--- a/src/HOL/Library/Polynomial.thy Sat Jun 15 17:19:23 2013 +0200
+++ b/src/HOL/Library/Polynomial.thy Sat Jun 15 17:19:23 2013 +0200
@@ -1,47 +1,201 @@
(* Title: HOL/Library/Polynomial.thy
Author: Brian Huffman
Author: Clemens Ballarin
+ Author: Florian Haftmann
*)
-header {* Univariate Polynomials *}
+header {* Polynomials as type over a ring structure *}
theory Polynomial
-imports Main
+imports Main GCD
begin
+subsection {* Auxiliary: operations for lists (later) representing coefficients *}
+
+definition strip_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "strip_while P = rev \<circ> dropWhile P \<circ> rev"
+
+lemma strip_while_Nil [simp]:
+ "strip_while P [] = []"
+ by (simp add: strip_while_def)
+
+lemma strip_while_append [simp]:
+ "\<not> P x \<Longrightarrow> strip_while P (xs @ [x]) = xs @ [x]"
+ by (simp add: strip_while_def)
+
+lemma strip_while_append_rec [simp]:
+ "P x \<Longrightarrow> strip_while P (xs @ [x]) = strip_while P xs"
+ by (simp add: strip_while_def)
+
+lemma strip_while_Cons [simp]:
+ "\<not> P x \<Longrightarrow> strip_while P (x # xs) = x # strip_while P xs"
+ by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
+
+lemma strip_while_eq_Nil [simp]:
+ "strip_while P xs = [] \<longleftrightarrow> (\<forall>x\<in>set xs. P x)"
+ by (simp add: strip_while_def)
+
+lemma strip_while_eq_Cons_rec:
+ "strip_while P (x # xs) = x # strip_while P xs \<longleftrightarrow> \<not> (P x \<and> (\<forall>x\<in>set xs. P x))"
+ by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
+
+lemma strip_while_not_last [simp]:
+ "\<not> P (last xs) \<Longrightarrow> strip_while P xs = xs"
+ by (cases xs rule: rev_cases) simp_all
+
+lemma split_strip_while_append:
+ fixes xs :: "'a list"
+ obtains ys zs :: "'a list"
+ where "strip_while P xs = ys" and "\<forall>x\<in>set zs. P x" and "xs = ys @ zs"
+proof (rule that)
+ show "strip_while P xs = strip_while P xs" ..
+ show "\<forall>x\<in>set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
+ have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
+ by (simp add: strip_while_def)
+ then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
+ by (simp only: rev_is_rev_conv)
+qed
+
+
+definition nth_default :: "'a \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a"
+where
+ "nth_default x xs n = (if n < length xs then xs ! n else x)"
+
+lemma nth_default_Nil [simp]:
+ "nth_default y [] n = y"
+ by (simp add: nth_default_def)
+
+lemma nth_default_Cons_0 [simp]:
+ "nth_default y (x # xs) 0 = x"
+ by (simp add: nth_default_def)
+
+lemma nth_default_Cons_Suc [simp]:
+ "nth_default y (x # xs) (Suc n) = nth_default y xs n"
+ by (simp add: nth_default_def)
+
+lemma nth_default_map_eq:
+ "f y = x \<Longrightarrow> nth_default x (map f xs) n = f (nth_default y xs n)"
+ by (simp add: nth_default_def)
+
+lemma nth_default_strip_while_eq [simp]:
+ "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
+proof -
+ from split_strip_while_append obtain ys zs
+ where "strip_while (HOL.eq x) xs = ys" and "\<forall>z\<in>set zs. x = z" and "xs = ys @ zs" by blast
+ then show ?thesis by (simp add: nth_default_def not_less nth_append)
+qed
+
+
+definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "##" 65)
+where
+ "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
+
+lemma cCons_0_Nil_eq [simp]:
+ "0 ## [] = []"
+ by (simp add: cCons_def)
+
+lemma cCons_Cons_eq [simp]:
+ "x ## y # ys = x # y # ys"
+ by (simp add: cCons_def)
+
+lemma cCons_append_Cons_eq [simp]:
+ "x ## xs @ y # ys = x # xs @ y # ys"
+ by (simp add: cCons_def)
+
+lemma cCons_not_0_eq [simp]:
+ "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
+ by (simp add: cCons_def)
+
+lemma strip_while_not_0_Cons_eq [simp]:
+ "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
+proof (cases "x = 0")
+ case False then show ?thesis by simp
+next
+ case True show ?thesis
+ proof (induct xs rule: rev_induct)
+ case Nil with True show ?case by simp
+ next
+ case (snoc y ys) then show ?case
+ by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
+ qed
+qed
+
+lemma tl_cCons [simp]:
+ "tl (x ## xs) = xs"
+ by (simp add: cCons_def)
+
+
+subsection {* Almost everywhere zero functions *}
+
+definition almost_everywhere_zero :: "(nat \<Rightarrow> 'a::zero) \<Rightarrow> bool"
+where
+ "almost_everywhere_zero f \<longleftrightarrow> (\<exists>n. \<forall>i>n. f i = 0)"
+
+lemma almost_everywhere_zeroI:
+ "(\<And>i. i > n \<Longrightarrow> f i = 0) \<Longrightarrow> almost_everywhere_zero f"
+ by (auto simp add: almost_everywhere_zero_def)
+
+lemma almost_everywhere_zeroE:
+ assumes "almost_everywhere_zero f"
+ obtains n where "\<And>i. i > n \<Longrightarrow> f i = 0"
+proof -
+ from assms have "\<exists>n. \<forall>i>n. f i = 0" by (simp add: almost_everywhere_zero_def)
+ then obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by blast
+ with that show thesis .
+qed
+
+lemma almost_everywhere_zero_nat_case:
+ assumes "almost_everywhere_zero f"
+ shows "almost_everywhere_zero (nat_case a f)"
+ using assms
+ by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
+ blast
+
+lemma almost_everywhere_zero_Suc:
+ assumes "almost_everywhere_zero f"
+ shows "almost_everywhere_zero (\<lambda>n. f (Suc n))"
+proof -
+ from assms obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by (erule almost_everywhere_zeroE)
+ then have "\<And>i. i > n \<Longrightarrow> f (Suc i) = 0" by auto
+ then show ?thesis by (rule almost_everywhere_zeroI)
+qed
+
+
subsection {* Definition of type @{text poly} *}
-definition "Poly = {f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
-
-typedef 'a poly = "Poly :: (nat => 'a::zero) set"
+typedef 'a poly = "{f :: nat \<Rightarrow> 'a::zero. almost_everywhere_zero f}"
morphisms coeff Abs_poly
- unfolding Poly_def by auto
+ unfolding almost_everywhere_zero_def by auto
-(* FIXME should be named poly_eq_iff *)
-lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
+setup_lifting (no_code) type_definition_poly
+
+lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
by (simp add: coeff_inject [symmetric] fun_eq_iff)
-(* FIXME should be named poly_eqI *)
-lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
- by (simp add: expand_poly_eq)
+lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
+ by (simp add: poly_eq_iff)
+
+lemma coeff_almost_everywhere_zero:
+ "almost_everywhere_zero (coeff p)"
+ using coeff [of p] by simp
subsection {* Degree of a polynomial *}
-definition
- degree :: "'a::zero poly \<Rightarrow> nat" where
+definition degree :: "'a::zero poly \<Rightarrow> nat"
+where
"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
-lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
+lemma coeff_eq_0:
+ assumes "degree p < n"
+ shows "coeff p n = 0"
proof -
- have "coeff p \<in> Poly"
- by (rule coeff)
- hence "\<exists>n. \<forall>i>n. coeff p i = 0"
- unfolding Poly_def by simp
- hence "\<forall>i>degree p. coeff p i = 0"
+ from coeff_almost_everywhere_zero
+ have "\<exists>n. \<forall>i>n. coeff p i = 0" by (blast intro: almost_everywhere_zeroE)
+ then have "\<forall>i>degree p. coeff p i = 0"
unfolding degree_def by (rule LeastI_ex)
- moreover assume "degree p < n"
- ultimately show ?thesis by simp
+ with assms show ?thesis by simp
qed
lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
@@ -59,25 +213,28 @@
instantiation poly :: (zero) zero
begin
-definition
- zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
+lift_definition zero_poly :: "'a poly"
+ is "\<lambda>_. 0" by (rule almost_everywhere_zeroI) simp
instance ..
+
end
-lemma coeff_0 [simp]: "coeff 0 n = 0"
- unfolding zero_poly_def
- by (simp add: Abs_poly_inverse Poly_def)
+lemma coeff_0 [simp]:
+ "coeff 0 n = 0"
+ by transfer rule
-lemma degree_0 [simp]: "degree 0 = 0"
+lemma degree_0 [simp]:
+ "degree 0 = 0"
by (rule order_antisym [OF degree_le le0]) simp
lemma leading_coeff_neq_0:
- assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
+ assumes "p \<noteq> 0"
+ shows "coeff p (degree p) \<noteq> 0"
proof (cases "degree p")
case 0
from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
then obtain n where "coeff p n \<noteq> 0" ..
hence "n \<le> degree p" by (rule le_degree)
with `coeff p n \<noteq> 0` and `degree p = 0`
@@ -93,68 +250,59 @@
with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
qed
-lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
+lemma leading_coeff_0_iff [simp]:
+ "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
subsection {* List-style constructor for polynomials *}
-definition
- pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
- "pCons a p = Abs_poly (nat_case a (coeff p))"
+lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+ is "\<lambda>a p. nat_case a (coeff p)"
+ using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_nat_case)
-syntax
- "_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
+lemmas coeff_pCons = pCons.rep_eq
-translations
- "[:x, xs:]" == "CONST pCons x [:xs:]"
- "[:x:]" == "CONST pCons x 0"
- "[:x:]" <= "CONST pCons x (_constrain 0 t)"
+lemma coeff_pCons_0 [simp]:
+ "coeff (pCons a p) 0 = a"
+ by transfer simp
-lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
- unfolding Poly_def by (auto split: nat.split)
-
-lemma coeff_pCons:
- "coeff (pCons a p) = nat_case a (coeff p)"
- unfolding pCons_def
- by (simp add: Abs_poly_inverse Poly_nat_case coeff)
-
-lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
+lemma coeff_pCons_Suc [simp]:
+ "coeff (pCons a p) (Suc n) = coeff p n"
by (simp add: coeff_pCons)
-lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
- by (simp add: coeff_pCons)
-
-lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
-by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
+lemma degree_pCons_le:
+ "degree (pCons a p) \<le> Suc (degree p)"
+ by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
lemma degree_pCons_eq:
"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
-apply (rule order_antisym [OF degree_pCons_le])
-apply (rule le_degree, simp)
-done
+ apply (rule order_antisym [OF degree_pCons_le])
+ apply (rule le_degree, simp)
+ done
-lemma degree_pCons_0: "degree (pCons a 0) = 0"
-apply (rule order_antisym [OF _ le0])
-apply (rule degree_le, simp add: coeff_pCons split: nat.split)
-done
+lemma degree_pCons_0:
+ "degree (pCons a 0) = 0"
+ apply (rule order_antisym [OF _ le0])
+ apply (rule degree_le, simp add: coeff_pCons split: nat.split)
+ done
lemma degree_pCons_eq_if [simp]:
"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
-apply (cases "p = 0", simp_all)
-apply (rule order_antisym [OF _ le0])
-apply (rule degree_le, simp add: coeff_pCons split: nat.split)
-apply (rule order_antisym [OF degree_pCons_le])
-apply (rule le_degree, simp)
-done
+ apply (cases "p = 0", simp_all)
+ apply (rule order_antisym [OF _ le0])
+ apply (rule degree_le, simp add: coeff_pCons split: nat.split)
+ apply (rule order_antisym [OF degree_pCons_le])
+ apply (rule le_degree, simp)
+ done
-lemma pCons_0_0 [simp, code_post]: "pCons 0 0 = 0"
-by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+lemma pCons_0_0 [simp]:
+ "pCons 0 0 = 0"
+ by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
lemma pCons_eq_iff [simp]:
"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
-proof (safe)
+proof safe
assume "pCons a p = pCons b q"
then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
then show "a = b" by simp
@@ -162,23 +310,19 @@
assume "pCons a p = pCons b q"
then have "\<forall>n. coeff (pCons a p) (Suc n) =
coeff (pCons b q) (Suc n)" by simp
- then show "p = q" by (simp add: expand_poly_eq)
+ then show "p = q" by (simp add: poly_eq_iff)
qed
-lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
+lemma pCons_eq_0_iff [simp]:
+ "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
using pCons_eq_iff [of a p 0 0] by simp
-lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
- unfolding Poly_def
- by (clarify, rule_tac x=n in exI, simp)
-
lemma pCons_cases [cases type: poly]:
obtains (pCons) a q where "p = pCons a q"
proof
show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
- by (rule poly_ext)
- (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
- split: nat.split)
+ by transfer
+ (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
qed
lemma pCons_induct [case_names 0 pCons, induct type: poly]:
@@ -208,52 +352,227 @@
qed
-subsection {* Recursion combinator for polynomials *}
+subsection {* List-style syntax for polynomials *}
+
+syntax
+ "_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
+
+translations
+ "[:x, xs:]" == "CONST pCons x [:xs:]"
+ "[:x:]" == "CONST pCons x 0"
+ "[:x:]" <= "CONST pCons x (_constrain 0 t)"
+
+
+subsection {* Representation of polynomials by lists of coefficients *}
+
+primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
+where
+ "Poly [] = 0"
+| "Poly (a # as) = pCons a (Poly as)"
+
+lemma Poly_replicate_0 [simp]:
+ "Poly (replicate n 0) = 0"
+ by (induct n) simp_all
+
+lemma Poly_eq_0:
+ "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
+ by (induct as) (auto simp add: Cons_replicate_eq)
+
+definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
+where
+ "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
+
+lemma coeffs_eq_Nil [simp]:
+ "coeffs p = [] \<longleftrightarrow> p = 0"
+ by (simp add: coeffs_def)
+
+lemma not_0_coeffs_not_Nil:
+ "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
+ by simp
+
+lemma coeffs_0_eq_Nil [simp]:
+ "coeffs 0 = []"
+ by simp
-function
- poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
-where
- poly_rec_pCons_eq_if [simp del]:
- "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
-by (case_tac x, rename_tac q, case_tac q, auto)
+lemma coeffs_pCons_eq_cCons [simp]:
+ "coeffs (pCons a p) = a ## coeffs p"
+proof -
+ { fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
+ assume "\<forall>m\<in>set ms. m > 0"
+ then have "map (nat_case x f) ms = map f (map (\<lambda>n. n - 1) ms)"
+ by (induct ms) (auto, metis Suc_pred' nat_case_Suc) }
+ note * = this
+ show ?thesis
+ by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
+qed
+
+lemma not_0_cCons_eq [simp]:
+ "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
+ by (simp add: cCons_def)
+
+lemma Poly_coeffs [simp, code abstype]:
+ "Poly (coeffs p) = p"
+ by (induct p) (simp_all add: cCons_def)
+
+lemma coeffs_Poly [simp]:
+ "coeffs (Poly as) = strip_while (HOL.eq 0) as"
+proof (induct as)
+ case Nil then show ?case by simp
+next
+ case (Cons a as)
+ have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
+ using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
+ with Cons show ?case by auto
+qed
+
+lemma last_coeffs_not_0:
+ "p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
+ by (induct p) (auto simp add: cCons_def)
+
+lemma strip_while_coeffs [simp]:
+ "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
+ by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)
+
+lemma coeffs_eq_iff:
+ "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
+proof
+ assume ?P then show ?Q by simp
+next
+ assume ?Q
+ then have "Poly (coeffs p) = Poly (coeffs q)" by simp
+ then show ?P by simp
+qed
+
+lemma coeff_Poly_eq:
+ "coeff (Poly xs) n = nth_default 0 xs n"
+ apply (induct xs arbitrary: n) apply simp_all
+ by (metis nat_case_0 nat_case_Suc not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)
-termination poly_rec
-by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
- (simp add: degree_pCons_eq)
+lemma nth_default_coeffs_eq:
+ "nth_default 0 (coeffs p) = coeff p"
+ by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
+
+lemma [code]:
+ "coeff p = nth_default 0 (coeffs p)"
+ by (simp add: nth_default_coeffs_eq)
+
+lemma coeffs_eqI:
+ assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
+ assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
+ shows "coeffs p = xs"
+proof -
+ from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
+ with zero show ?thesis by simp (cases xs, simp_all)
+qed
+
+lemma degree_eq_length_coeffs [code]:
+ "degree p = length (coeffs p) - 1"
+ by (simp add: coeffs_def)
+
+lemma length_coeffs_degree:
+ "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
+ by (induct p) (auto simp add: cCons_def)
+
+lemma [code abstract]:
+ "coeffs 0 = []"
+ by (fact coeffs_0_eq_Nil)
+
+lemma [code abstract]:
+ "coeffs (pCons a p) = a ## coeffs p"
+ by (fact coeffs_pCons_eq_cCons)
+
+instantiation poly :: ("{zero, equal}") equal
+begin
+
+definition
+ [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
+
+instance proof
+qed (simp add: equal equal_poly_def coeffs_eq_iff)
+
+end
+
+lemma [code nbe]:
+ "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
+ by (fact equal_refl)
-lemma poly_rec_0:
- "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
- using poly_rec_pCons_eq_if [of z f 0 0] by simp
+definition is_zero :: "'a::zero poly \<Rightarrow> bool"
+where
+ [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
+
+lemma is_zero_null [code_abbrev]:
+ "is_zero p \<longleftrightarrow> p = 0"
+ by (simp add: is_zero_def null_def)
+
+
+subsection {* Fold combinator for polynomials *}
+
+definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
+where
+ "fold_coeffs f p = foldr f (coeffs p)"
+
+lemma fold_coeffs_0_eq [simp]:
+ "fold_coeffs f 0 = id"
+ by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_eq [simp]:
+ "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+ by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
-lemma poly_rec_pCons:
- "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
- by (simp add: poly_rec_pCons_eq_if poly_rec_0)
+lemma fold_coeffs_pCons_0_0_eq [simp]:
+ "fold_coeffs f (pCons 0 0) = id"
+ by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
+ "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+ by (simp add: fold_coeffs_def)
+
+lemma fold_coeffs_pCons_not_0_0_eq [simp]:
+ "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
+ by (simp add: fold_coeffs_def)
+
+
+subsection {* Canonical morphism on polynomials -- evaluation *}
+
+definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
+where
+ "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" -- {* The Horner Schema *}
+
+lemma poly_0 [simp]:
+ "poly 0 x = 0"
+ by (simp add: poly_def)
+
+lemma poly_pCons [simp]:
+ "poly (pCons a p) x = a + x * poly p x"
+ by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
subsection {* Monomials *}
-definition
- monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
- "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
+lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
+ is "\<lambda>a m n. if m = n then a else 0"
+ by (auto intro!: almost_everywhere_zeroI)
+
+lemma coeff_monom [simp]:
+ "coeff (monom a m) n = (if m = n then a else 0)"
+ by transfer rule
-lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
- unfolding monom_def
- by (subst Abs_poly_inverse, auto simp add: Poly_def)
+lemma monom_0:
+ "monom a 0 = pCons a 0"
+ by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
-lemma monom_0: "monom a 0 = pCons a 0"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
-
-lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+lemma monom_Suc:
+ "monom a (Suc n) = pCons 0 (monom a n)"
+ by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
lemma monom_eq_0 [simp]: "monom 0 n = 0"
- by (rule poly_ext) simp
+ by (rule poly_eqI) simp
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
lemma degree_monom_le: "degree (monom a n) \<le> n"
by (rule degree_le, simp)
@@ -263,37 +582,47 @@
apply (rule le_degree, simp)
done
+lemma coeffs_monom [code abstract]:
+ "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
+ by (induct n) (simp_all add: monom_0 monom_Suc)
+
+lemma fold_coeffs_monom [simp]:
+ "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
+ by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
+
+lemma poly_monom:
+ fixes a x :: "'a::{comm_semiring_1}"
+ shows "poly (monom a n) x = a * x ^ n"
+ by (cases "a = 0", simp_all)
+ (induct n, simp_all add: mult.left_commute poly_def)
+
subsection {* Addition and subtraction *}
instantiation poly :: (comm_monoid_add) comm_monoid_add
begin
-definition
- plus_poly_def:
- "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
-
-lemma Poly_add:
- fixes f g :: "nat \<Rightarrow> 'a"
- shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
- unfolding Poly_def
- apply (clarify, rename_tac m n)
- apply (rule_tac x="max m n" in exI, simp)
- done
+lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+ is "\<lambda>p q n. coeff p n + coeff q n"
+proof (rule almost_everywhere_zeroI)
+ fix q p :: "'a poly" and i
+ assume "max (degree q) (degree p) < i"
+ then show "coeff p i + coeff q i = 0"
+ by (simp add: coeff_eq_0)
+qed
lemma coeff_add [simp]:
"coeff (p + q) n = coeff p n + coeff q n"
- unfolding plus_poly_def
- by (simp add: Abs_poly_inverse coeff Poly_add)
+ by (simp add: plus_poly.rep_eq)
instance proof
fix p q r :: "'a poly"
show "(p + q) + r = p + (q + r)"
- by (simp add: expand_poly_eq add_assoc)
+ by (simp add: poly_eq_iff add_assoc)
show "p + q = q + p"
- by (simp add: expand_poly_eq add_commute)
+ by (simp add: poly_eq_iff add_commute)
show "0 + p = p"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
qed
end
@@ -302,60 +631,58 @@
proof
fix p q r :: "'a poly"
assume "p + q = p + r" thus "q = r"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
qed
instantiation poly :: (ab_group_add) ab_group_add
begin
-definition
- uminus_poly_def:
- "- p = Abs_poly (\<lambda>n. - coeff p n)"
-
-definition
- minus_poly_def:
- "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
+lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
+ is "\<lambda>p n. - coeff p n"
+proof (rule almost_everywhere_zeroI)
+ fix p :: "'a poly" and i
+ assume "degree p < i"
+ then show "- coeff p i = 0"
+ by (simp add: coeff_eq_0)
+qed
-lemma Poly_minus:
- fixes f :: "nat \<Rightarrow> 'a"
- shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
- unfolding Poly_def by simp
-
-lemma Poly_diff:
- fixes f g :: "nat \<Rightarrow> 'a"
- shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
- unfolding diff_minus by (simp add: Poly_add Poly_minus)
+lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+ is "\<lambda>p q n. coeff p n - coeff q n"
+proof (rule almost_everywhere_zeroI)
+ fix q p :: "'a poly" and i
+ assume "max (degree q) (degree p) < i"
+ then show "coeff p i - coeff q i = 0"
+ by (simp add: coeff_eq_0)
+qed
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
- unfolding uminus_poly_def
- by (simp add: Abs_poly_inverse coeff Poly_minus)
+ by (simp add: uminus_poly.rep_eq)
lemma coeff_diff [simp]:
"coeff (p - q) n = coeff p n - coeff q n"
- unfolding minus_poly_def
- by (simp add: Abs_poly_inverse coeff Poly_diff)
+ by (simp add: minus_poly.rep_eq)
instance proof
fix p q :: "'a poly"
show "- p + p = 0"
- by (simp add: expand_poly_eq)
+ by (simp add: poly_eq_iff)
show "p - q = p + - q"
- by (simp add: expand_poly_eq diff_minus)
+ by (simp add: poly_eq_iff diff_minus)
qed
end
lemma add_pCons [simp]:
"pCons a p + pCons b q = pCons (a + b) (p + q)"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+ by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
lemma minus_pCons [simp]:
"- pCons a p = pCons (- a) (- p)"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+ by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
lemma diff_pCons [simp]:
"pCons a p - pCons b q = pCons (a - b) (p - q)"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+ by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
by (rule degree_le, auto simp add: coeff_eq_0)
@@ -398,75 +725,133 @@
by (simp add: diff_minus degree_add_less)
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
- by (rule poly_ext) simp
+ by (rule poly_eqI) simp
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
- by (rule poly_ext) simp
+ by (rule poly_eqI) simp
lemma minus_monom: "- monom a n = monom (-a) n"
- by (rule poly_ext) simp
+ by (rule poly_eqI) simp
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
by (cases "finite A", induct set: finite, simp_all)
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
- by (rule poly_ext) (simp add: coeff_setsum)
+ by (rule poly_eqI) (simp add: coeff_setsum)
+
+fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ "plus_coeffs xs [] = xs"
+| "plus_coeffs [] ys = ys"
+| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
+
+lemma coeffs_plus_eq_plus_coeffs [code abstract]:
+ "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
+proof -
+ { fix xs ys :: "'a list" and n
+ have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
+ proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
+ case (3 x xs y ys n) then show ?case by (cases n) (auto simp add: cCons_def)
+ qed simp_all }
+ note * = this
+ { fix xs ys :: "'a list"
+ assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
+ moreover assume "plus_coeffs xs ys \<noteq> []"
+ ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
+ proof (induct xs ys rule: plus_coeffs.induct)
+ case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
+ qed simp_all }
+ note ** = this
+ show ?thesis
+ apply (rule coeffs_eqI)
+ apply (simp add: * nth_default_coeffs_eq)
+ apply (rule **)
+ apply (auto dest: last_coeffs_not_0)
+ done
+qed
+
+lemma coeffs_uminus [code abstract]:
+ "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
+ by (rule coeffs_eqI)
+ (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
+
+lemma [code]:
+ fixes p q :: "'a::ab_group_add poly"
+ shows "p - q = p + - q"
+ by simp
+
+lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
+ apply (induct p arbitrary: q, simp)
+ apply (case_tac q, simp, simp add: algebra_simps)
+ done
+
+lemma poly_minus [simp]:
+ fixes x :: "'a::comm_ring"
+ shows "poly (- p) x = - poly p x"
+ by (induct p) simp_all
+
+lemma poly_diff [simp]:
+ fixes x :: "'a::comm_ring"
+ shows "poly (p - q) x = poly p x - poly q x"
+ by (simp add: diff_minus)
+
+lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
+ by (induct A rule: infinite_finite_induct) simp_all
-subsection {* Multiplication by a constant *}
-
-definition
- smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
- "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
+subsection {* Multiplication by a constant, polynomial multiplication and the unit polynomial *}
-lemma Poly_smult:
- fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
- shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
- unfolding Poly_def
- by (clarify, rule_tac x=n in exI, simp)
+lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+ is "\<lambda>a p n. a * coeff p n"
+proof (rule almost_everywhere_zeroI)
+ fix a :: 'a and p :: "'a poly" and i
+ assume "degree p < i"
+ then show "a * coeff p i = 0"
+ by (simp add: coeff_eq_0)
+qed
-lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
- unfolding smult_def
- by (simp add: Abs_poly_inverse Poly_smult coeff)
+lemma coeff_smult [simp]:
+ "coeff (smult a p) n = a * coeff p n"
+ by (simp add: smult.rep_eq)
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
by (rule degree_le, simp add: coeff_eq_0)
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
- by (rule poly_ext, simp add: mult_assoc)
+ by (rule poly_eqI, simp add: mult_assoc)
lemma smult_0_right [simp]: "smult a 0 = 0"
- by (rule poly_ext, simp)
+ by (rule poly_eqI, simp)
lemma smult_0_left [simp]: "smult 0 p = 0"
- by (rule poly_ext, simp)
+ by (rule poly_eqI, simp)
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
- by (rule poly_ext, simp)
+ by (rule poly_eqI, simp)
lemma smult_add_right:
"smult a (p + q) = smult a p + smult a q"
- by (rule poly_ext, simp add: algebra_simps)
+ by (rule poly_eqI, simp add: algebra_simps)
lemma smult_add_left:
"smult (a + b) p = smult a p + smult b p"
- by (rule poly_ext, simp add: algebra_simps)
+ by (rule poly_eqI, simp add: algebra_simps)
lemma smult_minus_right [simp]:
"smult (a::'a::comm_ring) (- p) = - smult a p"
- by (rule poly_ext, simp)
+ by (rule poly_eqI, simp)
lemma smult_minus_left [simp]:
"smult (- a::'a::comm_ring) p = - smult a p"
- by (rule poly_ext, simp)
+ by (rule poly_eqI, simp)
lemma smult_diff_right:
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
- by (rule poly_ext, simp add: algebra_simps)
+ by (rule poly_eqI, simp add: algebra_simps)
lemma smult_diff_left:
"smult (a - b::'a::comm_ring) p = smult a p - smult b p"
- by (rule poly_ext, simp add: algebra_simps)
+ by (rule poly_eqI, simp add: algebra_simps)
lemmas smult_distribs =
smult_add_left smult_add_right
@@ -474,7 +859,7 @@
lemma smult_pCons [simp]:
"smult a (pCons b p) = pCons (a * b) (smult a p)"
- by (rule poly_ext, simp add: coeff_pCons split: nat.split)
+ by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
by (induct n, simp add: monom_0, simp add: monom_Suc)
@@ -487,65 +872,48 @@
lemma smult_eq_0_iff [simp]:
fixes a :: "'a::idom"
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
- by (simp add: expand_poly_eq)
-
-
-subsection {* Multiplication of polynomials *}
+ by (simp add: poly_eq_iff)
-(* TODO: move to Set_Interval.thy *)
-lemma setsum_atMost_Suc_shift:
- fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
- shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
-proof (induct n)
- case 0 show ?case by simp
-next
- case (Suc n) note IH = this
- have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
- by (rule setsum_atMost_Suc)
- also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
- by (rule IH)
- also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
- f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
- by (rule add_assoc)
- also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
- by (rule setsum_atMost_Suc [symmetric])
- finally show ?case .
-qed
+lemma coeffs_smult [code abstract]:
+ fixes p :: "'a::idom poly"
+ shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
+ by (rule coeffs_eqI)
+ (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
instantiation poly :: (comm_semiring_0) comm_semiring_0
begin
definition
- times_poly_def:
- "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
+ "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
lemma mult_poly_0_left: "(0::'a poly) * q = 0"
- unfolding times_poly_def by (simp add: poly_rec_0)
+ by (simp add: times_poly_def)
lemma mult_pCons_left [simp]:
"pCons a p * q = smult a q + pCons 0 (p * q)"
- unfolding times_poly_def by (simp add: poly_rec_pCons)
+ by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
lemma mult_poly_0_right: "p * (0::'a poly) = 0"
- by (induct p, simp add: mult_poly_0_left, simp)
+ by (induct p) (simp add: mult_poly_0_left, simp)
lemma mult_pCons_right [simp]:
"p * pCons a q = smult a p + pCons 0 (p * q)"
- by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)
+ by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps)
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
-lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
- by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
+lemma mult_smult_left [simp]:
+ "smult a p * q = smult a (p * q)"
+ by (induct p) (simp add: mult_poly_0, simp add: smult_add_right)
-lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
- by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
+lemma mult_smult_right [simp]:
+ "p * smult a q = smult a (p * q)"
+ by (induct q) (simp add: mult_poly_0, simp add: smult_add_right)
lemma mult_poly_add_left:
fixes p q r :: "'a poly"
shows "(p + q) * r = p * r + q * r"
- by (induct r, simp add: mult_poly_0,
- simp add: smult_distribs algebra_simps)
+ by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps)
instance proof
fix p q r :: "'a poly"
@@ -585,20 +953,15 @@
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
-
-subsection {* The unit polynomial and exponentiation *}
-
instantiation poly :: (comm_semiring_1) comm_semiring_1
begin
-definition
- one_poly_def:
- "1 = pCons 1 0"
+definition one_poly_def:
+ "1 = pCons 1 0"
instance proof
fix p :: "'a poly" show "1 * p = p"
- unfolding one_poly_def
- by simp
+ unfolding one_poly_def by simp
next
show "0 \<noteq> (1::'a poly)"
unfolding one_poly_def by simp
@@ -608,6 +971,10 @@
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
+instance poly :: (comm_ring) comm_ring ..
+
+instance poly :: (comm_ring_1) comm_ring_1 ..
+
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
unfolding one_poly_def
by (simp add: coeff_pCons split: nat.split)
@@ -616,7 +983,33 @@
unfolding one_poly_def
by (rule degree_pCons_0)
-text {* Lemmas about divisibility *}
+lemma coeffs_1_eq [simp, code abstract]:
+ "coeffs 1 = [1]"
+ by (simp add: one_poly_def)
+
+lemma degree_power_le:
+ "degree (p ^ n) \<le> degree p * n"
+ by (induct n) (auto intro: order_trans degree_mult_le)
+
+lemma poly_smult [simp]:
+ "poly (smult a p) x = a * poly p x"
+ by (induct p, simp, simp add: algebra_simps)
+
+lemma poly_mult [simp]:
+ "poly (p * q) x = poly p x * poly q x"
+ by (induct p, simp_all, simp add: algebra_simps)
+
+lemma poly_1 [simp]:
+ "poly 1 x = 1"
+ by (simp add: one_poly_def)
+
+lemma poly_power [simp]:
+ fixes p :: "'a::{comm_semiring_1} poly"
+ shows "poly (p ^ n) x = poly p x ^ n"
+ by (induct n) simp_all
+
+
+subsection {* Lemmas about divisibility *}
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
proof -
@@ -655,13 +1048,6 @@
shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
by (auto elim: smult_dvd smult_dvd_cancel)
-lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
-by (induct n, simp, auto intro: order_trans degree_mult_le)
-
-instance poly :: (comm_ring) comm_ring ..
-
-instance poly :: (comm_ring_1) comm_ring_1 ..
-
subsection {* Polynomials form an integral domain *}
@@ -680,7 +1066,7 @@
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
using `p \<noteq> 0` and `q \<noteq> 0` by simp
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
- thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
+ thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
qed
lemma degree_mult_eq:
@@ -698,8 +1084,7 @@
subsection {* Polynomials form an ordered integral domain *}
-definition
- pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
+definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
where
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
@@ -725,6 +1110,20 @@
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
by (induct p) (auto simp add: pos_poly_pCons)
+lemma last_coeffs_eq_coeff_degree:
+ "p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"
+ by (simp add: coeffs_def)
+
+lemma pos_poly_coeffs [code]:
+ "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q")
+proof
+ assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
+next
+ assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def)
+ then have "p \<noteq> 0" by auto
+ with * show ?Q by (simp add: last_coeffs_eq_coeff_degree)
+qed
+
instantiation poly :: (linordered_idom) linordered_idom
begin
@@ -802,10 +1201,145 @@
text {* TODO: Simplification rules for comparisons *}
+subsection {* Synthetic division and polynomial roots *}
+
+text {*
+ Synthetic division is simply division by the linear polynomial @{term "x - c"}.
+*}
+
+definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
+where
+ "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
+
+definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
+where
+ "synthetic_div p c = fst (synthetic_divmod p c)"
+
+lemma synthetic_divmod_0 [simp]:
+ "synthetic_divmod 0 c = (0, 0)"
+ by (simp add: synthetic_divmod_def)
+
+lemma synthetic_divmod_pCons [simp]:
+ "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
+ by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
+
+lemma synthetic_div_0 [simp]:
+ "synthetic_div 0 c = 0"
+ unfolding synthetic_div_def by simp
+
+lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
+by (induct p arbitrary: a) simp_all
+
+lemma snd_synthetic_divmod:
+ "snd (synthetic_divmod p c) = poly p c"
+ by (induct p, simp, simp add: split_def)
+
+lemma synthetic_div_pCons [simp]:
+ "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
+ unfolding synthetic_div_def
+ by (simp add: split_def snd_synthetic_divmod)
+
+lemma synthetic_div_eq_0_iff:
+ "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
+ by (induct p, simp, case_tac p, simp)
+
+lemma degree_synthetic_div:
+ "degree (synthetic_div p c) = degree p - 1"
+ by (induct p, simp, simp add: synthetic_div_eq_0_iff)
+
+lemma synthetic_div_correct:
+ "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
+ by (induct p) simp_all
+
+lemma synthetic_div_unique:
+ "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
+apply (induct p arbitrary: q r)
+apply (simp, frule synthetic_div_unique_lemma, simp)
+apply (case_tac q, force)
+done
+
+lemma synthetic_div_correct':
+ fixes c :: "'a::comm_ring_1"
+ shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
+ using synthetic_div_correct [of p c]
+ by (simp add: algebra_simps)
+
+lemma poly_eq_0_iff_dvd:
+ fixes c :: "'a::idom"
+ shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
+proof
+ assume "poly p c = 0"
+ with synthetic_div_correct' [of c p]
+ have "p = [:-c, 1:] * synthetic_div p c" by simp
+ then show "[:-c, 1:] dvd p" ..
+next
+ assume "[:-c, 1:] dvd p"
+ then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
+ then show "poly p c = 0" by simp
+qed
+
+lemma dvd_iff_poly_eq_0:
+ fixes c :: "'a::idom"
+ shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
+ by (simp add: poly_eq_0_iff_dvd)
+
+lemma poly_roots_finite:
+ fixes p :: "'a::idom poly"
+ shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
+proof (induct n \<equiv> "degree p" arbitrary: p)
+ case (0 p)
+ then obtain a where "a \<noteq> 0" and "p = [:a:]"
+ by (cases p, simp split: if_splits)
+ then show "finite {x. poly p x = 0}" by simp
+next
+ case (Suc n p)
+ show "finite {x. poly p x = 0}"
+ proof (cases "\<exists>x. poly p x = 0")
+ case False
+ then show "finite {x. poly p x = 0}" by simp
+ next
+ case True
+ then obtain a where "poly p a = 0" ..
+ then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
+ then obtain k where k: "p = [:-a, 1:] * k" ..
+ with `p \<noteq> 0` have "k \<noteq> 0" by auto
+ with k have "degree p = Suc (degree k)"
+ by (simp add: degree_mult_eq del: mult_pCons_left)
+ with `Suc n = degree p` have "n = degree k" by simp
+ then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
+ then have "finite (insert a {x. poly k x = 0})" by simp
+ then show "finite {x. poly p x = 0}"
+ by (simp add: k uminus_add_conv_diff Collect_disj_eq
+ del: mult_pCons_left)
+ qed
+qed
+
+lemma poly_eq_poly_eq_iff:
+ fixes p q :: "'a::{idom,ring_char_0} poly"
+ shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
+proof
+ assume ?Q then show ?P by simp
+next
+ { fix p :: "'a::{idom,ring_char_0} poly"
+ have "poly p = poly 0 \<longleftrightarrow> p = 0"
+ apply (cases "p = 0", simp_all)
+ apply (drule poly_roots_finite)
+ apply (auto simp add: infinite_UNIV_char_0)
+ done
+ } note this [of "p - q"]
+ moreover assume ?P
+ ultimately show ?Q by auto
+qed
+
+lemma poly_all_0_iff_0:
+ fixes p :: "'a::{ring_char_0, idom} poly"
+ shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
+ by (auto simp add: poly_eq_poly_eq_iff [symmetric])
+
+
subsection {* Long division of polynomials *}
-definition
- pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
+definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
where
"pdivmod_rel x y q r \<longleftrightarrow>
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
@@ -1106,327 +1640,54 @@
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
done
-
-subsection {* GCD of polynomials *}
-
-function
- poly_gcd :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
- "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x"
-| "y \<noteq> 0 \<Longrightarrow> poly_gcd x y = poly_gcd y (x mod y)"
-by auto
-
-termination poly_gcd
-by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
- (auto dest: degree_mod_less)
-
-declare poly_gcd.simps [simp del]
-
-lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x"
- and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y"
- apply (induct x y rule: poly_gcd.induct)
- apply (simp_all add: poly_gcd.simps)
- apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
- apply (blast dest: dvd_mod_imp_dvd)
- done
-
-lemma poly_gcd_greatest: "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd poly_gcd x y"
- by (induct x y rule: poly_gcd.induct)
- (simp_all add: poly_gcd.simps dvd_mod dvd_smult)
-
-lemma dvd_poly_gcd_iff [iff]:
- "k dvd poly_gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
- by (blast intro!: poly_gcd_greatest intro: dvd_trans)
+definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
+where
+ "pdivmod p q = (p div q, p mod q)"
-lemma poly_gcd_monic:
- "coeff (poly_gcd x y) (degree (poly_gcd x y)) =
- (if x = 0 \<and> y = 0 then 0 else 1)"
- by (induct x y rule: poly_gcd.induct)
- (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero)
-
-lemma poly_gcd_zero_iff [simp]:
- "poly_gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
- by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
-
-lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0"
- by simp
+lemma div_poly_code [code]:
+ "p div q = fst (pdivmod p q)"
+ by (simp add: pdivmod_def)
-lemma poly_dvd_antisym:
- fixes p q :: "'a::idom poly"
- assumes coeff: "coeff p (degree p) = coeff q (degree q)"
- assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
-proof (cases "p = 0")
- case True with coeff show "p = q" by simp
-next
- case False with coeff have "q \<noteq> 0" by auto
- have degree: "degree p = degree q"
- using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
- by (intro order_antisym dvd_imp_degree_le)
-
- from `p dvd q` obtain a where a: "q = p * a" ..
- with `q \<noteq> 0` have "a \<noteq> 0" by auto
- with degree a `p \<noteq> 0` have "degree a = 0"
- by (simp add: degree_mult_eq)
- with coeff a show "p = q"
- by (cases a, auto split: if_splits)
-qed
+lemma mod_poly_code [code]:
+ "p mod q = snd (pdivmod p q)"
+ by (simp add: pdivmod_def)
-lemma poly_gcd_unique:
- assumes dvd1: "d dvd x" and dvd2: "d dvd y"
- and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
- and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
- shows "poly_gcd x y = d"
-proof -
- have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)"
- by (simp_all add: poly_gcd_monic monic)
- moreover have "poly_gcd x y dvd d"
- using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
- moreover have "d dvd poly_gcd x y"
- using dvd1 dvd2 by (rule poly_gcd_greatest)
- ultimately show ?thesis
- by (rule poly_dvd_antisym)
-qed
-
-interpretation poly_gcd: abel_semigroup poly_gcd
-proof
- fix x y z :: "'a poly"
- show "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)"
- by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
- show "poly_gcd x y = poly_gcd y x"
- by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-qed
-
-lemmas poly_gcd_assoc = poly_gcd.assoc
-lemmas poly_gcd_commute = poly_gcd.commute
-lemmas poly_gcd_left_commute = poly_gcd.left_commute
-
-lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
-
-lemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1"
-by (rule poly_gcd_unique) simp_all
+lemma pdivmod_0:
+ "pdivmod 0 q = (0, 0)"
+ by (simp add: pdivmod_def)
-lemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1"
-by (rule poly_gcd_unique) simp_all
-
-lemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-
-subsection {* Evaluation of polynomials *}
-
-definition
- poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
- "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
-
-lemma poly_0 [simp]: "poly 0 x = 0"
- unfolding poly_def by (simp add: poly_rec_0)
-
-lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
- unfolding poly_def by (simp add: poly_rec_pCons)
-
-lemma poly_1 [simp]: "poly 1 x = 1"
- unfolding one_poly_def by simp
-
-lemma poly_monom:
- fixes a x :: "'a::{comm_semiring_1}"
- shows "poly (monom a n) x = a * x ^ n"
- by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
-
-lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
- apply (induct p arbitrary: q, simp)
- apply (case_tac q, simp, simp add: algebra_simps)
+lemma pdivmod_pCons:
+ "pdivmod (pCons a p) q =
+ (if q = 0 then (0, pCons a p) else
+ (let (s, r) = pdivmod p q;
+ b = coeff (pCons a r) (degree q) / coeff q (degree q)
+ in (pCons b s, pCons a r - smult b q)))"
+ apply (simp add: pdivmod_def Let_def, safe)
+ apply (rule div_poly_eq)
+ apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
+ apply (rule mod_poly_eq)
+ apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
done
-lemma poly_minus [simp]:
- fixes x :: "'a::comm_ring"
- shows "poly (- p) x = - poly p x"
- by (induct p, simp_all)
-
-lemma poly_diff [simp]:
- fixes x :: "'a::comm_ring"
- shows "poly (p - q) x = poly p x - poly q x"
- by (simp add: diff_minus)
-
-lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
- by (cases "finite A", induct set: finite, simp_all)
-
-lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
- by (induct p, simp, simp add: algebra_simps)
-
-lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
- by (induct p, simp_all, simp add: algebra_simps)
-
-lemma poly_power [simp]:
- fixes p :: "'a::{comm_semiring_1} poly"
- shows "poly (p ^ n) x = poly p x ^ n"
- by (induct n, simp, simp add: power_Suc)
-
-
-subsection {* Synthetic division *}
-
-text {*
- Synthetic division is simply division by the
- linear polynomial @{term "x - c"}.
-*}
-
-definition
- synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
-where
- "synthetic_divmod p c =
- poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
-
-definition
- synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
-where
- "synthetic_div p c = fst (synthetic_divmod p c)"
-
-lemma synthetic_divmod_0 [simp]:
- "synthetic_divmod 0 c = (0, 0)"
- unfolding synthetic_divmod_def
- by (simp add: poly_rec_0)
-
-lemma synthetic_divmod_pCons [simp]:
- "synthetic_divmod (pCons a p) c =
- (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
- unfolding synthetic_divmod_def
- by (simp add: poly_rec_pCons)
-
-lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
- by (induct p, simp, simp add: split_def)
-
-lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
- unfolding synthetic_div_def by simp
-
-lemma synthetic_div_pCons [simp]:
- "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
- unfolding synthetic_div_def
- by (simp add: split_def snd_synthetic_divmod)
-
-lemma synthetic_div_eq_0_iff:
- "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
- by (induct p, simp, case_tac p, simp)
-
-lemma degree_synthetic_div:
- "degree (synthetic_div p c) = degree p - 1"
- by (induct p, simp, simp add: synthetic_div_eq_0_iff)
-
-lemma synthetic_div_correct:
- "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
- by (induct p) simp_all
-
-lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
-by (induct p arbitrary: a) simp_all
-
-lemma synthetic_div_unique:
- "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
-apply (induct p arbitrary: q r)
-apply (simp, frule synthetic_div_unique_lemma, simp)
-apply (case_tac q, force)
-done
-
-lemma synthetic_div_correct':
- fixes c :: "'a::comm_ring_1"
- shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
- using synthetic_div_correct [of p c]
- by (simp add: algebra_simps)
-
-lemma poly_eq_0_iff_dvd:
- fixes c :: "'a::idom"
- shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
-proof
- assume "poly p c = 0"
- with synthetic_div_correct' [of c p]
- have "p = [:-c, 1:] * synthetic_div p c" by simp
- then show "[:-c, 1:] dvd p" ..
-next
- assume "[:-c, 1:] dvd p"
- then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
- then show "poly p c = 0" by simp
-qed
-
-lemma dvd_iff_poly_eq_0:
- fixes c :: "'a::idom"
- shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
- by (simp add: poly_eq_0_iff_dvd)
-
-lemma poly_roots_finite:
- fixes p :: "'a::idom poly"
- shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
-proof (induct n \<equiv> "degree p" arbitrary: p)
- case (0 p)
- then obtain a where "a \<noteq> 0" and "p = [:a:]"
- by (cases p, simp split: if_splits)
- then show "finite {x. poly p x = 0}" by simp
-next
- case (Suc n p)
- show "finite {x. poly p x = 0}"
- proof (cases "\<exists>x. poly p x = 0")
- case False
- then show "finite {x. poly p x = 0}" by simp
- next
- case True
- then obtain a where "poly p a = 0" ..
- then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
- then obtain k where k: "p = [:-a, 1:] * k" ..
- with `p \<noteq> 0` have "k \<noteq> 0" by auto
- with k have "degree p = Suc (degree k)"
- by (simp add: degree_mult_eq del: mult_pCons_left)
- with `Suc n = degree p` have "n = degree k" by simp
- then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
- then have "finite (insert a {x. poly k x = 0})" by simp
- then show "finite {x. poly p x = 0}"
- by (simp add: k uminus_add_conv_diff Collect_disj_eq
- del: mult_pCons_left)
- qed
-qed
-
-lemma poly_zero:
- fixes p :: "'a::{idom,ring_char_0} poly"
- shows "poly p = poly 0 \<longleftrightarrow> p = 0"
-apply (cases "p = 0", simp_all)
-apply (drule poly_roots_finite)
-apply (auto simp add: infinite_UNIV_char_0)
-done
-
-lemma poly_eq_iff:
- fixes p q :: "'a::{idom,ring_char_0} poly"
- shows "poly p = poly q \<longleftrightarrow> p = q"
- using poly_zero [of "p - q"]
- by (simp add: fun_eq_iff)
-
-
-subsection {* Composition of polynomials *}
-
-definition
- pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
- "pcompose p q = poly_rec 0 (\<lambda>a _ c. [:a:] + q * c) p"
-
-lemma pcompose_0 [simp]: "pcompose 0 q = 0"
- unfolding pcompose_def by (simp add: poly_rec_0)
-
-lemma pcompose_pCons:
- "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
- unfolding pcompose_def by (simp add: poly_rec_pCons)
-
-lemma poly_pcompose: "poly (pcompose p q) x = poly p (poly q x)"
- by (induct p) (simp_all add: pcompose_pCons)
-
-lemma degree_pcompose_le:
- "degree (pcompose p q) \<le> degree p * degree q"
-apply (induct p, simp)
-apply (simp add: pcompose_pCons, clarify)
-apply (rule degree_add_le, simp)
-apply (rule order_trans [OF degree_mult_le], simp)
-done
+lemma pdivmod_fold_coeffs [code]:
+ "pdivmod p q = (if q = 0 then (0, p)
+ else fold_coeffs (\<lambda>a (s, r).
+ let b = coeff (pCons a r) (degree q) / coeff q (degree q)
+ in (pCons b s, pCons a r - smult b q)
+ ) p (0, 0))"
+ apply (cases "q = 0")
+ apply (simp add: pdivmod_def)
+ apply (rule sym)
+ apply (induct p)
+ apply (simp_all add: pdivmod_0 pdivmod_pCons)
+ apply (case_tac "a = 0 \<and> p = 0")
+ apply (auto simp add: pdivmod_def)
+ done
subsection {* Order of polynomial roots *}
-definition
- order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
+definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
where
"order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
@@ -1490,107 +1751,161 @@
done
-subsection {* Configuration of the code generator *}
-
-code_datatype "0::'a::zero poly" pCons
+subsection {* GCD of polynomials *}
-quickcheck_generator poly constructors: "0::'a::zero poly", pCons
-
-instantiation poly :: ("{zero, equal}") equal
+instantiation poly :: (field) gcd
begin
-definition
- "HOL.equal (p::'a poly) q \<longleftrightarrow> p = q"
+function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+where
+ "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
+| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
+by auto
-instance proof
-qed (rule equal_poly_def)
+termination "gcd :: _ poly \<Rightarrow> _"
+by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
+ (auto dest: degree_mod_less)
+
+declare gcd_poly.simps [simp del]
+
+instance ..
end
-lemma eq_poly_code [code]:
- "HOL.equal (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
- "HOL.equal (0::_ poly) (pCons b q) \<longleftrightarrow> HOL.equal 0 b \<and> HOL.equal 0 q"
- "HOL.equal (pCons a p) (0::_ poly) \<longleftrightarrow> HOL.equal a 0 \<and> HOL.equal p 0"
- "HOL.equal (pCons a p) (pCons b q) \<longleftrightarrow> HOL.equal a b \<and> HOL.equal p q"
- by (simp_all add: equal)
+lemma
+ fixes x y :: "_ poly"
+ shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
+ and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
+ apply (induct x y rule: gcd_poly.induct)
+ apply (simp_all add: gcd_poly.simps)
+ apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
+ apply (blast dest: dvd_mod_imp_dvd)
+ done
-lemma [code nbe]:
- "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
- by (fact equal_refl)
+lemma poly_gcd_greatest:
+ fixes k x y :: "_ poly"
+ shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
+ by (induct x y rule: gcd_poly.induct)
+ (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
-lemmas coeff_code [code] =
- coeff_0 coeff_pCons_0 coeff_pCons_Suc
+lemma dvd_poly_gcd_iff [iff]:
+ fixes k x y :: "_ poly"
+ shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
+ by (blast intro!: poly_gcd_greatest intro: dvd_trans)
-lemmas degree_code [code] =
- degree_0 degree_pCons_eq_if
+lemma poly_gcd_monic:
+ fixes x y :: "_ poly"
+ shows "coeff (gcd x y) (degree (gcd x y)) =
+ (if x = 0 \<and> y = 0 then 0 else 1)"
+ by (induct x y rule: gcd_poly.induct)
+ (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
-lemmas monom_poly_code [code] =
- monom_0 monom_Suc
+lemma poly_gcd_zero_iff [simp]:
+ fixes x y :: "_ poly"
+ shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+ by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
-lemma add_poly_code [code]:
- "0 + q = (q :: _ poly)"
- "p + 0 = (p :: _ poly)"
- "pCons a p + pCons b q = pCons (a + b) (p + q)"
-by simp_all
+lemma poly_gcd_0_0 [simp]:
+ "gcd (0::_ poly) 0 = 0"
+ by simp
-lemma minus_poly_code [code]:
- "- 0 = (0 :: _ poly)"
- "- pCons a p = pCons (- a) (- p)"
-by simp_all
+lemma poly_dvd_antisym:
+ fixes p q :: "'a::idom poly"
+ assumes coeff: "coeff p (degree p) = coeff q (degree q)"
+ assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
+proof (cases "p = 0")
+ case True with coeff show "p = q" by simp
+next
+ case False with coeff have "q \<noteq> 0" by auto
+ have degree: "degree p = degree q"
+ using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
+ by (intro order_antisym dvd_imp_degree_le)
-lemma diff_poly_code [code]:
- "0 - q = (- q :: _ poly)"
- "p - 0 = (p :: _ poly)"
- "pCons a p - pCons b q = pCons (a - b) (p - q)"
-by simp_all
+ from `p dvd q` obtain a where a: "q = p * a" ..
+ with `q \<noteq> 0` have "a \<noteq> 0" by auto
+ with degree a `p \<noteq> 0` have "degree a = 0"
+ by (simp add: degree_mult_eq)
+ with coeff a show "p = q"
+ by (cases a, auto split: if_splits)
+qed
-lemmas smult_poly_code [code] =
- smult_0_right smult_pCons
-
-lemma mult_poly_code [code]:
- "0 * q = (0 :: _ poly)"
- "pCons a p * q = smult a q + pCons 0 (p * q)"
-by simp_all
+lemma poly_gcd_unique:
+ fixes d x y :: "_ poly"
+ assumes dvd1: "d dvd x" and dvd2: "d dvd y"
+ and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
+ and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
+ shows "gcd x y = d"
+proof -
+ have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
+ by (simp_all add: poly_gcd_monic monic)
+ moreover have "gcd x y dvd d"
+ using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
+ moreover have "d dvd gcd x y"
+ using dvd1 dvd2 by (rule poly_gcd_greatest)
+ ultimately show ?thesis
+ by (rule poly_dvd_antisym)
+qed
-lemmas poly_code [code] =
- poly_0 poly_pCons
-
-lemmas synthetic_divmod_code [code] =
- synthetic_divmod_0 synthetic_divmod_pCons
+interpretation gcd_poly!: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
+proof
+ fix x y z :: "'a poly"
+ show "gcd (gcd x y) z = gcd x (gcd y z)"
+ by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
+ show "gcd x y = gcd y x"
+ by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+qed
-text {* code generator setup for div and mod *}
+lemmas poly_gcd_assoc = gcd_poly.assoc
+lemmas poly_gcd_commute = gcd_poly.commute
+lemmas poly_gcd_left_commute = gcd_poly.left_commute
-definition
- pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
-where
- "pdivmod x y = (x div y, x mod y)"
+lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
+
+lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
-lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
- unfolding pdivmod_def by simp
+lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
+
+lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
- unfolding pdivmod_def by simp
+lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
- unfolding pdivmod_def by simp
+lemma poly_gcd_code [code]:
+ "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
+ by (simp add: gcd_poly.simps)
+
+
+subsection {* Composition of polynomials *}
-lemma pdivmod_pCons [code]:
- "pdivmod (pCons a x) y =
- (if y = 0 then (0, pCons a x) else
- (let (q, r) = pdivmod x y;
- b = coeff (pCons a r) (degree y) / coeff y (degree y)
- in (pCons b q, pCons a r - smult b y)))"
-apply (simp add: pdivmod_def Let_def, safe)
-apply (rule div_poly_eq)
-apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
-apply (rule mod_poly_eq)
-apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
+definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+where
+ "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
+
+lemma pcompose_0 [simp]:
+ "pcompose 0 q = 0"
+ by (simp add: pcompose_def)
+
+lemma pcompose_pCons:
+ "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
+ by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
+
+lemma poly_pcompose:
+ "poly (pcompose p q) x = poly p (poly q x)"
+ by (induct p) (simp_all add: pcompose_pCons)
+
+lemma degree_pcompose_le:
+ "degree (pcompose p q) \<le> degree p * degree q"
+apply (induct p, simp)
+apply (simp add: pcompose_pCons, clarify)
+apply (rule degree_add_le, simp)
+apply (rule order_trans [OF degree_mult_le], simp)
done
-lemma poly_gcd_code [code]:
- "poly_gcd x y =
- (if y = 0 then smult (inverse (coeff x (degree x))) x
- else poly_gcd y (x mod y))"
- by (simp add: poly_gcd.simps)
+
+no_notation cCons (infixr "##" 65)
end
+