author | huffman |
Fri, 13 Feb 2009 14:12:00 -0800 | |
changeset 29904 | 856f16a3b436 |
parent 29878 | 06efd6e731c6 |
child 29977 | d76b830366bc |
permissions | -rw-r--r-- |
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(* Title: HOL/Polynomial.thy |
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Author: Brian Huffman |
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Based on an earlier development by Clemens Ballarin |
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*) |
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header {* Univariate Polynomials *} |
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theory Polynomial |
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Plain, Main form meeting points in import hierarchy
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imports Plain SetInterval Main |
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begin |
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subsection {* Definition of type @{text poly} *} |
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typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}" |
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morphisms coeff Abs_poly |
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by auto |
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lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)" |
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by (simp add: coeff_inject [symmetric] expand_fun_eq) |
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lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q" |
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by (simp add: expand_poly_eq) |
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subsection {* Degree of a polynomial *} |
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definition |
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degree :: "'a::zero poly \<Rightarrow> nat" where |
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"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)" |
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lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0" |
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proof - |
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have "coeff p \<in> Poly" |
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by (rule coeff) |
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hence "\<exists>n. \<forall>i>n. coeff p i = 0" |
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unfolding Poly_def by simp |
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hence "\<forall>i>degree p. coeff p i = 0" |
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unfolding degree_def by (rule LeastI_ex) |
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moreover assume "degree p < n" |
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ultimately show ?thesis by simp |
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qed |
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lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p" |
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by (erule contrapos_np, rule coeff_eq_0, simp) |
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lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n" |
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unfolding degree_def by (erule Least_le) |
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lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0" |
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unfolding degree_def by (drule not_less_Least, simp) |
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subsection {* The zero polynomial *} |
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instantiation poly :: (zero) zero |
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begin |
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definition |
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zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)" |
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instance .. |
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end |
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lemma coeff_0 [simp]: "coeff 0 n = 0" |
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unfolding zero_poly_def |
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by (simp add: Abs_poly_inverse Poly_def) |
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lemma degree_0 [simp]: "degree 0 = 0" |
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by (rule order_antisym [OF degree_le le0]) simp |
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lemma leading_coeff_neq_0: |
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assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0" |
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proof (cases "degree p") |
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case 0 |
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from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0" |
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by (simp add: expand_poly_eq) |
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then obtain n where "coeff p n \<noteq> 0" .. |
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hence "n \<le> degree p" by (rule le_degree) |
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with `coeff p n \<noteq> 0` and `degree p = 0` |
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show "coeff p (degree p) \<noteq> 0" by simp |
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next |
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case (Suc n) |
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from `degree p = Suc n` have "n < degree p" by simp |
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hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp) |
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then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast |
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from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp |
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also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree) |
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finally have "degree p = i" . |
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with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp |
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qed |
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lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0" |
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by (cases "p = 0", simp, simp add: leading_coeff_neq_0) |
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subsection {* List-style constructor for polynomials *} |
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definition |
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pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly" |
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where |
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[code del]: "pCons a p = Abs_poly (nat_case a (coeff p))" |
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syntax |
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"_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]") |
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translations |
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"[:x, xs:]" == "CONST pCons x [:xs:]" |
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"[:x:]" == "CONST pCons x 0" |
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lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly" |
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unfolding Poly_def by (auto split: nat.split) |
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lemma coeff_pCons: |
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"coeff (pCons a p) = nat_case a (coeff p)" |
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unfolding pCons_def |
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by (simp add: Abs_poly_inverse Poly_nat_case coeff) |
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lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a" |
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by (simp add: coeff_pCons) |
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lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" |
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by (simp add: coeff_pCons) |
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lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)" |
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by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split) |
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lemma degree_pCons_eq: |
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"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)" |
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apply (rule order_antisym [OF degree_pCons_le]) |
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apply (rule le_degree, simp) |
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done |
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lemma degree_pCons_0: "degree (pCons a 0) = 0" |
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apply (rule order_antisym [OF _ le0]) |
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apply (rule degree_le, simp add: coeff_pCons split: nat.split) |
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done |
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lemma degree_pCons_eq_if [simp]: |
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"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" |
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apply (cases "p = 0", simp_all) |
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apply (rule order_antisym [OF _ le0]) |
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apply (rule degree_le, simp add: coeff_pCons split: nat.split) |
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apply (rule order_antisym [OF degree_pCons_le]) |
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apply (rule le_degree, simp) |
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done |
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lemma pCons_0_0 [simp]: "pCons 0 0 = 0" |
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by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
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lemma pCons_eq_iff [simp]: |
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"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q" |
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proof (safe) |
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assume "pCons a p = pCons b q" |
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then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp |
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then show "a = b" by simp |
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next |
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assume "pCons a p = pCons b q" |
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then have "\<forall>n. coeff (pCons a p) (Suc n) = |
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coeff (pCons b q) (Suc n)" by simp |
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then show "p = q" by (simp add: expand_poly_eq) |
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qed |
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lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0" |
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using pCons_eq_iff [of a p 0 0] by simp |
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lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly" |
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unfolding Poly_def |
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by (clarify, rule_tac x=n in exI, simp) |
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lemma pCons_cases [cases type: poly]: |
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obtains (pCons) a q where "p = pCons a q" |
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proof |
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show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))" |
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by (rule poly_ext) |
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(simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons |
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split: nat.split) |
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qed |
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lemma pCons_induct [case_names 0 pCons, induct type: poly]: |
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assumes zero: "P 0" |
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assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)" |
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shows "P p" |
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proof (induct p rule: measure_induct_rule [where f=degree]) |
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case (less p) |
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obtain a q where "p = pCons a q" by (rule pCons_cases) |
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have "P q" |
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proof (cases "q = 0") |
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case True |
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then show "P q" by (simp add: zero) |
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next |
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case False |
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then have "degree (pCons a q) = Suc (degree q)" |
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by (rule degree_pCons_eq) |
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then have "degree q < degree p" |
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using `p = pCons a q` by simp |
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then show "P q" |
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by (rule less.hyps) |
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qed |
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then have "P (pCons a q)" |
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by (rule pCons) |
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then show ?case |
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using `p = pCons a q` by simp |
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qed |
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subsection {* Recursion combinator for polynomials *} |
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function |
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poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b" |
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where |
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poly_rec_pCons_eq_if [simp del, code del]: |
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"poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)" |
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by (case_tac x, rename_tac q, case_tac q, auto) |
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termination poly_rec |
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by (relation "measure (degree \<circ> snd \<circ> snd)", simp) |
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(simp add: degree_pCons_eq) |
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lemma poly_rec_0: |
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"f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z" |
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using poly_rec_pCons_eq_if [of z f 0 0] by simp |
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lemma poly_rec_pCons: |
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"f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)" |
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by (simp add: poly_rec_pCons_eq_if poly_rec_0) |
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subsection {* Monomials *} |
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definition |
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monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where |
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"monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)" |
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lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)" |
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unfolding monom_def |
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by (subst Abs_poly_inverse, auto simp add: Poly_def) |
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lemma monom_0: "monom a 0 = pCons a 0" |
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by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
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lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" |
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by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
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lemma monom_eq_0 [simp]: "monom 0 n = 0" |
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by (rule poly_ext) simp |
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lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0" |
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by (simp add: expand_poly_eq) |
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lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b" |
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by (simp add: expand_poly_eq) |
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lemma degree_monom_le: "degree (monom a n) \<le> n" |
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by (rule degree_le, simp) |
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lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n" |
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apply (rule order_antisym [OF degree_monom_le]) |
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apply (rule le_degree, simp) |
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done |
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subsection {* Addition and subtraction *} |
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instantiation poly :: (comm_monoid_add) comm_monoid_add |
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begin |
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definition |
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plus_poly_def [code del]: |
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"p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)" |
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lemma Poly_add: |
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fixes f g :: "nat \<Rightarrow> 'a" |
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shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly" |
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unfolding Poly_def |
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apply (clarify, rename_tac m n) |
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apply (rule_tac x="max m n" in exI, simp) |
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done |
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lemma coeff_add [simp]: |
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"coeff (p + q) n = coeff p n + coeff q n" |
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unfolding plus_poly_def |
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by (simp add: Abs_poly_inverse coeff Poly_add) |
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instance proof |
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fix p q r :: "'a poly" |
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show "(p + q) + r = p + (q + r)" |
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by (simp add: expand_poly_eq add_assoc) |
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show "p + q = q + p" |
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by (simp add: expand_poly_eq add_commute) |
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show "0 + p = p" |
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by (simp add: expand_poly_eq) |
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qed |
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end |
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instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add |
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proof |
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fix p q r :: "'a poly" |
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assume "p + q = p + r" thus "q = r" |
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by (simp add: expand_poly_eq) |
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qed |
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instantiation poly :: (ab_group_add) ab_group_add |
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begin |
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definition |
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uminus_poly_def [code del]: |
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"- p = Abs_poly (\<lambda>n. - coeff p n)" |
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definition |
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minus_poly_def [code del]: |
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"p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)" |
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lemma Poly_minus: |
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fixes f :: "nat \<Rightarrow> 'a" |
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shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly" |
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unfolding Poly_def by simp |
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lemma Poly_diff: |
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fixes f g :: "nat \<Rightarrow> 'a" |
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shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly" |
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unfolding diff_minus by (simp add: Poly_add Poly_minus) |
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lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n" |
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unfolding uminus_poly_def |
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by (simp add: Abs_poly_inverse coeff Poly_minus) |
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lemma coeff_diff [simp]: |
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"coeff (p - q) n = coeff p n - coeff q n" |
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unfolding minus_poly_def |
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by (simp add: Abs_poly_inverse coeff Poly_diff) |
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instance proof |
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fix p q :: "'a poly" |
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show "- p + p = 0" |
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by (simp add: expand_poly_eq) |
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show "p - q = p + - q" |
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by (simp add: expand_poly_eq diff_minus) |
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qed |
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end |
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lemma add_pCons [simp]: |
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"pCons a p + pCons b q = pCons (a + b) (p + q)" |
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by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
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lemma minus_pCons [simp]: |
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"- pCons a p = pCons (- a) (- p)" |
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by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
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lemma diff_pCons [simp]: |
|
352 |
"pCons a p - pCons b q = pCons (a - b) (p - q)" |
|
353 |
by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
|
354 |
||
29539 | 355 |
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)" |
29451 | 356 |
by (rule degree_le, auto simp add: coeff_eq_0) |
357 |
||
29539 | 358 |
lemma degree_add_le: |
359 |
"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n" |
|
360 |
by (auto intro: order_trans degree_add_le_max) |
|
361 |
||
29453 | 362 |
lemma degree_add_less: |
363 |
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n" |
|
29539 | 364 |
by (auto intro: le_less_trans degree_add_le_max) |
29453 | 365 |
|
29451 | 366 |
lemma degree_add_eq_right: |
367 |
"degree p < degree q \<Longrightarrow> degree (p + q) = degree q" |
|
368 |
apply (cases "q = 0", simp) |
|
369 |
apply (rule order_antisym) |
|
29539 | 370 |
apply (simp add: degree_add_le) |
29451 | 371 |
apply (rule le_degree) |
372 |
apply (simp add: coeff_eq_0) |
|
373 |
done |
|
374 |
||
375 |
lemma degree_add_eq_left: |
|
376 |
"degree q < degree p \<Longrightarrow> degree (p + q) = degree p" |
|
377 |
using degree_add_eq_right [of q p] |
|
378 |
by (simp add: add_commute) |
|
379 |
||
380 |
lemma degree_minus [simp]: "degree (- p) = degree p" |
|
381 |
unfolding degree_def by simp |
|
382 |
||
29539 | 383 |
lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)" |
29451 | 384 |
using degree_add_le [where p=p and q="-q"] |
385 |
by (simp add: diff_minus) |
|
386 |
||
29539 | 387 |
lemma degree_diff_le: |
388 |
"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n" |
|
389 |
by (simp add: diff_minus degree_add_le) |
|
390 |
||
29453 | 391 |
lemma degree_diff_less: |
392 |
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n" |
|
29539 | 393 |
by (simp add: diff_minus degree_add_less) |
29453 | 394 |
|
29451 | 395 |
lemma add_monom: "monom a n + monom b n = monom (a + b) n" |
396 |
by (rule poly_ext) simp |
|
397 |
||
398 |
lemma diff_monom: "monom a n - monom b n = monom (a - b) n" |
|
399 |
by (rule poly_ext) simp |
|
400 |
||
401 |
lemma minus_monom: "- monom a n = monom (-a) n" |
|
402 |
by (rule poly_ext) simp |
|
403 |
||
404 |
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)" |
|
405 |
by (cases "finite A", induct set: finite, simp_all) |
|
406 |
||
407 |
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)" |
|
408 |
by (rule poly_ext) (simp add: coeff_setsum) |
|
409 |
||
410 |
||
411 |
subsection {* Multiplication by a constant *} |
|
412 |
||
413 |
definition |
|
414 |
smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where |
|
415 |
"smult a p = Abs_poly (\<lambda>n. a * coeff p n)" |
|
416 |
||
417 |
lemma Poly_smult: |
|
418 |
fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0" |
|
419 |
shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly" |
|
420 |
unfolding Poly_def |
|
421 |
by (clarify, rule_tac x=n in exI, simp) |
|
422 |
||
423 |
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" |
|
424 |
unfolding smult_def |
|
425 |
by (simp add: Abs_poly_inverse Poly_smult coeff) |
|
426 |
||
427 |
lemma degree_smult_le: "degree (smult a p) \<le> degree p" |
|
428 |
by (rule degree_le, simp add: coeff_eq_0) |
|
429 |
||
29472 | 430 |
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" |
29451 | 431 |
by (rule poly_ext, simp add: mult_assoc) |
432 |
||
433 |
lemma smult_0_right [simp]: "smult a 0 = 0" |
|
434 |
by (rule poly_ext, simp) |
|
435 |
||
436 |
lemma smult_0_left [simp]: "smult 0 p = 0" |
|
437 |
by (rule poly_ext, simp) |
|
438 |
||
439 |
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" |
|
440 |
by (rule poly_ext, simp) |
|
441 |
||
442 |
lemma smult_add_right: |
|
443 |
"smult a (p + q) = smult a p + smult a q" |
|
29667 | 444 |
by (rule poly_ext, simp add: algebra_simps) |
29451 | 445 |
|
446 |
lemma smult_add_left: |
|
447 |
"smult (a + b) p = smult a p + smult b p" |
|
29667 | 448 |
by (rule poly_ext, simp add: algebra_simps) |
29451 | 449 |
|
29457 | 450 |
lemma smult_minus_right [simp]: |
29451 | 451 |
"smult (a::'a::comm_ring) (- p) = - smult a p" |
452 |
by (rule poly_ext, simp) |
|
453 |
||
29457 | 454 |
lemma smult_minus_left [simp]: |
29451 | 455 |
"smult (- a::'a::comm_ring) p = - smult a p" |
456 |
by (rule poly_ext, simp) |
|
457 |
||
458 |
lemma smult_diff_right: |
|
459 |
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q" |
|
29667 | 460 |
by (rule poly_ext, simp add: algebra_simps) |
29451 | 461 |
|
462 |
lemma smult_diff_left: |
|
463 |
"smult (a - b::'a::comm_ring) p = smult a p - smult b p" |
|
29667 | 464 |
by (rule poly_ext, simp add: algebra_simps) |
29451 | 465 |
|
29472 | 466 |
lemmas smult_distribs = |
467 |
smult_add_left smult_add_right |
|
468 |
smult_diff_left smult_diff_right |
|
469 |
||
29451 | 470 |
lemma smult_pCons [simp]: |
471 |
"smult a (pCons b p) = pCons (a * b) (smult a p)" |
|
472 |
by (rule poly_ext, simp add: coeff_pCons split: nat.split) |
|
473 |
||
474 |
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" |
|
475 |
by (induct n, simp add: monom_0, simp add: monom_Suc) |
|
476 |
||
29659 | 477 |
lemma degree_smult_eq [simp]: |
478 |
fixes a :: "'a::idom" |
|
479 |
shows "degree (smult a p) = (if a = 0 then 0 else degree p)" |
|
480 |
by (cases "a = 0", simp, simp add: degree_def) |
|
481 |
||
482 |
lemma smult_eq_0_iff [simp]: |
|
483 |
fixes a :: "'a::idom" |
|
484 |
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0" |
|
485 |
by (simp add: expand_poly_eq) |
|
486 |
||
29451 | 487 |
|
488 |
subsection {* Multiplication of polynomials *} |
|
489 |
||
29474 | 490 |
text {* TODO: move to SetInterval.thy *} |
29451 | 491 |
lemma setsum_atMost_Suc_shift: |
492 |
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" |
|
493 |
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" |
|
494 |
proof (induct n) |
|
495 |
case 0 show ?case by simp |
|
496 |
next |
|
497 |
case (Suc n) note IH = this |
|
498 |
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))" |
|
499 |
by (rule setsum_atMost_Suc) |
|
500 |
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" |
|
501 |
by (rule IH) |
|
502 |
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = |
|
503 |
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" |
|
504 |
by (rule add_assoc) |
|
505 |
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" |
|
506 |
by (rule setsum_atMost_Suc [symmetric]) |
|
507 |
finally show ?case . |
|
508 |
qed |
|
509 |
||
510 |
instantiation poly :: (comm_semiring_0) comm_semiring_0 |
|
511 |
begin |
|
512 |
||
513 |
definition |
|
29475 | 514 |
times_poly_def [code del]: |
29474 | 515 |
"p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p" |
516 |
||
517 |
lemma mult_poly_0_left: "(0::'a poly) * q = 0" |
|
518 |
unfolding times_poly_def by (simp add: poly_rec_0) |
|
519 |
||
520 |
lemma mult_pCons_left [simp]: |
|
521 |
"pCons a p * q = smult a q + pCons 0 (p * q)" |
|
522 |
unfolding times_poly_def by (simp add: poly_rec_pCons) |
|
523 |
||
524 |
lemma mult_poly_0_right: "p * (0::'a poly) = 0" |
|
525 |
by (induct p, simp add: mult_poly_0_left, simp) |
|
29451 | 526 |
|
29474 | 527 |
lemma mult_pCons_right [simp]: |
528 |
"p * pCons a q = smult a p + pCons 0 (p * q)" |
|
29667 | 529 |
by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps) |
29474 | 530 |
|
531 |
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right |
|
532 |
||
533 |
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" |
|
534 |
by (induct p, simp add: mult_poly_0, simp add: smult_add_right) |
|
535 |
||
536 |
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" |
|
537 |
by (induct q, simp add: mult_poly_0, simp add: smult_add_right) |
|
538 |
||
539 |
lemma mult_poly_add_left: |
|
540 |
fixes p q r :: "'a poly" |
|
541 |
shows "(p + q) * r = p * r + q * r" |
|
542 |
by (induct r, simp add: mult_poly_0, |
|
29667 | 543 |
simp add: smult_distribs algebra_simps) |
29451 | 544 |
|
545 |
instance proof |
|
546 |
fix p q r :: "'a poly" |
|
547 |
show 0: "0 * p = 0" |
|
29474 | 548 |
by (rule mult_poly_0_left) |
29451 | 549 |
show "p * 0 = 0" |
29474 | 550 |
by (rule mult_poly_0_right) |
29451 | 551 |
show "(p + q) * r = p * r + q * r" |
29474 | 552 |
by (rule mult_poly_add_left) |
29451 | 553 |
show "(p * q) * r = p * (q * r)" |
29474 | 554 |
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) |
29451 | 555 |
show "p * q = q * p" |
29474 | 556 |
by (induct p, simp add: mult_poly_0, simp) |
29451 | 557 |
qed |
558 |
||
559 |
end |
|
560 |
||
29540 | 561 |
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
562 |
||
29474 | 563 |
lemma coeff_mult: |
564 |
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))" |
|
565 |
proof (induct p arbitrary: n) |
|
566 |
case 0 show ?case by simp |
|
567 |
next |
|
568 |
case (pCons a p n) thus ?case |
|
569 |
by (cases n, simp, simp add: setsum_atMost_Suc_shift |
|
570 |
del: setsum_atMost_Suc) |
|
571 |
qed |
|
29451 | 572 |
|
29474 | 573 |
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" |
574 |
apply (rule degree_le) |
|
575 |
apply (induct p) |
|
576 |
apply simp |
|
577 |
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) |
|
29451 | 578 |
done |
579 |
||
580 |
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" |
|
581 |
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) |
|
582 |
||
583 |
||
584 |
subsection {* The unit polynomial and exponentiation *} |
|
585 |
||
586 |
instantiation poly :: (comm_semiring_1) comm_semiring_1 |
|
587 |
begin |
|
588 |
||
589 |
definition |
|
590 |
one_poly_def: |
|
591 |
"1 = pCons 1 0" |
|
592 |
||
593 |
instance proof |
|
594 |
fix p :: "'a poly" show "1 * p = p" |
|
595 |
unfolding one_poly_def |
|
596 |
by simp |
|
597 |
next |
|
598 |
show "0 \<noteq> (1::'a poly)" |
|
599 |
unfolding one_poly_def by simp |
|
600 |
qed |
|
601 |
||
602 |
end |
|
603 |
||
29540 | 604 |
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. |
605 |
||
29451 | 606 |
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" |
607 |
unfolding one_poly_def |
|
608 |
by (simp add: coeff_pCons split: nat.split) |
|
609 |
||
610 |
lemma degree_1 [simp]: "degree 1 = 0" |
|
611 |
unfolding one_poly_def |
|
612 |
by (rule degree_pCons_0) |
|
613 |
||
614 |
instantiation poly :: (comm_semiring_1) recpower |
|
615 |
begin |
|
616 |
||
617 |
primrec power_poly where |
|
618 |
power_poly_0: "(p::'a poly) ^ 0 = 1" |
|
619 |
| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" |
|
620 |
||
621 |
instance |
|
622 |
by default simp_all |
|
623 |
||
624 |
end |
|
625 |
||
626 |
instance poly :: (comm_ring) comm_ring .. |
|
627 |
||
628 |
instance poly :: (comm_ring_1) comm_ring_1 .. |
|
629 |
||
630 |
instantiation poly :: (comm_ring_1) number_ring |
|
631 |
begin |
|
632 |
||
633 |
definition |
|
634 |
"number_of k = (of_int k :: 'a poly)" |
|
635 |
||
636 |
instance |
|
637 |
by default (rule number_of_poly_def) |
|
638 |
||
639 |
end |
|
640 |
||
641 |
||
642 |
subsection {* Polynomials form an integral domain *} |
|
643 |
||
644 |
lemma coeff_mult_degree_sum: |
|
645 |
"coeff (p * q) (degree p + degree q) = |
|
646 |
coeff p (degree p) * coeff q (degree q)" |
|
29471 | 647 |
by (induct p, simp, simp add: coeff_eq_0) |
29451 | 648 |
|
649 |
instance poly :: (idom) idom |
|
650 |
proof |
|
651 |
fix p q :: "'a poly" |
|
652 |
assume "p \<noteq> 0" and "q \<noteq> 0" |
|
653 |
have "coeff (p * q) (degree p + degree q) = |
|
654 |
coeff p (degree p) * coeff q (degree q)" |
|
655 |
by (rule coeff_mult_degree_sum) |
|
656 |
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" |
|
657 |
using `p \<noteq> 0` and `q \<noteq> 0` by simp |
|
658 |
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. |
|
659 |
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) |
|
660 |
qed |
|
661 |
||
662 |
lemma degree_mult_eq: |
|
663 |
fixes p q :: "'a::idom poly" |
|
664 |
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" |
|
665 |
apply (rule order_antisym [OF degree_mult_le le_degree]) |
|
666 |
apply (simp add: coeff_mult_degree_sum) |
|
667 |
done |
|
668 |
||
669 |
lemma dvd_imp_degree_le: |
|
670 |
fixes p q :: "'a::idom poly" |
|
671 |
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" |
|
672 |
by (erule dvdE, simp add: degree_mult_eq) |
|
673 |
||
674 |
||
29878 | 675 |
subsection {* Polynomials form an ordered integral domain *} |
676 |
||
677 |
definition |
|
678 |
pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool" |
|
679 |
where |
|
680 |
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)" |
|
681 |
||
682 |
lemma pos_poly_pCons: |
|
683 |
"pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)" |
|
684 |
unfolding pos_poly_def by simp |
|
685 |
||
686 |
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0" |
|
687 |
unfolding pos_poly_def by simp |
|
688 |
||
689 |
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)" |
|
690 |
apply (induct p arbitrary: q, simp) |
|
691 |
apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos) |
|
692 |
done |
|
693 |
||
694 |
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)" |
|
695 |
unfolding pos_poly_def |
|
696 |
apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0") |
|
697 |
apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos) |
|
698 |
apply auto |
|
699 |
done |
|
700 |
||
701 |
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)" |
|
702 |
by (induct p) (auto simp add: pos_poly_pCons) |
|
703 |
||
704 |
instantiation poly :: (ordered_idom) ordered_idom |
|
705 |
begin |
|
706 |
||
707 |
definition |
|
708 |
[code del]: |
|
709 |
"x < y \<longleftrightarrow> pos_poly (y - x)" |
|
710 |
||
711 |
definition |
|
712 |
[code del]: |
|
713 |
"x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)" |
|
714 |
||
715 |
definition |
|
716 |
[code del]: |
|
717 |
"abs (x::'a poly) = (if x < 0 then - x else x)" |
|
718 |
||
719 |
definition |
|
720 |
[code del]: |
|
721 |
"sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
|
722 |
||
723 |
instance proof |
|
724 |
fix x y :: "'a poly" |
|
725 |
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
|
726 |
unfolding less_eq_poly_def less_poly_def |
|
727 |
apply safe |
|
728 |
apply simp |
|
729 |
apply (drule (1) pos_poly_add) |
|
730 |
apply simp |
|
731 |
done |
|
732 |
next |
|
733 |
fix x :: "'a poly" show "x \<le> x" |
|
734 |
unfolding less_eq_poly_def by simp |
|
735 |
next |
|
736 |
fix x y z :: "'a poly" |
|
737 |
assume "x \<le> y" and "y \<le> z" thus "x \<le> z" |
|
738 |
unfolding less_eq_poly_def |
|
739 |
apply safe |
|
740 |
apply (drule (1) pos_poly_add) |
|
741 |
apply (simp add: algebra_simps) |
|
742 |
done |
|
743 |
next |
|
744 |
fix x y :: "'a poly" |
|
745 |
assume "x \<le> y" and "y \<le> x" thus "x = y" |
|
746 |
unfolding less_eq_poly_def |
|
747 |
apply safe |
|
748 |
apply (drule (1) pos_poly_add) |
|
749 |
apply simp |
|
750 |
done |
|
751 |
next |
|
752 |
fix x y z :: "'a poly" |
|
753 |
assume "x \<le> y" thus "z + x \<le> z + y" |
|
754 |
unfolding less_eq_poly_def |
|
755 |
apply safe |
|
756 |
apply (simp add: algebra_simps) |
|
757 |
done |
|
758 |
next |
|
759 |
fix x y :: "'a poly" |
|
760 |
show "x \<le> y \<or> y \<le> x" |
|
761 |
unfolding less_eq_poly_def |
|
762 |
using pos_poly_total [of "x - y"] |
|
763 |
by auto |
|
764 |
next |
|
765 |
fix x y z :: "'a poly" |
|
766 |
assume "x < y" and "0 < z" |
|
767 |
thus "z * x < z * y" |
|
768 |
unfolding less_poly_def |
|
769 |
by (simp add: right_diff_distrib [symmetric] pos_poly_mult) |
|
770 |
next |
|
771 |
fix x :: "'a poly" |
|
772 |
show "\<bar>x\<bar> = (if x < 0 then - x else x)" |
|
773 |
by (rule abs_poly_def) |
|
774 |
next |
|
775 |
fix x :: "'a poly" |
|
776 |
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)" |
|
777 |
by (rule sgn_poly_def) |
|
778 |
qed |
|
779 |
||
780 |
end |
|
781 |
||
782 |
text {* TODO: Simplification rules for comparisons *} |
|
783 |
||
784 |
||
29451 | 785 |
subsection {* Long division of polynomials *} |
786 |
||
787 |
definition |
|
29537 | 788 |
pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" |
29451 | 789 |
where |
29475 | 790 |
[code del]: |
29537 | 791 |
"pdivmod_rel x y q r \<longleftrightarrow> |
29451 | 792 |
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" |
793 |
||
29537 | 794 |
lemma pdivmod_rel_0: |
795 |
"pdivmod_rel 0 y 0 0" |
|
796 |
unfolding pdivmod_rel_def by simp |
|
29451 | 797 |
|
29537 | 798 |
lemma pdivmod_rel_by_0: |
799 |
"pdivmod_rel x 0 0 x" |
|
800 |
unfolding pdivmod_rel_def by simp |
|
29451 | 801 |
|
802 |
lemma eq_zero_or_degree_less: |
|
803 |
assumes "degree p \<le> n" and "coeff p n = 0" |
|
804 |
shows "p = 0 \<or> degree p < n" |
|
805 |
proof (cases n) |
|
806 |
case 0 |
|
807 |
with `degree p \<le> n` and `coeff p n = 0` |
|
808 |
have "coeff p (degree p) = 0" by simp |
|
809 |
then have "p = 0" by simp |
|
810 |
then show ?thesis .. |
|
811 |
next |
|
812 |
case (Suc m) |
|
813 |
have "\<forall>i>n. coeff p i = 0" |
|
814 |
using `degree p \<le> n` by (simp add: coeff_eq_0) |
|
815 |
then have "\<forall>i\<ge>n. coeff p i = 0" |
|
816 |
using `coeff p n = 0` by (simp add: le_less) |
|
817 |
then have "\<forall>i>m. coeff p i = 0" |
|
818 |
using `n = Suc m` by (simp add: less_eq_Suc_le) |
|
819 |
then have "degree p \<le> m" |
|
820 |
by (rule degree_le) |
|
821 |
then have "degree p < n" |
|
822 |
using `n = Suc m` by (simp add: less_Suc_eq_le) |
|
823 |
then show ?thesis .. |
|
824 |
qed |
|
825 |
||
29537 | 826 |
lemma pdivmod_rel_pCons: |
827 |
assumes rel: "pdivmod_rel x y q r" |
|
29451 | 828 |
assumes y: "y \<noteq> 0" |
829 |
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" |
|
29537 | 830 |
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)" |
831 |
(is "pdivmod_rel ?x y ?q ?r") |
|
29451 | 832 |
proof - |
833 |
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" |
|
29537 | 834 |
using assms unfolding pdivmod_rel_def by simp_all |
29451 | 835 |
|
836 |
have 1: "?x = ?q * y + ?r" |
|
837 |
using b x by simp |
|
838 |
||
839 |
have 2: "?r = 0 \<or> degree ?r < degree y" |
|
840 |
proof (rule eq_zero_or_degree_less) |
|
29539 | 841 |
show "degree ?r \<le> degree y" |
842 |
proof (rule degree_diff_le) |
|
29451 | 843 |
show "degree (pCons a r) \<le> degree y" |
29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
844 |
using r by auto |
29451 | 845 |
show "degree (smult b y) \<le> degree y" |
846 |
by (rule degree_smult_le) |
|
847 |
qed |
|
848 |
next |
|
849 |
show "coeff ?r (degree y) = 0" |
|
850 |
using `y \<noteq> 0` unfolding b by simp |
|
851 |
qed |
|
852 |
||
853 |
from 1 2 show ?thesis |
|
29537 | 854 |
unfolding pdivmod_rel_def |
29451 | 855 |
using `y \<noteq> 0` by simp |
856 |
qed |
|
857 |
||
29537 | 858 |
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r" |
29451 | 859 |
apply (cases "y = 0") |
29537 | 860 |
apply (fast intro!: pdivmod_rel_by_0) |
29451 | 861 |
apply (induct x) |
29537 | 862 |
apply (fast intro!: pdivmod_rel_0) |
863 |
apply (fast intro!: pdivmod_rel_pCons) |
|
29451 | 864 |
done |
865 |
||
29537 | 866 |
lemma pdivmod_rel_unique: |
867 |
assumes 1: "pdivmod_rel x y q1 r1" |
|
868 |
assumes 2: "pdivmod_rel x y q2 r2" |
|
29451 | 869 |
shows "q1 = q2 \<and> r1 = r2" |
870 |
proof (cases "y = 0") |
|
871 |
assume "y = 0" with assms show ?thesis |
|
29537 | 872 |
by (simp add: pdivmod_rel_def) |
29451 | 873 |
next |
874 |
assume [simp]: "y \<noteq> 0" |
|
875 |
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" |
|
29537 | 876 |
unfolding pdivmod_rel_def by simp_all |
29451 | 877 |
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" |
29537 | 878 |
unfolding pdivmod_rel_def by simp_all |
29451 | 879 |
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1" |
29667 | 880 |
by (simp add: algebra_simps) |
29451 | 881 |
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y" |
29453 | 882 |
by (auto intro: degree_diff_less) |
29451 | 883 |
|
884 |
show "q1 = q2 \<and> r1 = r2" |
|
885 |
proof (rule ccontr) |
|
886 |
assume "\<not> (q1 = q2 \<and> r1 = r2)" |
|
887 |
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto |
|
888 |
with r3 have "degree (r2 - r1) < degree y" by simp |
|
889 |
also have "degree y \<le> degree (q1 - q2) + degree y" by simp |
|
890 |
also have "\<dots> = degree ((q1 - q2) * y)" |
|
891 |
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) |
|
892 |
also have "\<dots> = degree (r2 - r1)" |
|
893 |
using q3 by simp |
|
894 |
finally have "degree (r2 - r1) < degree (r2 - r1)" . |
|
895 |
then show "False" by simp |
|
896 |
qed |
|
897 |
qed |
|
898 |
||
29660 | 899 |
lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0" |
900 |
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0) |
|
901 |
||
902 |
lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x" |
|
903 |
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0) |
|
904 |
||
29537 | 905 |
lemmas pdivmod_rel_unique_div = |
906 |
pdivmod_rel_unique [THEN conjunct1, standard] |
|
29451 | 907 |
|
29537 | 908 |
lemmas pdivmod_rel_unique_mod = |
909 |
pdivmod_rel_unique [THEN conjunct2, standard] |
|
29451 | 910 |
|
911 |
instantiation poly :: (field) ring_div |
|
912 |
begin |
|
913 |
||
914 |
definition div_poly where |
|
29537 | 915 |
[code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)" |
29451 | 916 |
|
917 |
definition mod_poly where |
|
29537 | 918 |
[code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)" |
29451 | 919 |
|
920 |
lemma div_poly_eq: |
|
29537 | 921 |
"pdivmod_rel x y q r \<Longrightarrow> x div y = q" |
29451 | 922 |
unfolding div_poly_def |
29537 | 923 |
by (fast elim: pdivmod_rel_unique_div) |
29451 | 924 |
|
925 |
lemma mod_poly_eq: |
|
29537 | 926 |
"pdivmod_rel x y q r \<Longrightarrow> x mod y = r" |
29451 | 927 |
unfolding mod_poly_def |
29537 | 928 |
by (fast elim: pdivmod_rel_unique_mod) |
29451 | 929 |
|
29537 | 930 |
lemma pdivmod_rel: |
931 |
"pdivmod_rel x y (x div y) (x mod y)" |
|
29451 | 932 |
proof - |
29537 | 933 |
from pdivmod_rel_exists |
934 |
obtain q r where "pdivmod_rel x y q r" by fast |
|
29451 | 935 |
thus ?thesis |
936 |
by (simp add: div_poly_eq mod_poly_eq) |
|
937 |
qed |
|
938 |
||
939 |
instance proof |
|
940 |
fix x y :: "'a poly" |
|
941 |
show "x div y * y + x mod y = x" |
|
29537 | 942 |
using pdivmod_rel [of x y] |
943 |
by (simp add: pdivmod_rel_def) |
|
29451 | 944 |
next |
945 |
fix x :: "'a poly" |
|
29537 | 946 |
have "pdivmod_rel x 0 0 x" |
947 |
by (rule pdivmod_rel_by_0) |
|
29451 | 948 |
thus "x div 0 = 0" |
949 |
by (rule div_poly_eq) |
|
950 |
next |
|
951 |
fix y :: "'a poly" |
|
29537 | 952 |
have "pdivmod_rel 0 y 0 0" |
953 |
by (rule pdivmod_rel_0) |
|
29451 | 954 |
thus "0 div y = 0" |
955 |
by (rule div_poly_eq) |
|
956 |
next |
|
957 |
fix x y z :: "'a poly" |
|
958 |
assume "y \<noteq> 0" |
|
29537 | 959 |
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" |
960 |
using pdivmod_rel [of x y] |
|
961 |
by (simp add: pdivmod_rel_def left_distrib) |
|
29451 | 962 |
thus "(x + z * y) div y = z + x div y" |
963 |
by (rule div_poly_eq) |
|
964 |
qed |
|
965 |
||
966 |
end |
|
967 |
||
968 |
lemma degree_mod_less: |
|
969 |
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" |
|
29537 | 970 |
using pdivmod_rel [of x y] |
971 |
unfolding pdivmod_rel_def by simp |
|
29451 | 972 |
|
973 |
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" |
|
974 |
proof - |
|
975 |
assume "degree x < degree y" |
|
29537 | 976 |
hence "pdivmod_rel x y 0 x" |
977 |
by (simp add: pdivmod_rel_def) |
|
29451 | 978 |
thus "x div y = 0" by (rule div_poly_eq) |
979 |
qed |
|
980 |
||
981 |
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" |
|
982 |
proof - |
|
983 |
assume "degree x < degree y" |
|
29537 | 984 |
hence "pdivmod_rel x y 0 x" |
985 |
by (simp add: pdivmod_rel_def) |
|
29451 | 986 |
thus "x mod y = x" by (rule mod_poly_eq) |
987 |
qed |
|
988 |
||
29659 | 989 |
lemma pdivmod_rel_smult_left: |
990 |
"pdivmod_rel x y q r |
|
991 |
\<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)" |
|
992 |
unfolding pdivmod_rel_def by (simp add: smult_add_right) |
|
993 |
||
994 |
lemma div_smult_left: "(smult a x) div y = smult a (x div y)" |
|
995 |
by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) |
|
996 |
||
997 |
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" |
|
998 |
by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) |
|
999 |
||
1000 |
lemma pdivmod_rel_smult_right: |
|
1001 |
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk> |
|
1002 |
\<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r" |
|
1003 |
unfolding pdivmod_rel_def by simp |
|
1004 |
||
1005 |
lemma div_smult_right: |
|
1006 |
"a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)" |
|
1007 |
by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) |
|
1008 |
||
1009 |
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y" |
|
1010 |
by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) |
|
1011 |
||
29660 | 1012 |
lemma pdivmod_rel_mult: |
1013 |
"\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk> |
|
1014 |
\<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)" |
|
1015 |
apply (cases "z = 0", simp add: pdivmod_rel_def) |
|
1016 |
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff) |
|
1017 |
apply (cases "r = 0") |
|
1018 |
apply (cases "r' = 0") |
|
1019 |
apply (simp add: pdivmod_rel_def) |
|
1020 |
apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq) |
|
1021 |
apply (cases "r' = 0") |
|
1022 |
apply (simp add: pdivmod_rel_def degree_mult_eq) |
|
1023 |
apply (simp add: pdivmod_rel_def ring_simps) |
|
1024 |
apply (simp add: degree_mult_eq degree_add_less) |
|
1025 |
done |
|
1026 |
||
1027 |
lemma poly_div_mult_right: |
|
1028 |
fixes x y z :: "'a::field poly" |
|
1029 |
shows "x div (y * z) = (x div y) div z" |
|
1030 |
by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) |
|
1031 |
||
1032 |
lemma poly_mod_mult_right: |
|
1033 |
fixes x y z :: "'a::field poly" |
|
1034 |
shows "x mod (y * z) = y * (x div y mod z) + x mod y" |
|
1035 |
by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) |
|
1036 |
||
29451 | 1037 |
lemma mod_pCons: |
1038 |
fixes a and x |
|
1039 |
assumes y: "y \<noteq> 0" |
|
1040 |
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" |
|
1041 |
shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)" |
|
1042 |
unfolding b |
|
1043 |
apply (rule mod_poly_eq) |
|
29537 | 1044 |
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) |
29451 | 1045 |
done |
1046 |
||
1047 |
||
1048 |
subsection {* Evaluation of polynomials *} |
|
1049 |
||
1050 |
definition |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1051 |
poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where |
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1052 |
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" |
29451 | 1053 |
|
1054 |
lemma poly_0 [simp]: "poly 0 x = 0" |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1055 |
unfolding poly_def by (simp add: poly_rec_0) |
29451 | 1056 |
|
1057 |
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1058 |
unfolding poly_def by (simp add: poly_rec_pCons) |
29451 | 1059 |
|
1060 |
lemma poly_1 [simp]: "poly 1 x = 1" |
|
1061 |
unfolding one_poly_def by simp |
|
1062 |
||
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1063 |
lemma poly_monom: |
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1064 |
fixes a x :: "'a::{comm_semiring_1,recpower}" |
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1065 |
shows "poly (monom a n) x = a * x ^ n" |
29451 | 1066 |
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) |
1067 |
||
1068 |
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" |
|
1069 |
apply (induct p arbitrary: q, simp) |
|
29667 | 1070 |
apply (case_tac q, simp, simp add: algebra_simps) |
29451 | 1071 |
done |
1072 |
||
1073 |
lemma poly_minus [simp]: |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1074 |
fixes x :: "'a::comm_ring" |
29451 | 1075 |
shows "poly (- p) x = - poly p x" |
1076 |
by (induct p, simp_all) |
|
1077 |
||
1078 |
lemma poly_diff [simp]: |
|
29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset
|
1079 |
fixes x :: "'a::comm_ring" |
29451 | 1080 |
shows "poly (p - q) x = poly p x - poly q x" |
1081 |
by (simp add: diff_minus) |
|
1082 |
||
1083 |
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" |
|
1084 |
by (cases "finite A", induct set: finite, simp_all) |
|
1085 |
||
1086 |
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" |
|
29667 | 1087 |
by (induct p, simp, simp add: algebra_simps) |
29451 | 1088 |
|
1089 |
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" |
|
29667 | 1090 |
by (induct p, simp_all, simp add: algebra_simps) |
29451 | 1091 |
|
29462 | 1092 |
lemma poly_power [simp]: |
1093 |
fixes p :: "'a::{comm_semiring_1,recpower} poly" |
|
1094 |
shows "poly (p ^ n) x = poly p x ^ n" |
|
1095 |
by (induct n, simp, simp add: power_Suc) |
|
1096 |
||
29456 | 1097 |
|
1098 |
subsection {* Synthetic division *} |
|
1099 |
||
1100 |
definition |
|
1101 |
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" |
|
29478 | 1102 |
where [code del]: |
29456 | 1103 |
"synthetic_divmod p c = |
1104 |
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p" |
|
1105 |
||
1106 |
definition |
|
1107 |
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" |
|
1108 |
where |
|
1109 |
"synthetic_div p c = fst (synthetic_divmod p c)" |
|
1110 |
||
1111 |
lemma synthetic_divmod_0 [simp]: |
|
1112 |
"synthetic_divmod 0 c = (0, 0)" |
|
1113 |
unfolding synthetic_divmod_def |
|
1114 |
by (simp add: poly_rec_0) |
|
1115 |
||
1116 |
lemma synthetic_divmod_pCons [simp]: |
|
1117 |
"synthetic_divmod (pCons a p) c = |
|
1118 |
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" |
|
1119 |
unfolding synthetic_divmod_def |
|
1120 |
by (simp add: poly_rec_pCons) |
|
1121 |
||
1122 |
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" |
|
1123 |
by (induct p, simp, simp add: split_def) |
|
1124 |
||
1125 |
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" |
|
1126 |
unfolding synthetic_div_def by simp |
|
1127 |
||
1128 |
lemma synthetic_div_pCons [simp]: |
|
1129 |
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" |
|
1130 |
unfolding synthetic_div_def |
|
1131 |
by (simp add: split_def snd_synthetic_divmod) |
|
1132 |
||
29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1133 |
lemma synthetic_div_eq_0_iff: |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1134 |
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1135 |
by (induct p, simp, case_tac p, simp) |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1136 |
|
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1137 |
lemma degree_synthetic_div: |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1138 |
"degree (synthetic_div p c) = degree p - 1" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1139 |
by (induct p, simp, simp add: synthetic_div_eq_0_iff) |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1140 |
|
29457 | 1141 |
lemma synthetic_div_correct: |
29456 | 1142 |
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" |
1143 |
by (induct p) simp_all |
|
1144 |
||
29457 | 1145 |
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" |
1146 |
by (induct p arbitrary: a) simp_all |
|
1147 |
||
1148 |
lemma synthetic_div_unique: |
|
1149 |
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" |
|
1150 |
apply (induct p arbitrary: q r) |
|
1151 |
apply (simp, frule synthetic_div_unique_lemma, simp) |
|
1152 |
apply (case_tac q, force) |
|
1153 |
done |
|
1154 |
||
1155 |
lemma synthetic_div_correct': |
|
1156 |
fixes c :: "'a::comm_ring_1" |
|
1157 |
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p" |
|
1158 |
using synthetic_div_correct [of p c] |
|
29667 | 1159 |
by (simp add: algebra_simps) |
29457 | 1160 |
|
29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1161 |
lemma poly_eq_0_iff_dvd: |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1162 |
fixes c :: "'a::idom" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1163 |
shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1164 |
proof |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1165 |
assume "poly p c = 0" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1166 |
with synthetic_div_correct' [of c p] |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1167 |
have "p = [:-c, 1:] * synthetic_div p c" by simp |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1168 |
then show "[:-c, 1:] dvd p" .. |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1169 |
next |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1170 |
assume "[:-c, 1:] dvd p" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1171 |
then obtain k where "p = [:-c, 1:] * k" by (rule dvdE) |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1172 |
then show "poly p c = 0" by simp |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1173 |
qed |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1174 |
|
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1175 |
lemma dvd_iff_poly_eq_0: |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1176 |
fixes c :: "'a::idom" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1177 |
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0" |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1178 |
by (simp add: poly_eq_0_iff_dvd) |
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset
|
1179 |
|
29462 | 1180 |
lemma poly_roots_finite: |
1181 |
fixes p :: "'a::idom poly" |
|
1182 |
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" |
|
1183 |
proof (induct n \<equiv> "degree p" arbitrary: p) |
|
1184 |
case (0 p) |
|
1185 |
then obtain a where "a \<noteq> 0" and "p = [:a:]" |
|
1186 |
by (cases p, simp split: if_splits) |
|
1187 |
then show "finite {x. poly p x = 0}" by simp |
|
1188 |
next |
|
1189 |
case (Suc n p) |
|
1190 |
show "finite {x. poly p x = 0}" |
|
1191 |
proof (cases "\<exists>x. poly p x = 0") |
|
1192 |
case False |
|
1193 |
then show "finite {x. poly p x = 0}" by simp |
|
1194 |
next |
|
1195 |
case True |
|
1196 |
then obtain a where "poly p a = 0" .. |
|
1197 |
then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) |
|
1198 |
then obtain k where k: "p = [:-a, 1:] * k" .. |
|
1199 |
with `p \<noteq> 0` have "k \<noteq> 0" by auto |
|
1200 |
with k have "degree p = Suc (degree k)" |
|
1201 |
by (simp add: degree_mult_eq del: mult_pCons_left) |
|
1202 |
with `Suc n = degree p` have "n = degree k" by simp |
|
1203 |
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps) |
|
1204 |
then have "finite (insert a {x. poly k x = 0})" by simp |
|
1205 |
then show "finite {x. poly p x = 0}" |
|
1206 |
by (simp add: k uminus_add_conv_diff Collect_disj_eq |
|
1207 |
del: mult_pCons_left) |
|
1208 |
qed |
|
1209 |
qed |
|
1210 |
||
29478 | 1211 |
|
1212 |
subsection {* Configuration of the code generator *} |
|
1213 |
||
1214 |
code_datatype "0::'a::zero poly" pCons |
|
1215 |
||
29480 | 1216 |
declare pCons_0_0 [code post] |
1217 |
||
29478 | 1218 |
instantiation poly :: ("{zero,eq}") eq |
1219 |
begin |
|
1220 |
||
1221 |
definition [code del]: |
|
1222 |
"eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q" |
|
1223 |
||
1224 |
instance |
|
1225 |
by default (rule eq_poly_def) |
|
1226 |
||
29451 | 1227 |
end |
29478 | 1228 |
|
1229 |
lemma eq_poly_code [code]: |
|
1230 |
"eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True" |
|
1231 |
"eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q" |
|
1232 |
"eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0" |
|
1233 |
"eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q" |
|
1234 |
unfolding eq by simp_all |
|
1235 |
||
1236 |
lemmas coeff_code [code] = |
|
1237 |
coeff_0 coeff_pCons_0 coeff_pCons_Suc |
|
1238 |
||
1239 |
lemmas degree_code [code] = |
|
1240 |
degree_0 degree_pCons_eq_if |
|
1241 |
||
1242 |
lemmas monom_poly_code [code] = |
|
1243 |
monom_0 monom_Suc |
|
1244 |
||
1245 |
lemma add_poly_code [code]: |
|
1246 |
"0 + q = (q :: _ poly)" |
|
1247 |
"p + 0 = (p :: _ poly)" |
|
1248 |
"pCons a p + pCons b q = pCons (a + b) (p + q)" |
|
1249 |
by simp_all |
|
1250 |
||
1251 |
lemma minus_poly_code [code]: |
|
1252 |
"- 0 = (0 :: _ poly)" |
|
1253 |
"- pCons a p = pCons (- a) (- p)" |
|
1254 |
by simp_all |
|
1255 |
||
1256 |
lemma diff_poly_code [code]: |
|
1257 |
"0 - q = (- q :: _ poly)" |
|
1258 |
"p - 0 = (p :: _ poly)" |
|
1259 |
"pCons a p - pCons b q = pCons (a - b) (p - q)" |
|
1260 |
by simp_all |
|
1261 |
||
1262 |
lemmas smult_poly_code [code] = |
|
1263 |
smult_0_right smult_pCons |
|
1264 |
||
1265 |
lemma mult_poly_code [code]: |
|
1266 |
"0 * q = (0 :: _ poly)" |
|
1267 |
"pCons a p * q = smult a q + pCons 0 (p * q)" |
|
1268 |
by simp_all |
|
1269 |
||
1270 |
lemmas poly_code [code] = |
|
1271 |
poly_0 poly_pCons |
|
1272 |
||
1273 |
lemmas synthetic_divmod_code [code] = |
|
1274 |
synthetic_divmod_0 synthetic_divmod_pCons |
|
1275 |
||
1276 |
text {* code generator setup for div and mod *} |
|
1277 |
||
1278 |
definition |
|
29537 | 1279 |
pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" |
29478 | 1280 |
where |
29537 | 1281 |
[code del]: "pdivmod x y = (x div y, x mod y)" |
29478 | 1282 |
|
29537 | 1283 |
lemma div_poly_code [code]: "x div y = fst (pdivmod x y)" |
1284 |
unfolding pdivmod_def by simp |
|
29478 | 1285 |
|
29537 | 1286 |
lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)" |
1287 |
unfolding pdivmod_def by simp |
|
29478 | 1288 |
|
29537 | 1289 |
lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)" |
1290 |
unfolding pdivmod_def by simp |
|
29478 | 1291 |
|
29537 | 1292 |
lemma pdivmod_pCons [code]: |
1293 |
"pdivmod (pCons a x) y = |
|
29478 | 1294 |
(if y = 0 then (0, pCons a x) else |
29537 | 1295 |
(let (q, r) = pdivmod x y; |
29478 | 1296 |
b = coeff (pCons a r) (degree y) / coeff y (degree y) |
1297 |
in (pCons b q, pCons a r - smult b y)))" |
|
29537 | 1298 |
apply (simp add: pdivmod_def Let_def, safe) |
29478 | 1299 |
apply (rule div_poly_eq) |
29537 | 1300 |
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) |
29478 | 1301 |
apply (rule mod_poly_eq) |
29537 | 1302 |
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) |
29478 | 1303 |
done |
1304 |
||
1305 |
end |