Suc n) = (let |
|
3254 |
rr = smult lc r; |
|
3255 |
qq = coeff r dr; |
|
3256 |
rrr = rr - monom qq n * d; |
|
3257 |
qqq = smult lc q + monom qq n |
|
3258 |
in pseudo_divmod_main lc qqq rrr d (dr - 1) n)" |
|
3259 |
| "pseudo_divmod_main lc q r d dr 0 = (q,r)" |
|
3260 |
||
3261 |
definition pseudo_divmod :: "'a :: comm_ring_1 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" where |
|
3262 |
"pseudo_divmod p q \<equiv> if q = 0 then (0,p) else |
|
3263 |
pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p) (1 + length (coeffs p) - length (coeffs q))" |
|
3264 |
||
3265 |
lemma pseudo_divmod_main: assumes d: "d \<noteq> 0" "lc = coeff d (degree d)" |
|
3266 |
and *: "degree r \<le> dr" "pseudo_divmod_main lc q r d dr n = (q',r')" |
|
3267 |
"n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> r = 0" |
|
3268 |
shows "(r' = 0 \<or> degree r' < degree d) \<and> smult (lc^n) (d * q + r) = d * q' + r'" |
|
3269 |
using * |
|
3270 |
proof (induct n arbitrary: q r dr) |
|
3271 |
case (Suc n q r dr) |
|
3272 |
let ?rr = "smult lc r" |
|
3273 |
let ?qq = "coeff r dr" |
|
3274 |
define b where [simp]: "b = monom ?qq n" |
|
3275 |
let ?rrr = "?rr - b * d" |
|
3276 |
let ?qqq = "smult lc q + b" |
|
3277 |
note res = Suc(3) |
|
3278 |
from res[unfolded pseudo_divmod_main.simps[of lc q] Let_def] |
|
3279 |
have res: "pseudo_divmod_main lc ?qqq ?rrr d (dr - 1) n = (q',r')" |
|
3280 |
by (simp del: pseudo_divmod_main.simps) |
|
3281 |
have dr: "dr = n + degree d" using Suc(4) by auto |
|
3282 |
have "coeff (b * d) dr = coeff b n * coeff d (degree d)" |
|
3283 |
proof (cases "?qq = 0") |
|
3284 |
case False |
|
3285 |
hence n: "n = degree b" by (simp add: degree_monom_eq) |
|
3286 |
show ?thesis unfolding n dr by (simp add: coeff_mult_degree_sum) |
|
3287 |
qed auto |
|
3288 |
also have "\<dots> = lc * coeff b n" unfolding d by simp |
|
3289 |
finally have "coeff (b * d) dr = lc * coeff b n" . |
|
3290 |
moreover have "coeff ?rr dr = lc * coeff r dr" by simp |
|
3291 |
ultimately have c0: "coeff ?rrr dr = 0" by auto |
|
3292 |
have dr: "dr = n + degree d" using Suc(4) by auto |
|
3293 |
have deg_rr: "degree ?rr \<le> dr" using Suc(2) |
|
3294 |
using degree_smult_le dual_order.trans by blast |
|
3295 |
have deg_bd: "degree (b * d) \<le> dr" |
|
3296 |
unfolding dr |
|
3297 |
by(rule order.trans[OF degree_mult_le], auto simp: degree_monom_le) |
|
3298 |
have "degree ?rrr \<le> dr" |
|
3299 |
using degree_diff_le[OF deg_rr deg_bd] by auto |
|
3300 |
with c0 have deg_rrr: "degree ?rrr \<le> (dr - 1)" by (rule coeff_0_degree_minus_1) |
|
3301 |
have "n = 1 + (dr - 1) - degree d \<or> dr - 1 = 0 \<and> n = 0 \<and> ?rrr = 0" |
|
3302 |
proof (cases dr) |
|
3303 |
case 0 |
|
3304 |
with Suc(4) have 0: "dr = 0" "n = 0" "degree d = 0" by auto |
|
3305 |
with deg_rrr have "degree ?rrr = 0" by simp |
|
3306 |
hence "\<exists> a. ?rrr = [: a :]" by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases) |
|
3307 |
from this obtain a where rrr: "?rrr = [:a:]" by auto |
|
3308 |
show ?thesis unfolding 0 using c0 unfolding rrr 0 by simp |
|
3309 |
qed (insert Suc(4), auto) |
|
3310 |
note IH = Suc(1)[OF deg_rrr res this] |
|
3311 |
show ?case |
|
3312 |
proof (intro conjI) |
|
3313 |
show "r' = 0 \<or> degree r' < degree d" using IH by blast |
|
3314 |
show "smult (lc ^ Suc n) (d * q + r) = d * q' + r'" |
|
3315 |
unfolding IH[THEN conjunct2,symmetric] |
|
3316 |
by (simp add: field_simps smult_add_right) |
|
3317 |
qed |
|
3318 |
qed auto |
|
3319 |
||
3320 |
lemma pseudo_divmod: |
|
3321 |
assumes g: "g \<noteq> 0" and *: "pseudo_divmod f g = (q,r)" |
|
3322 |
shows "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" (is ?A) |
|
3323 |
and "r = 0 \<or> degree r < degree g" (is ?B) |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3324 |
proof - |
64795 | 3325 |
from *[unfolded pseudo_divmod_def Let_def] |
3326 |
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) |
|
3327 |
note main = pseudo_divmod_main[OF _ _ _ this, OF g refl le_refl] |
|
3328 |
have "1 + length (coeffs f) - length (coeffs g) = 1 + degree f - degree g \<or> |
|
3329 |
degree f = 0 \<and> 1 + length (coeffs f) - length (coeffs g) = 0 \<and> f = 0" using g |
|
3330 |
by (cases "f = 0"; cases "coeffs g", auto simp: degree_eq_length_coeffs) |
|
3331 |
note main = main[OF this] |
|
3332 |
from main show "r = 0 \<or> degree r < degree g" by auto |
|
3333 |
show "smult (coeff g (degree g) ^ (Suc (degree f) - degree g)) f = g * q + r" |
|
3334 |
by (subst main[THEN conjunct2, symmetric], simp add: degree_eq_length_coeffs, |
|
3335 |
insert g, cases "f = 0"; cases "coeffs g", auto) |
|
3336 |
qed |
|
3337 |
||
3338 |
definition "pseudo_mod_main lc r d dr n = snd (pseudo_divmod_main lc 0 r d dr n)" |
|
3339 |
||
3340 |
lemma snd_pseudo_divmod_main: |
|
3341 |
"snd (pseudo_divmod_main lc q r d dr n) = snd (pseudo_divmod_main lc q' r d dr n)" |
|
3342 |
by (induct n arbitrary: q q' lc r d dr; simp add: Let_def) |
|
3343 |
||
3344 |
definition pseudo_mod |
|
3345 |
:: "'a :: {comm_ring_1,semiring_1_no_zero_divisors} poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
3346 |
"pseudo_mod f g = snd (pseudo_divmod f g)" |
|
3347 |
||
3348 |
lemma pseudo_mod: |
|
3349 |
fixes f g |
|
3350 |
defines "r \<equiv> pseudo_mod f g" |
|
3351 |
assumes g: "g \<noteq> 0" |
|
3352 |
shows "\<exists> a q. a \<noteq> 0 \<and> smult a f = g * q + r" "r = 0 \<or> degree r < degree g" |
|
3353 |
proof - |
|
3354 |
let ?cg = "coeff g (degree g)" |
|
3355 |
let ?cge = "?cg ^ (Suc (degree f) - degree g)" |
|
3356 |
define a where "a = ?cge" |
|
3357 |
obtain q where pdm: "pseudo_divmod f g = (q,r)" using r_def[unfolded pseudo_mod_def] |
|
3358 |
by (cases "pseudo_divmod f g", auto) |
|
3359 |
from pseudo_divmod[OF g pdm] have id: "smult a f = g * q + r" and "r = 0 \<or> degree r < degree g" |
|
3360 |
unfolding a_def by auto |
|
3361 |
show "r = 0 \<or> degree r < degree g" by fact |
|
3362 |
from g have "a \<noteq> 0" unfolding a_def by auto |
|
3363 |
thus "\<exists> a q. a \<noteq> 0 \<and> smult a f = g * q + r" using id by auto |
|
3364 |
qed |
|
3365 |
||
3366 |
lemma fst_pseudo_divmod_main_as_divide_poly_main: |
|
3367 |
assumes d: "d \<noteq> 0" |
|
3368 |
defines lc: "lc \<equiv> coeff d (degree d)" |
|
3369 |
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" |
|
3370 |
proof(induct n arbitrary: q r dr) |
|
3371 |
case 0 then show ?case by simp |
|
3372 |
next |
|
3373 |
case (Suc n) |
|
3374 |
note lc0 = leading_coeff_neq_0[OF d, folded lc] |
|
3375 |
then have "pseudo_divmod_main lc q r d dr (Suc n) = |
|
3376 |
pseudo_divmod_main lc (smult lc q + monom (coeff r dr) n) |
|
3377 |
(smult lc r - monom (coeff r dr) n * d) d (dr - 1) n" |
|
3378 |
by (simp add: Let_def ac_simps) |
|
3379 |
also have "fst ... = divide_poly_main lc |
|
3380 |
(smult (lc^n) (smult lc q + monom (coeff r dr) n)) |
|
3381 |
(smult (lc^n) (smult lc r - monom (coeff r dr) n * d)) |
|
3382 |
d (dr - 1) n" |
|
3383 |
unfolding Suc[unfolded divide_poly_main.simps Let_def].. |
|
3384 |
also have "... = divide_poly_main lc (smult (lc ^ Suc n) q) |
|
3385 |
(smult (lc ^ Suc n) r) d dr (Suc n)" |
|
3386 |
unfolding smult_monom smult_distribs mult_smult_left[symmetric] |
|
3387 |
using lc0 by (simp add: Let_def ac_simps) |
|
3388 |
finally show ?case. |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3389 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3390 |
|
64795 | 3391 |
|
3392 |
subsubsection \<open>Division in polynomials over fields\<close> |
|
3393 |
||
3394 |
lemma pseudo_divmod_field: |
|
3395 |
assumes g: "(g::'a::field poly) \<noteq> 0" and *: "pseudo_divmod f g = (q,r)" |
|
3396 |
defines "c \<equiv> coeff g (degree g) ^ (Suc (degree f) - degree g)" |
|
3397 |
shows "f = g * smult (1/c) q + smult (1/c) r" |
|
3398 |
proof - |
|
3399 |
from leading_coeff_neq_0[OF g] have c0: "c \<noteq> 0" unfolding c_def by auto |
|
3400 |
from pseudo_divmod(1)[OF g *, folded c_def] |
|
3401 |
have "smult c f = g * q + r" by auto |
|
3402 |
also have "smult (1/c) ... = g * smult (1/c) q + smult (1/c) r" by (simp add: smult_add_right) |
|
3403 |
finally show ?thesis using c0 by auto |
|
3404 |
qed |
|
3405 |
||
3406 |
lemma divide_poly_main_field: |
|
3407 |
assumes d: "(d::'a::field poly) \<noteq> 0" |
|
3408 |
defines lc: "lc \<equiv> coeff d (degree d)" |
|
3409 |
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)" |
|
3410 |
unfolding lc |
|
3411 |
by(subst fst_pseudo_divmod_main_as_divide_poly_main, auto simp: d power_one_over) |
|
3412 |
||
3413 |
lemma divide_poly_field: |
|
3414 |
fixes f g :: "'a :: field poly" |
|
3415 |
defines "f' \<equiv> smult ((1 / coeff g (degree g)) ^ (Suc (degree f) - degree g)) f" |
|
3416 |
shows "(f::'a::field poly) div g = fst (pseudo_divmod f' g)" |
|
3417 |
proof (cases "g = 0") |
|
3418 |
case True show ?thesis |
|
3419 |
unfolding divide_poly_def pseudo_divmod_def Let_def f'_def True by (simp add: divide_poly_main_0) |
|
3420 |
next |
|
3421 |
case False |
|
3422 |
from leading_coeff_neq_0[OF False] have "degree f' = degree f" unfolding f'_def by auto |
|
3423 |
then show ?thesis |
|
3424 |
using length_coeffs_degree[of f'] length_coeffs_degree[of f] |
|
3425 |
unfolding divide_poly_def pseudo_divmod_def Let_def |
|
3426 |
divide_poly_main_field[OF False] |
|
3427 |
length_coeffs_degree[OF False] |
|
3428 |
f'_def |
|
3429 |
by force |
|
3430 |
qed |
|
3431 |
||
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3432 |
instantiation poly :: ("{semidom_divide_unit_factor, idom_divide}") normalization_semidom |
64795 | 3433 |
begin |
3434 |
||
3435 |
definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3436 |
where "unit_factor_poly p = [:unit_factor (lead_coeff p):]" |
64795 | 3437 |
|
3438 |
definition normalize_poly :: "'a poly \<Rightarrow> 'a poly" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3439 |
where "normalize p = p div [:unit_factor (lead_coeff p):]" |
64795 | 3440 |
|
3441 |
instance proof |
|
3442 |
fix p :: "'a poly" |
|
3443 |
show "unit_factor p * normalize p = p" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3444 |
proof (cases "p = 0") |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3445 |
case True |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3446 |
then show ?thesis |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3447 |
by (simp add: unit_factor_poly_def normalize_poly_def) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3448 |
next |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3449 |
case False |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3450 |
then have "lead_coeff p \<noteq> 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3451 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3452 |
then have *: "unit_factor (lead_coeff p) \<noteq> 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3453 |
using unit_factor_is_unit [of "lead_coeff p"] by auto |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3454 |
then have "unit_factor (lead_coeff p) dvd 1" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3455 |
by (auto intro: unit_factor_is_unit) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3456 |
then have **: "unit_factor (lead_coeff p) dvd c" for c |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3457 |
by (rule dvd_trans) simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3458 |
have ***: "unit_factor (lead_coeff p) * (c div unit_factor (lead_coeff p)) = c" for c |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3459 |
proof - |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3460 |
from ** obtain b where "c = unit_factor (lead_coeff p) * b" .. |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3461 |
then show ?thesis |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3462 |
using False * by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3463 |
qed |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3464 |
have "p div [:unit_factor (lead_coeff p):] = |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3465 |
map_poly (\<lambda>c. c div unit_factor (lead_coeff p)) p" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3466 |
by (simp add: const_poly_dvd_iff div_const_poly_conv_map_poly **) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3467 |
then show ?thesis |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3468 |
by (simp add: normalize_poly_def unit_factor_poly_def |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3469 |
smult_conv_map_poly map_poly_map_poly o_def ***) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3470 |
qed |
64795 | 3471 |
next |
3472 |
fix p :: "'a poly" |
|
3473 |
assume "is_unit p" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3474 |
then obtain c where p: "p = [:c:]" "c dvd 1" |
64795 | 3475 |
by (auto simp: is_unit_poly_iff) |
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3476 |
then show "unit_factor p = p" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3477 |
by (simp add: unit_factor_poly_def monom_0 is_unit_unit_factor) |
64795 | 3478 |
next |
3479 |
fix p :: "'a poly" assume "p \<noteq> 0" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3480 |
then show "is_unit (unit_factor p)" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3481 |
by (simp add: unit_factor_poly_def monom_0 is_unit_poly_iff unit_factor_is_unit) |
64795 | 3482 |
qed (simp_all add: normalize_poly_def unit_factor_poly_def monom_0 lead_coeff_mult unit_factor_mult) |
3483 |
||
3484 |
end |
|
3485 |
||
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3486 |
lemma normalize_poly_eq_map_poly: |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3487 |
"normalize p = map_poly (\<lambda>x. x div unit_factor (lead_coeff p)) p" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3488 |
proof - |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3489 |
have "[:unit_factor (lead_coeff p):] dvd p" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3490 |
by (metis unit_factor_poly_def unit_factor_self) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3491 |
then show ?thesis |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3492 |
by (simp add: normalize_poly_def div_const_poly_conv_map_poly) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3493 |
qed |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3494 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3495 |
lemma coeff_normalize [simp]: |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3496 |
"coeff (normalize p) n = coeff p n div unit_factor (lead_coeff p)" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3497 |
by (simp add: normalize_poly_eq_map_poly coeff_map_poly) |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3498 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3499 |
class field_unit_factor = field + unit_factor + |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3500 |
assumes unit_factor_field [simp]: "unit_factor = id" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3501 |
begin |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3502 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3503 |
subclass semidom_divide_unit_factor |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3504 |
proof |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3505 |
fix a |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3506 |
assume "a \<noteq> 0" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3507 |
then have "1 = a * inverse a" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3508 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3509 |
then have "a dvd 1" .. |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3510 |
then show "unit_factor a dvd 1" |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3511 |
by simp |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3512 |
qed simp_all |
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3513 |
|
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3514 |
end |
64795 | 3515 |
|
3516 |
lemma unit_factor_pCons: |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3517 |
"unit_factor (pCons a p) = (if p = 0 then [:unit_factor a:] else unit_factor p)" |
64795 | 3518 |
by (simp add: unit_factor_poly_def) |
3519 |
||
3520 |
lemma normalize_monom [simp]: |
|
3521 |
"normalize (monom a n) = monom (normalize a) n" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3522 |
by (cases "a = 0") (simp_all add: map_poly_monom normalize_poly_eq_map_poly degree_monom_eq) |
64795 | 3523 |
|
3524 |
lemma unit_factor_monom [simp]: |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3525 |
"unit_factor (monom a n) = [:unit_factor a:]" |
64795 | 3526 |
by (cases "a = 0") (simp_all add: unit_factor_poly_def degree_monom_eq) |
3527 |
||
3528 |
lemma normalize_const_poly: "normalize [:c:] = [:normalize c:]" |
|
64848
c50db2128048
slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents:
64811
diff
changeset
|
3529 |
by (simp add: normalize_poly_eq_map_poly map_poly_pCons) |
64795 | 3530 |
|
3531 |
lemma normalize_smult: "normalize (smult c p) = smult (normalize c) (normalize p)" |
|
3532 |
proof - |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3533 |
have "smult c p = [:c:] * p" by simp |
64795 | 3534 |
also have "normalize \<dots> = smult (normalize c) (normalize p)" |
3535 |
by (subst normalize_mult) (simp add: normalize_const_poly) |
|
3536 |
finally show ?thesis . |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3537 |
qed |
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3538 |
|
64795 | 3539 |
lemma smult_content_normalize_primitive_part [simp]: |
3540 |
"smult (content p) (normalize (primitive_part p)) = normalize p" |
|
62352
35a9e1cbb5b3
separated potentially conflicting type class instance into separate theory
haftmann
parents:
62351
diff
changeset
|
3541 |
proof - |
64795 | 3542 |
have "smult (content p) (normalize (primitive_part p)) = |
3543 |
normalize ([:content p:] * primitive_part p)" |
|
3544 |
by (subst normalize_mult) (simp_all add: normalize_const_poly) |
|
3545 |
also have "[:content p:] * primitive_part p = p" by simp |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3546 |
finally show ?thesis . |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3547 |
qed |
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3548 |
|
64795 | 3549 |
inductive eucl_rel_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly \<Rightarrow> bool" |
3550 |
where eucl_rel_poly_by0: "eucl_rel_poly x 0 (0, x)" |
|
3551 |
| eucl_rel_poly_dividesI: "y \<noteq> 0 \<Longrightarrow> x = q * y \<Longrightarrow> eucl_rel_poly x y (q, 0)" |
|
3552 |
| eucl_rel_poly_remainderI: "y \<noteq> 0 \<Longrightarrow> degree r < degree y |
|
3553 |
\<Longrightarrow> x = q * y + r \<Longrightarrow> eucl_rel_poly x y (q, r)" |
|
3554 |
||
3555 |
lemma eucl_rel_poly_iff: |
|
3556 |
"eucl_rel_poly x y (q, r) \<longleftrightarrow> |
|
3557 |
x = q * y + r \<and> |
|
3558 |
(if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" |
|
3559 |
by (auto elim: eucl_rel_poly.cases |
|
3560 |
intro: eucl_rel_poly_by0 eucl_rel_poly_dividesI eucl_rel_poly_remainderI) |
|
3561 |
||
3562 |
lemma eucl_rel_poly_0: |
|
3563 |
"eucl_rel_poly 0 y (0, 0)" |
|
3564 |
unfolding eucl_rel_poly_iff by simp |
|
3565 |
||
3566 |
lemma eucl_rel_poly_by_0: |
|
3567 |
"eucl_rel_poly x 0 (0, x)" |
|
3568 |
unfolding eucl_rel_poly_iff by simp |
|
3569 |
||
3570 |
lemma eucl_rel_poly_pCons: |
|
3571 |
assumes rel: "eucl_rel_poly x y (q, r)" |
|
3572 |
assumes y: "y \<noteq> 0" |
|
3573 |
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" |
|
3574 |
shows "eucl_rel_poly (pCons a x) y (pCons b q, pCons a r - smult b y)" |
|
3575 |
(is "eucl_rel_poly ?x y (?q, ?r)") |
|
3576 |
proof - |
|
3577 |
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" |
|
3578 |
using assms unfolding eucl_rel_poly_iff by simp_all |
|
3579 |
||
3580 |
have 1: "?x = ?q * y + ?r" |
|
3581 |
using b x by simp |
|
3582 |
||
3583 |
have 2: "?r = 0 \<or> degree ?r < degree y" |
|
3584 |
proof (rule eq_zero_or_degree_less) |
|
3585 |
show "degree ?r \<le> degree y" |
|
3586 |
proof (rule degree_diff_le) |
|
3587 |
show "degree (pCons a r) \<le> degree y" |
|
3588 |
using r by auto |
|
3589 |
show "degree (smult b y) \<le> degree y" |
|
3590 |
by (rule degree_smult_le) |
|
3591 |
qed |
|
3592 |
next |
|
3593 |
show "coeff ?r (degree y) = 0" |
|
3594 |
using \<open>y \<noteq> 0\<close> unfolding b by simp |
|
3595 |
qed |
|
3596 |
||
3597 |
from 1 2 show ?thesis |
|
3598 |
unfolding eucl_rel_poly_iff |
|
3599 |
using \<open>y \<noteq> 0\<close> by simp |
|
3600 |
qed |
|
3601 |
||
3602 |
lemma eucl_rel_poly_exists: "\<exists>q r. eucl_rel_poly x y (q, r)" |
|
3603 |
apply (cases "y = 0") |
|
3604 |
apply (fast intro!: eucl_rel_poly_by_0) |
|
3605 |
apply (induct x) |
|
3606 |
apply (fast intro!: eucl_rel_poly_0) |
|
3607 |
apply (fast intro!: eucl_rel_poly_pCons) |
|
3608 |
done |
|
3609 |
||
3610 |
lemma eucl_rel_poly_unique: |
|
3611 |
assumes 1: "eucl_rel_poly x y (q1, r1)" |
|
3612 |
assumes 2: "eucl_rel_poly x y (q2, r2)" |
|
3613 |
shows "q1 = q2 \<and> r1 = r2" |
|
3614 |
proof (cases "y = 0") |
|
3615 |
assume "y = 0" with assms show ?thesis |
|
3616 |
by (simp add: eucl_rel_poly_iff) |
|
3617 |
next |
|
3618 |
assume [simp]: "y \<noteq> 0" |
|
3619 |
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" |
|
3620 |
unfolding eucl_rel_poly_iff by simp_all |
|
3621 |
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" |
|
3622 |
unfolding eucl_rel_poly_iff by simp_all |
|
3623 |
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" |
|
3624 |
by (simp add: algebra_simps) |
|
3625 |
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y" |
|
3626 |
by (auto intro: degree_diff_less) |
|
3627 |
||
3628 |
show "q1 = q2 \<and> r1 = r2" |
|
3629 |
proof (rule ccontr) |
|
3630 |
assume "\<not> (q1 = q2 \<and> r1 = r2)" |
|
3631 |
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto |
|
3632 |
with r3 have "degree (r2 - r1) < degree y" by simp |
|
3633 |
also have "degree y \<le> degree (q1 - q2) + degree y" by simp |
|
3634 |
also have "\<dots> = degree ((q1 - q2) * y)" |
|
3635 |
using \<open>q1 \<noteq> q2\<close> by (simp add: degree_mult_eq) |
|
3636 |
also have "\<dots> = degree (r2 - r1)" |
|
3637 |
using q3 by simp |
|
3638 |
finally have "degree (r2 - r1) < degree (r2 - r1)" . |
|
3639 |
then show "False" by simp |
|
3640 |
qed |
|
3641 |
qed |
|
3642 |
||
3643 |
lemma eucl_rel_poly_0_iff: "eucl_rel_poly 0 y (q, r) \<longleftrightarrow> q = 0 \<and> r = 0" |
|
3644 |
by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_0) |
|
3645 |
||
3646 |
lemma eucl_rel_poly_by_0_iff: "eucl_rel_poly x 0 (q, r) \<longleftrightarrow> q = 0 \<and> r = x" |
|
3647 |
by (auto dest: eucl_rel_poly_unique intro: eucl_rel_poly_by_0) |
|
3648 |
||
3649 |
lemmas eucl_rel_poly_unique_div = eucl_rel_poly_unique [THEN conjunct1] |
|
3650 |
||
3651 |
lemmas eucl_rel_poly_unique_mod = eucl_rel_poly_unique [THEN conjunct2] |
|
3652 |
||
64861 | 3653 |
instantiation poly :: (field) semidom_modulo |
64795 | 3654 |
begin |
64861 | 3655 |
|
3656 |
definition modulo_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
|
3657 |
where mod_poly_def: "f mod g = (if g = 0 then f |
|
3658 |
else pseudo_mod (smult ((1 / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)" |
|
3659 |
||
3660 |
instance proof |
|
3661 |
fix x y :: "'a poly" |
|
3662 |
show "x div y * y + x mod y = x" |
|
3663 |
proof (cases "y = 0") |
|
3664 |
case True then show ?thesis |
|
3665 |
by (simp add: divide_poly_0 mod_poly_def) |
|
64795 | 3666 |
next |
3667 |
case False |
|
64861 | 3668 |
then have "pseudo_divmod (smult ((1 / lead_coeff y) ^ (Suc (degree x) - degree y)) x) y = |
3669 |
(x div y, x mod y)" |
|
3670 |
by (simp add: divide_poly_field mod_poly_def pseudo_mod_def) |
|
3671 |
from pseudo_divmod [OF False this] |
|
64795 | 3672 |
show ?thesis using False |
64861 | 3673 |
by (simp add: power_mult_distrib [symmetric] ac_simps) |
64795 | 3674 |
qed |
64861 | 3675 |
qed |
3676 |
||
3677 |
end |
|
3678 |
||
3679 |
lemma eucl_rel_poly: "eucl_rel_poly x y (x div y, x mod y)" |
|
3680 |
unfolding eucl_rel_poly_iff proof |
|
3681 |
show "x = x div y * y + x mod y" |
|
3682 |
by (simp add: div_mult_mod_eq) |
|
64795 | 3683 |
show "if y = 0 then x div y = 0 else x mod y = 0 \<or> degree (x mod y) < degree y" |
3684 |
proof (cases "y = 0") |
|
3685 |
case True then show ?thesis by auto |
|
3686 |
next |
|
3687 |
case False |
|
3688 |
with pseudo_mod[OF this] show ?thesis unfolding mod_poly_def by simp |
|
3689 |
qed |
|
3690 |
qed |
|
3691 |
||
3692 |
lemma div_poly_eq: |
|
3693 |
"eucl_rel_poly (x::'a::field poly) y (q, r) \<Longrightarrow> x div y = q" |
|
64861 | 3694 |
by(rule eucl_rel_poly_unique_div [OF eucl_rel_poly]) |
64795 | 3695 |
|
3696 |
lemma mod_poly_eq: |
|
3697 |
"eucl_rel_poly (x::'a::field poly) y (q, r) \<Longrightarrow> x mod y = r" |
|
64861 | 3698 |
by (rule eucl_rel_poly_unique_mod [OF eucl_rel_poly]) |
3699 |
||
3700 |
instance poly :: (field) ring_div |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3701 |
proof |
64795 | 3702 |
fix x y z :: "'a poly" |
3703 |
assume "y \<noteq> 0" |
|
3704 |
hence "eucl_rel_poly (x + z * y) y (z + x div y, x mod y)" |
|
3705 |
using eucl_rel_poly [of x y] |
|
3706 |
by (simp add: eucl_rel_poly_iff distrib_right) |
|
3707 |
thus "(x + z * y) div y = z + x div y" |
|
3708 |
by (rule div_poly_eq) |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
3709 |
next |
64795 | 3710 |
fix x y z :: "'a poly" |
3711 |
assume "x \<noteq> 0" |
|
3712 |
show "(x * y) div (x * z) = y div z" |
|
3713 |
proof (cases "y \<noteq> 0 \<and> z \<noteq> 0") |
|
3714 |
have "\<And>x::'a poly. eucl_rel_poly x 0 (0, x)" |
|
3715 |
by (rule eucl_rel_poly_by_0) |
|
3716 |
then have [simp]: "\<And>x::'a poly. x div 0 = 0" |
|
3717 |
by (rule div_poly_eq) |
|
3718 |
have "\<And>x::'a poly. eucl_rel_poly 0 x (0, 0)" |
|
3719 |
by (rule eucl_rel_poly_0) |
|
3720 |
then have [simp]: "\<And>x::'a poly. 0 div x = 0" |
|
3721 |
by (rule div_poly_eq) |
|
3722 |
case False then show ?thesis by auto |
|
3723 |
next |
|
3724 |
case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto |
|
3725 |
with \<open>x \<noteq> 0\<close> |
|
3726 |
have "\<And>q r. eucl_rel_poly y z (q, r) \<Longrightarrow> eucl_rel_poly (x * y) (x * z) (q, x * r)" |
|
3727 |
by (auto simp add: eucl_rel_poly_iff algebra_simps) |
|
3728 |
(rule classical, simp add: degree_mult_eq) |
|
3729 |
moreover from eucl_rel_poly have "eucl_rel_poly y z (y div z, y mod z)" . |
|
3730 |
ultimately have "eucl_rel_poly (x * y) (x * z) (y div z, x * (y mod z))" . |
|
3731 |
then show ?thesis by (simp add: div_poly_eq) |
|
3732 |
qed |
|
3733 |
qed |
|
3734 |
||
64811 | 3735 |
lemma div_pCons_eq: |
3736 |
"pCons a p div q = (if q = 0 then 0 |
|
3737 |
else pCons (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) |
|
3738 |
(p div q))" |
|
3739 |
using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p] |
|
3740 |
by (auto intro: div_poly_eq) |
|
3741 |
||
3742 |
lemma mod_pCons_eq: |
|
3743 |
"pCons a p mod q = (if q = 0 then pCons a p |
|
3744 |
else pCons a (p mod q) - smult (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) |
|
3745 |
q)" |
|
3746 |
using eucl_rel_poly_pCons [OF eucl_rel_poly _ refl, of q a p] |
|
3747 |
by (auto intro: mod_poly_eq) |
|
3748 |
||
3749 |
lemma div_mod_fold_coeffs: |
|
3750 |
"(p div q, p mod q) = (if q = 0 then (0, p) |
|
3751 |
else fold_coeffs (\<lambda>a (s, r). |
|
3752 |
let b = coeff (pCons a r) (degree q) / coeff q (degree q) |
|
3753 |
in (pCons b s, pCons a r - smult b q) |
|
3754 |
) p (0, 0))" |
|
3755 |
by (rule sym, induct p) (auto simp add: div_pCons_eq mod_pCons_eq Let_def) |
|
3756 |
||
64795 | 3757 |
lemma degree_mod_less: |
3758 |
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" |
|
3759 |
using eucl_rel_poly [of x y] |
|
3760 |
unfolding eucl_rel_poly_iff by simp |
|
3761 |
||
3762 |
lemma degree_mod_less': "b \<noteq> 0 \<Longrightarrow> a mod b \<noteq> 0 \<Longrightarrow> degree (a mod b) < degree b" |
|
3763 |
using degree_mod_less[of b a] by auto |
|
3764 |
||
3765 |
lemma div_poly_less: "degree (x::'a::field poly) < degree y \<Longrightarrow> x div y = 0" |
|
3766 |
proof - |
|
3767 |
assume "degree x < degree y" |
|
3768 |
hence "eucl_rel_poly x y (0, x)" |
|
3769 |
by (simp add: eucl_rel_poly_iff) |
|
3770 |
thus "x div y = 0" by (rule div_poly_eq) |
|
3771 |
qed |
|
3772 |
||
3773 |
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" |
|
3774 |
proof - |
|
3775 |
assume "degree x < degree y" |
|
3776 |
hence "eucl_rel_poly x y (0, x)" |
|
3777 |
by (simp add: eucl_rel_poly_iff) |
|
3778 |
thus "x mod y = x" by (rule mod_poly_eq) |
|
3779 |
qed |
|
3780 |
||
3781 |
lemma eucl_rel_poly_smult_left: |
|
3782 |
"eucl_rel_poly x y (q, r) |
|
3783 |
\<Longrightarrow> eucl_rel_poly (smult a x) y (smult a q, smult a r)" |
|
3784 |
unfolding eucl_rel_poly_iff by (simp add: smult_add_right) |
|
3785 |
||
3786 |
lemma div_smult_left: "(smult (a::'a::field) x) div y = smult a (x div y)" |
|
3787 |
by (rule div_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly) |
|
3788 |
||
3789 |
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" |
|
3790 |
by (rule mod_poly_eq, rule eucl_rel_poly_smult_left, rule eucl_rel_poly) |
|
3791 |
||
3792 |
lemma poly_div_minus_left [simp]: |
|
3793 |
fixes x y :: "'a::field poly" |
|
3794 |
shows "(- x) div y = - (x div y)" |
|
3795 |
using div_smult_left [of "- 1::'a"] by simp |
|
3796 |
||
3797 |
lemma poly_mod_minus_left [simp]: |
|
3798 |
fixes x y :: "'a::field poly" |
|
3799 |
shows "(- x) mod y = - (x mod y)" |
|
3800 |
using mod_smult_left [of "- 1::'a"] by simp |
|
3801 |
||
3802 |
lemma eucl_rel_poly_add_left: |
|
3803 |
assumes "eucl_rel_poly x y (q, r)" |
|
3804 |
assumes "eucl_rel_poly x' y (q', r')" |
|
3805 |
shows "eucl_rel_poly (x + x') y (q + q', r + r')" |
|
3806 |
using assms unfolding eucl_rel_poly_iff |
|
3807 |
by (auto simp add: algebra_simps degree_add_less) |
|
3808 |
||
3809 |
lemma poly_div_add_left: |
|
3810 |
fixes x y z :: "'a::field poly" |
|
3811 |
shows "(x + y) div z = x div z + y div z" |
|
3812 |
using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly] |
|
3813 |
by (rule div_poly_eq) |
|
3814 |
||
3815 |
lemma poly_mod_add_left: |
|
3816 |
fixes x y z :: "'a::field poly" |
|
3817 |
shows "(x + y) mod z = x mod z + y mod z" |
|
3818 |
using eucl_rel_poly_add_left [OF eucl_rel_poly eucl_rel_poly] |
|
3819 |
by (rule mod_poly_eq) |
|
3820 |
||
3821 |
lemma poly_div_diff_left: |
|
3822 |
fixes x y z :: "'a::field poly" |
|
3823 |
shows "(x - y) div z = x div z - y div z" |
|
3824 |
by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left) |
|
3825 |
||
3826 |
lemma poly_mod_diff_left: |
|
3827 |
fixes x y z :: "'a::field poly" |
|
3828 |
shows "(x - y) mod z = x mod z - y mod z" |
|
3829 |
by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left) |
|
3830 |
||
3831 |
lemma eucl_rel_poly_smult_right: |
|
3832 |
"a \<noteq> 0 \<Longrightarrow> eucl_rel_poly x y (q, r) |
|
3833 |
\<Longrightarrow> eucl_rel_poly x (smult a y) (smult (inverse a) q, r)" |
|
3834 |
unfolding eucl_rel_poly_iff by simp |
|
3835 |
||
3836 |
lemma div_smult_right: |
|
3837 |
"(a::'a::field) \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)" |
|
3838 |
by (rule div_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly) |
|
3839 |
||
3840 |
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y" |
|
3841 |
by (rule mod_poly_eq, erule eucl_rel_poly_smult_right, rule eucl_rel_poly) |
|
3842 |
||
3843 |
lemma poly_div_minus_right [simp]: |
|
3844 |
fixes x y :: "'a::field poly" |
|
3845 |
shows "x div (- y) = - (x div y)" |
|
3846 |
using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq) |
|
3847 |
||
3848 |
lemma poly_mod_minus_right [simp]: |
|
3849 |
fixes x y :: "'a::field poly" |
|
3850 |
shows "x mod (- y) = x mod y" |
|
3851 |
using mod_smult_right [of "- 1::'a"] by simp |
|
3852 |
||
3853 |
lemma eucl_rel_poly_mult: |
|
3854 |
"eucl_rel_poly x y (q, r) \<Longrightarrow> eucl_rel_poly q z (q', r') |
|
3855 |
\<Longrightarrow> eucl_rel_poly x (y * z) (q', y * r' + r)" |
|
3856 |
apply (cases "z = 0", simp add: eucl_rel_poly_iff) |
|
3857 |
apply (cases "y = 0", simp add: eucl_rel_poly_by_0_iff eucl_rel_poly_0_iff) |
|
3858 |
apply (cases "r = 0") |
|
3859 |
apply (cases "r' = 0") |
|
3860 |
apply (simp add: eucl_rel_poly_iff) |
|
3861 |
apply (simp add: eucl_rel_poly_iff field_simps degree_mult_eq) |
|
3862 |
apply (cases "r' = 0") |
|
3863 |
apply (simp add: eucl_rel_poly_iff degree_mult_eq) |
|
3864 |
apply (simp add: eucl_rel_poly_iff field_simps) |
|
3865 |
apply (simp add: degree_mult_eq degree_add_less) |
|
3866 |
done |
|
3867 |
||
3868 |
lemma poly_div_mult_right: |
|
3869 |
fixes x y z :: "'a::field poly" |
|
3870 |
shows "x div (y * z) = (x div y) div z" |
|
3871 |
by (rule div_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+) |
|
3872 |
||
3873 |
lemma poly_mod_mult_right: |
|
3874 |
fixes x y z :: "'a::field poly" |
|
3875 |
shows "x mod (y * z) = y * (x div y mod z) + x mod y" |
|
3876 |
by (rule mod_poly_eq, rule eucl_rel_poly_mult, (rule eucl_rel_poly)+) |
|
3877 |
||
3878 |
lemma mod_pCons: |
|
3879 |
fixes a and x |
|
3880 |
assumes y: "y \<noteq> 0" |
|
3881 |
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" |
|
3882 |
shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" |
|
3883 |
unfolding b |
|
3884 |
apply (rule mod_poly_eq) |
|
3885 |
apply (rule eucl_rel_poly_pCons [OF eucl_rel_poly y refl]) |
|
3886 |
done |
|
3887 |
||
3888 |
||
3889 |
subsubsection \<open>List-based versions for fast implementation\<close> |
|
3890 |
(* Subsection by: |
|
3891 |
Sebastiaan Joosten |
|
3892 |
René Thiemann |
|
3893 |
Akihisa Yamada |
|
3894 |
*) |
|
3895 |
fun minus_poly_rev_list :: "'a :: group_add list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
3896 |
"minus_poly_rev_list (x # xs) (y # ys) = (x - y) # (minus_poly_rev_list xs ys)" |
|
3897 |
| "minus_poly_rev_list xs [] = xs" |
|
3898 |
| "minus_poly_rev_list [] (y # ys) = []" |
|
3899 |
||
3900 |
fun pseudo_divmod_main_list :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list |
|
3901 |
\<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list" where |
|
3902 |
"pseudo_divmod_main_list lc q r d (Suc n) = (let |
|
3903 |
rr = map (op * lc) r; |
|
3904 |
a = hd r; |
|
3905 |
qqq = cCons a (map (op * lc) q); |
|
3906 |
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d)) |
|
3907 |
in pseudo_divmod_main_list lc qqq rrr d n)" |
|
3908 |
| "pseudo_divmod_main_list lc q r d 0 = (q,r)" |
|
3909 |
||
3910 |
fun pseudo_mod_main_list :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list |
|
3911 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
|
3912 |
"pseudo_mod_main_list lc r d (Suc n) = (let |
|
3913 |
rr = map (op * lc) r; |
|
3914 |
a = hd r; |
|
3915 |
rrr = tl (if a = 0 then rr else minus_poly_rev_list rr (map (op * a) d)) |
|
3916 |
in pseudo_mod_main_list lc rrr d n)" |
|
3917 |
| "pseudo_mod_main_list lc r d 0 = r" |
|
3918 |
||
3919 |
||
3920 |
fun divmod_poly_one_main_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list |
|
3921 |
\<Rightarrow> nat \<Rightarrow> 'a list \<times> 'a list" where |
|
3922 |
"divmod_poly_one_main_list q r d (Suc n) = (let |
|
3923 |
a = hd r; |
|
3924 |
qqq = cCons a q; |
|
3925 |
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d)) |
|
3926 |
in divmod_poly_one_main_list qqq rr d n)" |
|
3927 |
| "divmod_poly_one_main_list q r d 0 = (q,r)" |
|
3928 |
||
3929 |
fun mod_poly_one_main_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list |
|
3930 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
|
3931 |
"mod_poly_one_main_list r d (Suc n) = (let |
|
3932 |
a = hd r; |
|
3933 |
rr = tl (if a = 0 then r else minus_poly_rev_list r (map (op * a) d)) |
|
3934 |
in mod_poly_one_main_list rr d n)" |
|
3935 |
| "mod_poly_one_main_list r d 0 = r" |
|
3936 |
||
3937 |
definition pseudo_divmod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list \<times> 'a list" where |
|
3938 |
"pseudo_divmod_list p q = |
|
3939 |
(if q = [] then ([],p) else |
|
3940 |
(let rq = rev q; |
|
3941 |
(qu,re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in |
|
3942 |
(qu,rev re)))" |
|
3943 |
||
3944 |
definition pseudo_mod_list :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
|
3945 |
"pseudo_mod_list p q = |
|
3946 |
(if q = [] then p else |
|
3947 |
(let rq = rev q; |
|
3948 |
re = pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q) in |
|
3949 |
(rev re)))" |
|
3950 |
||
3951 |
lemma minus_zero_does_nothing: |
|
3952 |
"(minus_poly_rev_list x (map (op * 0) y)) = (x :: 'a :: ring list)" |
|
3953 |
by(induct x y rule: minus_poly_rev_list.induct, auto) |
|
3954 |
||
3955 |
lemma length_minus_poly_rev_list[simp]: |
|
3956 |
"length (minus_poly_rev_list xs ys) = length xs" |
|
3957 |
by(induct xs ys rule: minus_poly_rev_list.induct, auto) |
|
3958 |
||
3959 |
lemma if_0_minus_poly_rev_list: |
|
3960 |
"(if a = 0 then x else minus_poly_rev_list x (map (op * a) y)) |
|
3961 |
= minus_poly_rev_list x (map (op * (a :: 'a :: ring)) y)" |
|
3962 |
by(cases "a=0",simp_all add:minus_zero_does_nothing) |
|
3963 |
||
3964 |
lemma Poly_append: |
|
3965 |
"Poly ((a::'a::comm_semiring_1 list) @ b) = Poly a + monom 1 (length a) * Poly b" |
|
3966 |
by (induct a,auto simp: monom_0 monom_Suc) |
|
3967 |
||
3968 |
lemma minus_poly_rev_list: "length p \<ge> length (q :: 'a :: comm_ring_1 list) \<Longrightarrow> |
|
3969 |
Poly (rev (minus_poly_rev_list (rev p) (rev q))) |
|
3970 |
= Poly p - monom 1 (length p - length q) * Poly q" |
|
3971 |
proof (induct "rev p" "rev q" arbitrary: p q rule: minus_poly_rev_list.induct) |
|
3972 |
case (1 x xs y ys) |
|
3973 |
have "length (rev q) \<le> length (rev p)" using 1 by simp |
|
3974 |
from this[folded 1(2,3)] have ys_xs:"length ys \<le> length xs" by simp |
|
3975 |
hence a:"Poly (rev (minus_poly_rev_list xs ys)) = |
|
3976 |
Poly (rev xs) - monom 1 (length xs - length ys) * Poly (rev ys)" |
|
3977 |
by(subst "1.hyps"(1)[of "rev xs" "rev ys", unfolded rev_rev_ident length_rev],auto) |
|
3978 |
have "Poly p - monom 1 (length p - length q) * Poly q |
|
3979 |
= Poly (rev (rev p)) - monom 1 (length (rev (rev p)) - length (rev (rev q))) * Poly (rev (rev q))" |
|
3980 |
by simp |
|
3981 |
also have "\<dots> = Poly (rev (x # xs)) - monom 1 (length (x # xs) - length (y # ys)) * Poly (rev (y # ys))" |
|
3982 |
unfolding 1(2,3) by simp |
|
3983 |
also have "\<dots> = Poly (rev xs) + monom x (length xs) - |
|
3984 |
(monom 1 (length xs - length ys) * Poly (rev ys) + monom y (length xs))" using ys_xs |
|
3985 |
by (simp add:Poly_append distrib_left mult_monom smult_monom) |
|
3986 |
also have "\<dots> = Poly (rev (minus_poly_rev_list xs ys)) + monom (x - y) (length xs)" |
|
3987 |
unfolding a diff_monom[symmetric] by(simp) |
|
3988 |
finally show ?case |
|
3989 |
unfolding 1(2,3)[symmetric] by (simp add: smult_monom Poly_append) |
|
3990 |
qed auto |
|
3991 |
||
3992 |
lemma smult_monom_mult: "smult a (monom b n * f) = monom (a * b) n * f" |
|
3993 |
using smult_monom [of a _ n] by (metis mult_smult_left) |
|
3994 |
||
3995 |
lemma head_minus_poly_rev_list: |
|
3996 |
"length d \<le> length r \<Longrightarrow> d\<noteq>[] \<Longrightarrow> |
|
3997 |
hd (minus_poly_rev_list (map (op * (last d :: 'a :: comm_ring)) r) (map (op * (hd r)) (rev d))) = 0" |
|
3998 |
proof(induct r) |
|
3999 |
case (Cons a rs) |
|
4000 |
thus ?case by(cases "rev d", simp_all add:ac_simps) |
|
4001 |
qed simp |
|
4002 |
||
4003 |
lemma Poly_map: "Poly (map (op * a) p) = smult a (Poly p)" |
|
4004 |
proof (induct p) |
|
4005 |
case(Cons x xs) thus ?case by (cases "Poly xs = 0",auto) |
|
4006 |
qed simp |
|
4007 |
||
4008 |
lemma last_coeff_is_hd: "xs \<noteq> [] \<Longrightarrow> coeff (Poly xs) (length xs - 1) = hd (rev xs)" |
|
4009 |
by (simp_all add: hd_conv_nth rev_nth nth_default_nth nth_append) |
|
4010 |
||
4011 |
lemma pseudo_divmod_main_list_invar : |
|
4012 |
assumes leading_nonzero:"last d \<noteq> 0" |
|
4013 |
and lc:"last d = lc" |
|
4014 |
and dNonempty:"d \<noteq> []" |
|
4015 |
and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q',rev r')" |
|
4016 |
and "n = (1 + length r - length d)" |
|
4017 |
shows |
|
4018 |
"pseudo_divmod_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n = |
|
4019 |
(Poly q', Poly r')" |
|
4020 |
using assms(4-) |
|
4021 |
proof(induct "n" arbitrary: r q) |
|
4022 |
case (Suc n r q) |
|
4023 |
have ifCond: "\<not> Suc (length r) \<le> length d" using Suc.prems by simp |
|
4024 |
have rNonempty:"r \<noteq> []" |
|
4025 |
using ifCond dNonempty using Suc_leI length_greater_0_conv list.size(3) by fastforce |
|
4026 |
let ?a = "(hd (rev r))" |
|
4027 |
let ?rr = "map (op * lc) (rev r)" |
|
4028 |
let ?rrr = "rev (tl (minus_poly_rev_list ?rr (map (op * ?a) (rev d))))" |
|
4029 |
let ?qq = "cCons ?a (map (op * lc) q)" |
|
4030 |
have n: "n = (1 + length r - length d - 1)" |
|
4031 |
using ifCond Suc(3) by simp |
|
4032 |
have rr_val:"(length ?rrr) = (length r - 1)" using ifCond by auto |
|
4033 |
hence rr_smaller: "(1 + length r - length d - 1) = (1 + length ?rrr - length d)" |
|
4034 |
using rNonempty ifCond unfolding One_nat_def by auto |
|
4035 |
from ifCond have id: "Suc (length r) - length d = Suc (length r - length d)" by auto |
|
4036 |
have "pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) (1 + length r - length d - 1) = (q', rev r')" |
|
4037 |
using Suc.prems ifCond by (simp add:Let_def if_0_minus_poly_rev_list id) |
|
4038 |
hence v:"pseudo_divmod_main_list lc ?qq (rev ?rrr) (rev d) n = (q', rev r')" |
|
4039 |
using n by auto |
|
4040 |
have sucrr:"Suc (length r) - length d = Suc (length r - length d)" |
|
4041 |
using Suc_diff_le ifCond not_less_eq_eq by blast |
|
4042 |
have n_ok : "n = 1 + (length ?rrr) - length d" using Suc(3) rNonempty by simp |
|
4043 |
have cong: "\<And> x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow> |
|
4044 |
pseudo_divmod_main lc x1 x2 x3 x4 n = pseudo_divmod_main lc y1 y2 y3 y4 n" by simp |
|
4045 |
have hd_rev:"coeff (Poly r) (length r - Suc 0) = hd (rev r)" |
|
4046 |
using last_coeff_is_hd[OF rNonempty] by simp |
|
4047 |
show ?case unfolding Suc.hyps(1)[OF v n_ok, symmetric] pseudo_divmod_main.simps Let_def |
|
4048 |
proof (rule cong[OF _ _ refl], goal_cases) |
|
4049 |
case 1 |
|
4050 |
show ?case unfolding monom_Suc hd_rev[symmetric] |
|
4051 |
by (simp add: smult_monom Poly_map) |
|
4052 |
next |
|
4053 |
case 2 |
|
4054 |
show ?case |
|
4055 |
proof (subst Poly_on_rev_starting_with_0, goal_cases) |
|
4056 |
show "hd (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))) = 0" |
|
4057 |
by (fold lc, subst head_minus_poly_rev_list, insert ifCond dNonempty,auto) |
|
4058 |
from ifCond have "length d \<le> length r" by simp |
|
4059 |
then show "smult lc (Poly r) - monom (coeff (Poly r) (length r - 1)) n * Poly d = |
|
4060 |
Poly (rev (minus_poly_rev_list (map (op * lc) (rev r)) (map (op * (hd (rev r))) (rev d))))" |
|
4061 |
by (fold rev_map) (auto simp add: n smult_monom_mult Poly_map hd_rev [symmetric] |
|
4062 |
minus_poly_rev_list) |
|
4063 |
qed |
|
4064 |
qed simp |
|
4065 |
qed simp |
|
4066 |
||
4067 |
lemma pseudo_divmod_impl[code]: "pseudo_divmod (f::'a::comm_ring_1 poly) g = |
|
4068 |
map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))" |
|
4069 |
proof (cases "g=0") |
|
4070 |
case False |
|
4071 |
hence coeffs_g_nonempty:"(coeffs g) \<noteq> []" by simp |
|
4072 |
hence lastcoeffs:"last (coeffs g) = coeff g (degree g)" |
|
4073 |
by (simp add: hd_rev last_coeffs_eq_coeff_degree not_0_coeffs_not_Nil) |
|
4074 |
obtain q r where qr: "pseudo_divmod_main_list |
|
4075 |
(last (coeffs g)) (rev []) |
|
4076 |
(rev (coeffs f)) (rev (coeffs g)) |
|
4077 |
(1 + length (coeffs f) - |
|
4078 |
length (coeffs g)) = (q,rev (rev r))" by force |
|
4079 |
then have qr': "pseudo_divmod_main_list |
|
4080 |
(hd (rev (coeffs g))) [] |
|
4081 |
(rev (coeffs f)) (rev (coeffs g)) |
|
4082 |
(1 + length (coeffs f) - |
|
4083 |
length (coeffs g)) = (q,r)" using hd_rev[OF coeffs_g_nonempty] by(auto) |
|
4084 |
from False have cg: "(coeffs g = []) = False" by auto |
|
4085 |
have last_non0:"last (coeffs g) \<noteq> 0" using False by (simp add:last_coeffs_not_0) |
|
4086 |
show ?thesis |
|
4087 |
unfolding pseudo_divmod_def pseudo_divmod_list_def Let_def qr' map_prod_def split cg if_False |
|
4088 |
pseudo_divmod_main_list_invar[OF last_non0 _ _ qr,unfolded lastcoeffs,simplified,symmetric,OF False] |
|
4089 |
poly_of_list_def |
|
4090 |
using False by (auto simp: degree_eq_length_coeffs) |
|
4091 |
next |
|
4092 |
case True |
|
4093 |
show ?thesis unfolding True unfolding pseudo_divmod_def pseudo_divmod_list_def |
|
4094 |
by auto |
|
4095 |
qed |
|
4096 |
||
4097 |
lemma pseudo_mod_main_list: "snd (pseudo_divmod_main_list l q |
|
4098 |
xs ys n) = pseudo_mod_main_list l xs ys n" |
|
4099 |
by (induct n arbitrary: l q xs ys, auto simp: Let_def) |
|
4100 |
||
4101 |
lemma pseudo_mod_impl[code]: "pseudo_mod f g = |
|
4102 |
poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))" |
|
4103 |
proof - |
|
4104 |
have snd_case: "\<And> f g p. snd ((\<lambda> (x,y). (f x, g y)) p) = g (snd p)" |
|
4105 |
by auto |
|
4106 |
show ?thesis |
|
4107 |
unfolding pseudo_mod_def pseudo_divmod_impl pseudo_divmod_list_def |
|
4108 |
pseudo_mod_list_def Let_def |
|
4109 |
by (simp add: snd_case pseudo_mod_main_list) |
|
4110 |
qed |
|
4111 |
||
4112 |
||
4113 |
(* *************** *) |
|
4114 |
subsubsection \<open>Improved Code-Equations for Polynomial (Pseudo) Division\<close> |
|
4115 |
||
64811 | 4116 |
lemma pdivmod_pdivmodrel: "eucl_rel_poly p q (r, s) \<longleftrightarrow> (p div q, p mod q) = (r, s)" |
4117 |
by (metis eucl_rel_poly eucl_rel_poly_unique) |
|
4118 |
||
4119 |
lemma pdivmod_via_pseudo_divmod: "(f div g, f mod g) = (if g = 0 then (0,f) |
|
64795 | 4120 |
else let |
4121 |
ilc = inverse (coeff g (degree g)); |
|
4122 |
h = smult ilc g; |
|
4123 |
(q,r) = pseudo_divmod f h |
|
4124 |
in (smult ilc q, r))" (is "?l = ?r") |
|
4125 |
proof (cases "g = 0") |
|
4126 |
case False |
|
4127 |
define lc where "lc = inverse (coeff g (degree g))" |
|
4128 |
define h where "h = smult lc g" |
|
4129 |
from False have h1: "coeff h (degree h) = 1" and lc: "lc \<noteq> 0" unfolding h_def lc_def by auto |
|
4130 |
hence h0: "h \<noteq> 0" by auto |
|
4131 |
obtain q r where p: "pseudo_divmod f h = (q,r)" by force |
|
4132 |
from False have id: "?r = (smult lc q, r)" |
|
4133 |
unfolding Let_def h_def[symmetric] lc_def[symmetric] p by auto |
|
4134 |
from pseudo_divmod[OF h0 p, unfolded h1] |
|
4135 |
have f: "f = h * q + r" and r: "r = 0 \<or> degree r < degree h" by auto |
|
4136 |
have "eucl_rel_poly f h (q, r)" unfolding eucl_rel_poly_iff using f r h0 by auto |
|
64811 | 4137 |
hence "(f div h, f mod h) = (q,r)" by (simp add: pdivmod_pdivmodrel) |
4138 |
hence "(f div g, f mod g) = (smult lc q, r)" |
|
4139 |
unfolding h_def div_smult_right[OF lc] mod_smult_right[OF lc] |
|
64795 | 4140 |
using lc by auto |
4141 |
with id show ?thesis by auto |
|
64811 | 4142 |
qed simp |
4143 |
||
4144 |
lemma pdivmod_via_pseudo_divmod_list: "(f div g, f mod g) = (let |
|
64795 | 4145 |
cg = coeffs g |
4146 |
in if cg = [] then (0,f) |
|
4147 |
else let |
|
4148 |
cf = coeffs f; |
|
4149 |
ilc = inverse (last cg); |
|
4150 |
ch = map (op * ilc) cg; |
|
4151 |
(q,r) = pseudo_divmod_main_list 1 [] (rev cf) (rev ch) (1 + length cf - length cg) |
|
4152 |
in (poly_of_list (map (op * ilc) q), poly_of_list (rev r)))" |
|
4153 |
proof - |
|
4154 |
note d = pdivmod_via_pseudo_divmod |
|
4155 |
pseudo_divmod_impl pseudo_divmod_list_def |
|
4156 |
show ?thesis |
|
4157 |
proof (cases "g = 0") |
|
4158 |
case True thus ?thesis unfolding d by auto |
|
4159 |
next |
|
4160 |
case False |
|
4161 |
define ilc where "ilc = inverse (coeff g (degree g))" |
|
4162 |
from False have ilc: "ilc \<noteq> 0" unfolding ilc_def by auto |
|
4163 |
with False have id: "(g = 0) = False" "(coeffs g = []) = False" |
|
4164 |
"last (coeffs g) = coeff g (degree g)" |
|
4165 |
"(coeffs (smult ilc g) = []) = False" |
|
4166 |
by (auto simp: last_coeffs_eq_coeff_degree) |
|
4167 |
have id2: "hd (rev (coeffs (smult ilc g))) = 1" |
|
4168 |
by (subst hd_rev, insert id ilc, auto simp: coeffs_smult, subst last_map, auto simp: id ilc_def) |
|
4169 |
have id3: "length (coeffs (smult ilc g)) = length (coeffs g)" |
|
4170 |
"rev (coeffs (smult ilc g)) = rev (map (op * ilc) (coeffs g))" unfolding coeffs_smult using ilc by auto |
|
4171 |
obtain q r where pair: "pseudo_divmod_main_list 1 [] (rev (coeffs f)) (rev (map (op * ilc) (coeffs g))) |
|
4172 |
(1 + length (coeffs f) - length (coeffs g)) = (q,r)" by force |
|
4173 |
show ?thesis unfolding d Let_def id if_False ilc_def[symmetric] map_prod_def[symmetric] id2 |
|
4174 |
unfolding id3 pair map_prod_def split by (auto simp: Poly_map) |
|
4175 |
qed |
|
4176 |
qed |
|
4177 |
||
4178 |
lemma pseudo_divmod_main_list_1: "pseudo_divmod_main_list 1 = divmod_poly_one_main_list" |
|
4179 |
proof (intro ext, goal_cases) |
|
4180 |
case (1 q r d n) |
|
4181 |
{ |
|
4182 |
fix xs :: "'a list" |
|
4183 |
have "map (op * 1) xs = xs" by (induct xs, auto) |
|
4184 |
} note [simp] = this |
|
4185 |
show ?case by (induct n arbitrary: q r d, auto simp: Let_def) |
|
4186 |
qed |
|
4187 |
||
4188 |
fun divide_poly_main_list :: "'a::idom_divide \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list |
|
4189 |
\<Rightarrow> nat \<Rightarrow> 'a list" where |
|
4190 |
"divide_poly_main_list lc q r d (Suc n) = (let |
|
4191 |
cr = hd r |
|
4192 |
in if cr = 0 then divide_poly_main_list lc (cCons cr q) (tl r) d n else let |
|
4193 |
a = cr div lc; |
|
4194 |
qq = cCons a q; |
|
4195 |
rr = minus_poly_rev_list r (map (op * a) d) |
|
4196 |
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" |
|
4197 |
| "divide_poly_main_list lc q r d 0 = q" |
|
4198 |
||
4199 |
lemma divide_poly_main_list_simp[simp]: "divide_poly_main_list lc q r d (Suc n) = (let |
|
4200 |
cr = hd r; |
|
4201 |
a = cr div lc; |
|
4202 |
qq = cCons a q; |
|
4203 |
rr = minus_poly_rev_list r (map (op * a) d) |
|
4204 |
in if hd rr = 0 then divide_poly_main_list lc qq (tl rr) d n else [])" |
|
4205 |
by (simp add: Let_def minus_zero_does_nothing) |
|
4206 |
||
4207 |
declare divide_poly_main_list.simps(1)[simp del] |
|
4208 |
||
4209 |
definition divide_poly_list :: "'a::idom_divide poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
4210 |
"divide_poly_list f g = |
|
4211 |
(let cg = coeffs g |
|
4212 |
in if cg = [] then g |
|
4213 |
else let cf = coeffs f; cgr = rev cg |
|
4214 |
in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))" |
|
4215 |
||
64811 | 4216 |
lemmas pdivmod_via_divmod_list = pdivmod_via_pseudo_divmod_list[unfolded pseudo_divmod_main_list_1] |
64795 | 4217 |
|
4218 |
lemma mod_poly_one_main_list: "snd (divmod_poly_one_main_list q r d n) = mod_poly_one_main_list r d n" |
|
4219 |
by (induct n arbitrary: q r d, auto simp: Let_def) |
|
4220 |
||
4221 |
lemma mod_poly_code[code]: "f mod g = |
|
4222 |
(let cg = coeffs g |
|
4223 |
in if cg = [] then f |
|
4224 |
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg; |
|
4225 |
r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg) |
|
4226 |
in poly_of_list (rev r))" (is "?l = ?r") |
|
4227 |
proof - |
|
64811 | 4228 |
have "snd (f div g, f mod g) = ?r" unfolding pdivmod_via_divmod_list Let_def |
4229 |
mod_poly_one_main_list [symmetric, of _ _ _ Nil] by (auto split: prod.splits) |
|
4230 |
then show ?thesis |
|
4231 |
by simp |
|
64795 | 4232 |
qed |
4233 |
||
4234 |
definition div_field_poly_impl :: "'a :: field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
4235 |
"div_field_poly_impl f g = ( |
|
4236 |
let cg = coeffs g |
|
4237 |
in if cg = [] then 0 |
|
4238 |
else let cf = coeffs f; ilc = inverse (last cg); ch = map (op * ilc) cg; |
|
4239 |
q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg)) |
|
4240 |
in poly_of_list ((map (op * ilc) q)))" |
|
4241 |
||
4242 |
text \<open>We do not declare the following lemma as code equation, since then polynomial division |
|
4243 |
on non-fields will no longer be executable. However, a code-unfold is possible, since |
|
4244 |
\<open>div_field_poly_impl\<close> is a bit more efficient than the generic polynomial division.\<close> |
|
4245 |
lemma div_field_poly_impl[code_unfold]: "op div = div_field_poly_impl" |
|
4246 |
proof (intro ext) |
|
4247 |
fix f g :: "'a poly" |
|
64811 | 4248 |
have "fst (f div g, f mod g) = div_field_poly_impl f g" unfolding |
64795 | 4249 |
div_field_poly_impl_def pdivmod_via_divmod_list Let_def by (auto split: prod.splits) |
64811 | 4250 |
then show "f div g = div_field_poly_impl f g" |
4251 |
by simp |
|
64795 | 4252 |
qed |
4253 |
||
4254 |
lemma divide_poly_main_list: |
|
4255 |
assumes lc0: "lc \<noteq> 0" |
|
4256 |
and lc:"last d = lc" |
|
4257 |
and d:"d \<noteq> []" |
|
4258 |
and "n = (1 + length r - length d)" |
|
4259 |
shows |
|
4260 |
"Poly (divide_poly_main_list lc q (rev r) (rev d) n) = |
|
4261 |
divide_poly_main lc (monom 1 n * Poly q) (Poly r) (Poly d) (length r - 1) n" |
|
4262 |
using assms(4-) |
|
4263 |
proof(induct "n" arbitrary: r q) |
|
4264 |
case (Suc n r q) |
|
4265 |
have ifCond: "\<not> Suc (length r) \<le> length d" using Suc.prems by simp |
|
4266 |
have r: "r \<noteq> []" |
|
4267 |
using ifCond d using Suc_leI length_greater_0_conv list.size(3) by fastforce |
|
4268 |
then obtain rr lcr where r: "r = rr @ [lcr]" by (cases r rule: rev_cases, auto) |
|
4269 |
from d lc obtain dd where d: "d = dd @ [lc]" |
|
4270 |
by (cases d rule: rev_cases, auto) |
|
4271 |
from Suc(2) ifCond have n: "n = 1 + length rr - length d" by (auto simp: r) |
|
4272 |
from ifCond have len: "length dd \<le> length rr" by (simp add: r d) |
|
4273 |
show ?case |
|
4274 |
proof (cases "lcr div lc * lc = lcr") |
|
4275 |
case False |
|
4276 |
thus ?thesis unfolding Suc(2)[symmetric] using r d |
|
4277 |
by (auto simp add: Let_def nth_default_append) |
|
4278 |
next |
|
4279 |
case True |
|
4280 |
hence id: |
|
4281 |
"?thesis = (Poly (divide_poly_main_list lc (cCons (lcr div lc) q) |
|
4282 |
(rev (rev (minus_poly_rev_list (rev rr) (rev (map (op * (lcr div lc)) dd))))) (rev d) n) = |
|
4283 |
divide_poly_main lc |
|
4284 |
(monom 1 (Suc n) * Poly q + monom (lcr div lc) n) |
|
4285 |
(Poly r - monom (lcr div lc) n * Poly d) |
|
4286 |
(Poly d) (length rr - 1) n)" |
|
4287 |
using r d |
|
4288 |
by (cases r rule: rev_cases; cases "d" rule: rev_cases; |
|
4289 |
auto simp add: Let_def rev_map nth_default_append) |
|
4290 |
have cong: "\<And> x1 x2 x3 x4 y1 y2 y3 y4. x1 = y1 \<Longrightarrow> x2 = y2 \<Longrightarrow> x3 = y3 \<Longrightarrow> x4 = y4 \<Longrightarrow> |
|
4291 |
divide_poly_main lc x1 x2 x3 x4 n = divide_poly_main lc y1 y2 y3 y4 n" by simp |
|
4292 |
show ?thesis unfolding id |
|
4293 |
proof (subst Suc(1), simp add: n, |
|
4294 |
subst minus_poly_rev_list, force simp: len, rule cong[OF _ _ refl], goal_cases) |
|
4295 |
case 2 |
|
4296 |
have "monom lcr (length rr) = monom (lcr div lc) (length rr - length dd) * monom lc (length dd)" |
|
4297 |
by (simp add: mult_monom len True) |
|
4298 |
thus ?case unfolding r d Poly_append n ring_distribs |
|
4299 |
by (auto simp: Poly_map smult_monom smult_monom_mult) |
|
4300 |
qed (auto simp: len monom_Suc smult_monom) |
|
4301 |
qed |
|
4302 |
qed simp |
|
4303 |
||
4304 |
||
4305 |
lemma divide_poly_list[code]: "f div g = divide_poly_list f g" |
|
4306 |
proof - |
|
4307 |
note d = divide_poly_def divide_poly_list_def |
|
4308 |
show ?thesis |
|
4309 |
proof (cases "g = 0") |
|
4310 |
case True |
|
4311 |
show ?thesis unfolding d True by auto |
|
4312 |
next |
|
4313 |
case False |
|
4314 |
then obtain cg lcg where cg: "coeffs g = cg @ [lcg]" by (cases "coeffs g" rule: rev_cases, auto) |
|
4315 |
with False have id: "(g = 0) = False" "(cg @ [lcg] = []) = False" by auto |
|
4316 |
from cg False have lcg: "coeff g (degree g) = lcg" |
|
4317 |
using last_coeffs_eq_coeff_degree last_snoc by force |
|
4318 |
with False have lcg0: "lcg \<noteq> 0" by auto |
|
4319 |
from cg have ltp: "Poly (cg @ [lcg]) = g" |
|
4320 |
using Poly_coeffs [of g] by auto |
|
4321 |
show ?thesis unfolding d cg Let_def id if_False poly_of_list_def |
|
4322 |
by (subst divide_poly_main_list, insert False cg lcg0, auto simp: lcg ltp, |
|
4323 |
simp add: degree_eq_length_coeffs) |
|
4324 |
qed |
|
63317
ca187a9f66da
Various additions to polynomials, FPSs, Gamma function
eberlm
parents:
63145
diff
changeset
|
4325 |
qed |
52380 | 4326 |
|
4327 |
no_notation cCons (infixr "##" 65) |
|
31663 | 4328 |
|
29478 | 4329 |
end |