--- a/src/HOL/Deriv.thy Wed Feb 18 19:51:39 2009 -0800
+++ b/src/HOL/Deriv.thy Wed Feb 18 20:14:45 2009 -0800
@@ -9,7 +9,7 @@
header{* Differentiation *}
theory Deriv
-imports Lim Polynomial
+imports Lim
begin
text{*Standard Definitions*}
--- a/src/HOL/IsaMakefile Wed Feb 18 19:51:39 2009 -0800
+++ b/src/HOL/IsaMakefile Wed Feb 18 20:14:45 2009 -0800
@@ -284,7 +284,6 @@
GCD.thy \
Parity.thy \
Lubs.thy \
- Polynomial.thy \
PReal.thy \
Rational.thy \
RComplete.thy \
@@ -337,6 +336,7 @@
Library/RBT.thy Library/Univ_Poly.thy \
Library/Random.thy Library/Quickcheck.thy \
Library/Poly_Deriv.thy \
+ Library/Polynomial.thy \
Library/Enum.thy Library/Float.thy $(SRC)/Tools/float.ML $(SRC)/HOL/Tools/float_arith.ML \
Library/reify_data.ML Library/reflection.ML
@cd Library; $(ISABELLE_TOOL) usedir $(OUT)/HOL Library
--- a/src/HOL/Library/Library.thy Wed Feb 18 19:51:39 2009 -0800
+++ b/src/HOL/Library/Library.thy Wed Feb 18 20:14:45 2009 -0800
@@ -37,6 +37,7 @@
Permutation
Pocklington
Poly_Deriv
+ Polynomial
Primes
Quickcheck
Quicksort
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Polynomial.thy Wed Feb 18 20:14:45 2009 -0800
@@ -0,0 +1,1441 @@
+(* Title: HOL/Polynomial.thy
+ Author: Brian Huffman
+ Based on an earlier development by Clemens Ballarin
+*)
+
+header {* Univariate Polynomials *}
+
+theory Polynomial
+imports Plain SetInterval Main
+begin
+
+subsection {* Definition of type @{text poly} *}
+
+typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
+ morphisms coeff Abs_poly
+ by auto
+
+lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
+by (simp add: coeff_inject [symmetric] expand_fun_eq)
+
+lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
+by (simp add: expand_poly_eq)
+
+
+subsection {* Degree of a polynomial *}
+
+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"
+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"
+ unfolding degree_def by (rule LeastI_ex)
+ moreover assume "degree p < n"
+ ultimately show ?thesis by simp
+qed
+
+lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
+ by (erule contrapos_np, rule coeff_eq_0, simp)
+
+lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
+ unfolding degree_def by (erule Least_le)
+
+lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
+ unfolding degree_def by (drule not_less_Least, simp)
+
+
+subsection {* The zero polynomial *}
+
+instantiation poly :: (zero) zero
+begin
+
+definition
+ zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
+
+instance ..
+end
+
+lemma coeff_0 [simp]: "coeff 0 n = 0"
+ unfolding zero_poly_def
+ by (simp add: Abs_poly_inverse Poly_def)
+
+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"
+proof (cases "degree p")
+ case 0
+ from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
+ by (simp add: expand_poly_eq)
+ 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`
+ show "coeff p (degree p) \<noteq> 0" by simp
+next
+ case (Suc n)
+ from `degree p = Suc n` have "n < degree p" by simp
+ hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
+ then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
+ from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
+ also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
+ finally have "degree p = i" .
+ 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"
+ 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
+ [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
+
+syntax
+ "_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
+
+translations
+ "[:x, xs:]" == "CONST pCons x [:xs:]"
+ "[:x:]" == "CONST pCons x 0"
+
+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"
+ 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_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
+
+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
+
+lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
+by (rule poly_ext, 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)
+ 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
+next
+ 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)
+qed
+
+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)
+qed
+
+lemma pCons_induct [case_names 0 pCons, induct type: poly]:
+ assumes zero: "P 0"
+ assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
+ shows "P p"
+proof (induct p rule: measure_induct_rule [where f=degree])
+ case (less p)
+ obtain a q where "p = pCons a q" by (rule pCons_cases)
+ have "P q"
+ proof (cases "q = 0")
+ case True
+ then show "P q" by (simp add: zero)
+ next
+ case False
+ then have "degree (pCons a q) = Suc (degree q)"
+ by (rule degree_pCons_eq)
+ then have "degree q < degree p"
+ using `p = pCons a q` by simp
+ then show "P q"
+ by (rule less.hyps)
+ qed
+ then have "P (pCons a q)"
+ by (rule pCons)
+ then show ?case
+ using `p = pCons a q` by simp
+qed
+
+
+subsection {* Recursion combinator for polynomials *}
+
+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, code 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)
+
+termination poly_rec
+by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
+ (simp add: degree_pCons_eq)
+
+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
+
+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)
+
+
+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)"
+
+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_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_eq_0 [simp]: "monom 0 n = 0"
+ by (rule poly_ext) simp
+
+lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
+ by (simp add: expand_poly_eq)
+
+lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
+ by (simp add: expand_poly_eq)
+
+lemma degree_monom_le: "degree (monom a n) \<le> n"
+ by (rule degree_le, simp)
+
+lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
+ apply (rule order_antisym [OF degree_monom_le])
+ apply (rule le_degree, simp)
+ done
+
+
+subsection {* Addition and subtraction *}
+
+instantiation poly :: (comm_monoid_add) comm_monoid_add
+begin
+
+definition
+ plus_poly_def [code del]:
+ "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
+
+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)
+
+instance proof
+ fix p q r :: "'a poly"
+ show "(p + q) + r = p + (q + r)"
+ by (simp add: expand_poly_eq add_assoc)
+ show "p + q = q + p"
+ by (simp add: expand_poly_eq add_commute)
+ show "0 + p = p"
+ by (simp add: expand_poly_eq)
+qed
+
+end
+
+instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
+proof
+ fix p q r :: "'a poly"
+ assume "p + q = p + r" thus "q = r"
+ by (simp add: expand_poly_eq)
+qed
+
+instantiation poly :: (ab_group_add) ab_group_add
+begin
+
+definition
+ uminus_poly_def [code del]:
+ "- p = Abs_poly (\<lambda>n. - coeff p n)"
+
+definition
+ minus_poly_def [code del]:
+ "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
+
+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)
+
+lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
+ unfolding uminus_poly_def
+ by (simp add: Abs_poly_inverse coeff Poly_minus)
+
+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)
+
+instance proof
+ fix p q :: "'a poly"
+ show "- p + p = 0"
+ by (simp add: expand_poly_eq)
+ show "p - q = p + - q"
+ by (simp add: expand_poly_eq 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)
+
+lemma minus_pCons [simp]:
+ "- pCons a p = pCons (- a) (- p)"
+ by (rule poly_ext, 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)
+
+lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
+ by (rule degree_le, auto simp add: coeff_eq_0)
+
+lemma degree_add_le:
+ "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
+ by (auto intro: order_trans degree_add_le_max)
+
+lemma degree_add_less:
+ "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
+ by (auto intro: le_less_trans degree_add_le_max)
+
+lemma degree_add_eq_right:
+ "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
+ apply (cases "q = 0", simp)
+ apply (rule order_antisym)
+ apply (simp add: degree_add_le)
+ apply (rule le_degree)
+ apply (simp add: coeff_eq_0)
+ done
+
+lemma degree_add_eq_left:
+ "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
+ using degree_add_eq_right [of q p]
+ by (simp add: add_commute)
+
+lemma degree_minus [simp]: "degree (- p) = degree p"
+ unfolding degree_def by simp
+
+lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
+ using degree_add_le [where p=p and q="-q"]
+ by (simp add: diff_minus)
+
+lemma degree_diff_le:
+ "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
+ by (simp add: diff_minus degree_add_le)
+
+lemma degree_diff_less:
+ "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
+ 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
+
+lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
+ by (rule poly_ext) simp
+
+lemma minus_monom: "- monom a n = monom (-a) n"
+ by (rule poly_ext) 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)
+
+
+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)"
+
+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)
+
+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 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)
+
+lemma smult_0_right [simp]: "smult a 0 = 0"
+ by (rule poly_ext, simp)
+
+lemma smult_0_left [simp]: "smult 0 p = 0"
+ by (rule poly_ext, simp)
+
+lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
+ by (rule poly_ext, simp)
+
+lemma smult_add_right:
+ "smult a (p + q) = smult a p + smult a q"
+ by (rule poly_ext, 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)
+
+lemma smult_minus_right [simp]:
+ "smult (a::'a::comm_ring) (- p) = - smult a p"
+ by (rule poly_ext, simp)
+
+lemma smult_minus_left [simp]:
+ "smult (- a::'a::comm_ring) p = - smult a p"
+ by (rule poly_ext, 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)
+
+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)
+
+lemmas smult_distribs =
+ smult_add_left smult_add_right
+ smult_diff_left smult_diff_right
+
+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)
+
+lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
+ by (induct n, simp add: monom_0, simp add: monom_Suc)
+
+lemma degree_smult_eq [simp]:
+ fixes a :: "'a::idom"
+ shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
+ by (cases "a = 0", simp, simp add: degree_def)
+
+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 *}
+
+text {* TODO: move to SetInterval.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
+
+instantiation poly :: (comm_semiring_0) comm_semiring_0
+begin
+
+definition
+ times_poly_def [code del]:
+ "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
+
+lemma mult_poly_0_left: "(0::'a poly) * q = 0"
+ unfolding times_poly_def by (simp add: poly_rec_0)
+
+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)
+
+lemma mult_poly_0_right: "p * (0::'a poly) = 0"
+ 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)
+
+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_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)
+
+instance proof
+ fix p q r :: "'a poly"
+ show 0: "0 * p = 0"
+ by (rule mult_poly_0_left)
+ show "p * 0 = 0"
+ by (rule mult_poly_0_right)
+ show "(p + q) * r = p * r + q * r"
+ by (rule mult_poly_add_left)
+ show "(p * q) * r = p * (q * r)"
+ by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
+ show "p * q = q * p"
+ by (induct p, simp add: mult_poly_0, simp)
+qed
+
+end
+
+instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
+
+lemma coeff_mult:
+ "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
+proof (induct p arbitrary: n)
+ case 0 show ?case by simp
+next
+ case (pCons a p n) thus ?case
+ by (cases n, simp, simp add: setsum_atMost_Suc_shift
+ del: setsum_atMost_Suc)
+qed
+
+lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
+apply (rule degree_le)
+apply (induct p)
+apply simp
+apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
+done
+
+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"
+
+instance proof
+ fix p :: "'a poly" show "1 * p = p"
+ unfolding one_poly_def
+ by simp
+next
+ show "0 \<noteq> (1::'a poly)"
+ unfolding one_poly_def by simp
+qed
+
+end
+
+instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
+
+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)
+
+lemma degree_1 [simp]: "degree 1 = 0"
+ unfolding one_poly_def
+ by (rule degree_pCons_0)
+
+text {* Lemmas about divisibility *}
+
+lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
+proof -
+ assume "p dvd q"
+ then obtain k where "q = p * k" ..
+ then have "smult a q = p * smult a k" by simp
+ then show "p dvd smult a q" ..
+qed
+
+lemma dvd_smult_cancel:
+ fixes a :: "'a::field"
+ shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
+ by (drule dvd_smult [where a="inverse a"]) simp
+
+lemma dvd_smult_iff:
+ fixes a :: "'a::field"
+ shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
+ by (safe elim!: dvd_smult dvd_smult_cancel)
+
+instantiation poly :: (comm_semiring_1) recpower
+begin
+
+primrec power_poly where
+ power_poly_0: "(p::'a poly) ^ 0 = 1"
+| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
+
+instance
+ by default simp_all
+
+end
+
+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 ..
+
+instantiation poly :: (comm_ring_1) number_ring
+begin
+
+definition
+ "number_of k = (of_int k :: 'a poly)"
+
+instance
+ by default (rule number_of_poly_def)
+
+end
+
+
+subsection {* Polynomials form an integral domain *}
+
+lemma coeff_mult_degree_sum:
+ "coeff (p * q) (degree p + degree q) =
+ coeff p (degree p) * coeff q (degree q)"
+ by (induct p, simp, simp add: coeff_eq_0)
+
+instance poly :: (idom) idom
+proof
+ fix p q :: "'a poly"
+ assume "p \<noteq> 0" and "q \<noteq> 0"
+ have "coeff (p * q) (degree p + degree q) =
+ coeff p (degree p) * coeff q (degree q)"
+ by (rule coeff_mult_degree_sum)
+ 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)
+qed
+
+lemma degree_mult_eq:
+ fixes p q :: "'a::idom poly"
+ shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
+apply (rule order_antisym [OF degree_mult_le le_degree])
+apply (simp add: coeff_mult_degree_sum)
+done
+
+lemma dvd_imp_degree_le:
+ fixes p q :: "'a::idom poly"
+ shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
+ by (erule dvdE, simp add: degree_mult_eq)
+
+
+subsection {* Polynomials form an ordered integral domain *}
+
+definition
+ pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool"
+where
+ "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
+
+lemma pos_poly_pCons:
+ "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
+ unfolding pos_poly_def by simp
+
+lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
+ unfolding pos_poly_def by simp
+
+lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
+ apply (induct p arbitrary: q, simp)
+ apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
+ done
+
+lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
+ unfolding pos_poly_def
+ apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
+ apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)
+ apply auto
+ done
+
+lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
+by (induct p) (auto simp add: pos_poly_pCons)
+
+instantiation poly :: (ordered_idom) ordered_idom
+begin
+
+definition
+ [code del]:
+ "x < y \<longleftrightarrow> pos_poly (y - x)"
+
+definition
+ [code del]:
+ "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
+
+definition
+ [code del]:
+ "abs (x::'a poly) = (if x < 0 then - x else x)"
+
+definition
+ [code del]:
+ "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
+
+instance proof
+ fix x y :: "'a poly"
+ show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+ unfolding less_eq_poly_def less_poly_def
+ apply safe
+ apply simp
+ apply (drule (1) pos_poly_add)
+ apply simp
+ done
+next
+ fix x :: "'a poly" show "x \<le> x"
+ unfolding less_eq_poly_def by simp
+next
+ fix x y z :: "'a poly"
+ assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
+ unfolding less_eq_poly_def
+ apply safe
+ apply (drule (1) pos_poly_add)
+ apply (simp add: algebra_simps)
+ done
+next
+ fix x y :: "'a poly"
+ assume "x \<le> y" and "y \<le> x" thus "x = y"
+ unfolding less_eq_poly_def
+ apply safe
+ apply (drule (1) pos_poly_add)
+ apply simp
+ done
+next
+ fix x y z :: "'a poly"
+ assume "x \<le> y" thus "z + x \<le> z + y"
+ unfolding less_eq_poly_def
+ apply safe
+ apply (simp add: algebra_simps)
+ done
+next
+ fix x y :: "'a poly"
+ show "x \<le> y \<or> y \<le> x"
+ unfolding less_eq_poly_def
+ using pos_poly_total [of "x - y"]
+ by auto
+next
+ fix x y z :: "'a poly"
+ assume "x < y" and "0 < z"
+ thus "z * x < z * y"
+ unfolding less_poly_def
+ by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
+next
+ fix x :: "'a poly"
+ show "\<bar>x\<bar> = (if x < 0 then - x else x)"
+ by (rule abs_poly_def)
+next
+ fix x :: "'a poly"
+ show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
+ by (rule sgn_poly_def)
+qed
+
+end
+
+text {* TODO: Simplification rules for comparisons *}
+
+
+subsection {* Long division of polynomials *}
+
+definition
+ pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
+where
+ [code del]:
+ "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)"
+
+lemma pdivmod_rel_0:
+ "pdivmod_rel 0 y 0 0"
+ unfolding pdivmod_rel_def by simp
+
+lemma pdivmod_rel_by_0:
+ "pdivmod_rel x 0 0 x"
+ unfolding pdivmod_rel_def by simp
+
+lemma eq_zero_or_degree_less:
+ assumes "degree p \<le> n" and "coeff p n = 0"
+ shows "p = 0 \<or> degree p < n"
+proof (cases n)
+ case 0
+ with `degree p \<le> n` and `coeff p n = 0`
+ have "coeff p (degree p) = 0" by simp
+ then have "p = 0" by simp
+ then show ?thesis ..
+next
+ case (Suc m)
+ have "\<forall>i>n. coeff p i = 0"
+ using `degree p \<le> n` by (simp add: coeff_eq_0)
+ then have "\<forall>i\<ge>n. coeff p i = 0"
+ using `coeff p n = 0` by (simp add: le_less)
+ then have "\<forall>i>m. coeff p i = 0"
+ using `n = Suc m` by (simp add: less_eq_Suc_le)
+ then have "degree p \<le> m"
+ by (rule degree_le)
+ then have "degree p < n"
+ using `n = Suc m` by (simp add: less_Suc_eq_le)
+ then show ?thesis ..
+qed
+
+lemma pdivmod_rel_pCons:
+ assumes rel: "pdivmod_rel x y q r"
+ assumes y: "y \<noteq> 0"
+ assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
+ shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
+ (is "pdivmod_rel ?x y ?q ?r")
+proof -
+ have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
+ using assms unfolding pdivmod_rel_def by simp_all
+
+ have 1: "?x = ?q * y + ?r"
+ using b x by simp
+
+ have 2: "?r = 0 \<or> degree ?r < degree y"
+ proof (rule eq_zero_or_degree_less)
+ show "degree ?r \<le> degree y"
+ proof (rule degree_diff_le)
+ show "degree (pCons a r) \<le> degree y"
+ using r by auto
+ show "degree (smult b y) \<le> degree y"
+ by (rule degree_smult_le)
+ qed
+ next
+ show "coeff ?r (degree y) = 0"
+ using `y \<noteq> 0` unfolding b by simp
+ qed
+
+ from 1 2 show ?thesis
+ unfolding pdivmod_rel_def
+ using `y \<noteq> 0` by simp
+qed
+
+lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
+apply (cases "y = 0")
+apply (fast intro!: pdivmod_rel_by_0)
+apply (induct x)
+apply (fast intro!: pdivmod_rel_0)
+apply (fast intro!: pdivmod_rel_pCons)
+done
+
+lemma pdivmod_rel_unique:
+ assumes 1: "pdivmod_rel x y q1 r1"
+ assumes 2: "pdivmod_rel x y q2 r2"
+ shows "q1 = q2 \<and> r1 = r2"
+proof (cases "y = 0")
+ assume "y = 0" with assms show ?thesis
+ by (simp add: pdivmod_rel_def)
+next
+ assume [simp]: "y \<noteq> 0"
+ from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
+ unfolding pdivmod_rel_def by simp_all
+ from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
+ unfolding pdivmod_rel_def by simp_all
+ from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
+ by (simp add: algebra_simps)
+ from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
+ by (auto intro: degree_diff_less)
+
+ show "q1 = q2 \<and> r1 = r2"
+ proof (rule ccontr)
+ assume "\<not> (q1 = q2 \<and> r1 = r2)"
+ with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
+ with r3 have "degree (r2 - r1) < degree y" by simp
+ also have "degree y \<le> degree (q1 - q2) + degree y" by simp
+ also have "\<dots> = degree ((q1 - q2) * y)"
+ using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
+ also have "\<dots> = degree (r2 - r1)"
+ using q3 by simp
+ finally have "degree (r2 - r1) < degree (r2 - r1)" .
+ then show "False" by simp
+ qed
+qed
+
+lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
+by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
+
+lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
+by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
+
+lemmas pdivmod_rel_unique_div =
+ pdivmod_rel_unique [THEN conjunct1, standard]
+
+lemmas pdivmod_rel_unique_mod =
+ pdivmod_rel_unique [THEN conjunct2, standard]
+
+instantiation poly :: (field) ring_div
+begin
+
+definition div_poly where
+ [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
+
+definition mod_poly where
+ [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
+
+lemma div_poly_eq:
+ "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
+unfolding div_poly_def
+by (fast elim: pdivmod_rel_unique_div)
+
+lemma mod_poly_eq:
+ "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
+unfolding mod_poly_def
+by (fast elim: pdivmod_rel_unique_mod)
+
+lemma pdivmod_rel:
+ "pdivmod_rel x y (x div y) (x mod y)"
+proof -
+ from pdivmod_rel_exists
+ obtain q r where "pdivmod_rel x y q r" by fast
+ thus ?thesis
+ by (simp add: div_poly_eq mod_poly_eq)
+qed
+
+instance proof
+ fix x y :: "'a poly"
+ show "x div y * y + x mod y = x"
+ using pdivmod_rel [of x y]
+ by (simp add: pdivmod_rel_def)
+next
+ fix x :: "'a poly"
+ have "pdivmod_rel x 0 0 x"
+ by (rule pdivmod_rel_by_0)
+ thus "x div 0 = 0"
+ by (rule div_poly_eq)
+next
+ fix y :: "'a poly"
+ have "pdivmod_rel 0 y 0 0"
+ by (rule pdivmod_rel_0)
+ thus "0 div y = 0"
+ by (rule div_poly_eq)
+next
+ fix x y z :: "'a poly"
+ assume "y \<noteq> 0"
+ hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
+ using pdivmod_rel [of x y]
+ by (simp add: pdivmod_rel_def left_distrib)
+ thus "(x + z * y) div y = z + x div y"
+ by (rule div_poly_eq)
+qed
+
+end
+
+lemma degree_mod_less:
+ "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
+ using pdivmod_rel [of x y]
+ unfolding pdivmod_rel_def by simp
+
+lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
+proof -
+ assume "degree x < degree y"
+ hence "pdivmod_rel x y 0 x"
+ by (simp add: pdivmod_rel_def)
+ thus "x div y = 0" by (rule div_poly_eq)
+qed
+
+lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
+proof -
+ assume "degree x < degree y"
+ hence "pdivmod_rel x y 0 x"
+ by (simp add: pdivmod_rel_def)
+ thus "x mod y = x" by (rule mod_poly_eq)
+qed
+
+lemma pdivmod_rel_smult_left:
+ "pdivmod_rel x y q r
+ \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
+ unfolding pdivmod_rel_def by (simp add: smult_add_right)
+
+lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
+ by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
+
+lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
+ by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
+
+lemma pdivmod_rel_smult_right:
+ "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
+ \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
+ unfolding pdivmod_rel_def by simp
+
+lemma div_smult_right:
+ "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
+ by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
+
+lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
+ by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
+
+lemma pdivmod_rel_mult:
+ "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
+ \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
+apply (cases "z = 0", simp add: pdivmod_rel_def)
+apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
+apply (cases "r = 0")
+apply (cases "r' = 0")
+apply (simp add: pdivmod_rel_def)
+apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq)
+apply (cases "r' = 0")
+apply (simp add: pdivmod_rel_def degree_mult_eq)
+apply (simp add: pdivmod_rel_def ring_simps)
+apply (simp add: degree_mult_eq degree_add_less)
+done
+
+lemma poly_div_mult_right:
+ fixes x y z :: "'a::field poly"
+ shows "x div (y * z) = (x div y) div z"
+ by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
+
+lemma poly_mod_mult_right:
+ fixes x y z :: "'a::field poly"
+ shows "x mod (y * z) = y * (x div y mod z) + x mod y"
+ by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
+
+lemma mod_pCons:
+ fixes a and x
+ assumes y: "y \<noteq> 0"
+ defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
+ shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
+unfolding b
+apply (rule mod_poly_eq)
+apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
+done
+
+
+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,recpower}"
+ 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)
+ 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,recpower} 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 [code del]:
+ "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
+ with `k \<noteq> 0` have "finite {x. poly k x = 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: expand_fun_eq)
+
+
+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
+
+
+subsection {* Order of polynomial roots *}
+
+definition
+ order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
+where
+ [code del]:
+ "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
+
+lemma coeff_linear_power:
+ fixes a :: "'a::comm_semiring_1"
+ shows "coeff ([:a, 1:] ^ n) n = 1"
+apply (induct n, simp_all)
+apply (subst coeff_eq_0)
+apply (auto intro: le_less_trans degree_power_le)
+done
+
+lemma degree_linear_power:
+ fixes a :: "'a::comm_semiring_1"
+ shows "degree ([:a, 1:] ^ n) = n"
+apply (rule order_antisym)
+apply (rule ord_le_eq_trans [OF degree_power_le], simp)
+apply (rule le_degree, simp add: coeff_linear_power)
+done
+
+lemma order_1: "[:-a, 1:] ^ order a p dvd p"
+apply (cases "p = 0", simp)
+apply (cases "order a p", simp)
+apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
+apply (drule not_less_Least, simp)
+apply (fold order_def, simp)
+done
+
+lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
+unfolding order_def
+apply (rule LeastI_ex)
+apply (rule_tac x="degree p" in exI)
+apply (rule notI)
+apply (drule (1) dvd_imp_degree_le)
+apply (simp only: degree_linear_power)
+done
+
+lemma order:
+ "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
+by (rule conjI [OF order_1 order_2])
+
+lemma order_degree:
+ assumes p: "p \<noteq> 0"
+ shows "order a p \<le> degree p"
+proof -
+ have "order a p = degree ([:-a, 1:] ^ order a p)"
+ by (simp only: degree_linear_power)
+ also have "\<dots> \<le> degree p"
+ using order_1 p by (rule dvd_imp_degree_le)
+ finally show ?thesis .
+qed
+
+lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
+apply (cases "p = 0", simp_all)
+apply (rule iffI)
+apply (rule ccontr, simp)
+apply (frule order_2 [where a=a], simp)
+apply (simp add: poly_eq_0_iff_dvd)
+apply (simp add: poly_eq_0_iff_dvd)
+apply (simp only: order_def)
+apply (drule not_less_Least, simp)
+done
+
+
+subsection {* Configuration of the code generator *}
+
+code_datatype "0::'a::zero poly" pCons
+
+declare pCons_0_0 [code post]
+
+instantiation poly :: ("{zero,eq}") eq
+begin
+
+definition [code del]:
+ "eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q"
+
+instance
+ by default (rule eq_poly_def)
+
+end
+
+lemma eq_poly_code [code]:
+ "eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
+ "eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q"
+ "eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0"
+ "eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q"
+unfolding eq by simp_all
+
+lemmas coeff_code [code] =
+ coeff_0 coeff_pCons_0 coeff_pCons_Suc
+
+lemmas degree_code [code] =
+ degree_0 degree_pCons_eq_if
+
+lemmas monom_poly_code [code] =
+ monom_0 monom_Suc
+
+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 minus_poly_code [code]:
+ "- 0 = (0 :: _ poly)"
+ "- pCons a p = pCons (- a) (- p)"
+by simp_all
+
+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
+
+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
+
+lemmas poly_code [code] =
+ poly_0 poly_pCons
+
+lemmas synthetic_divmod_code [code] =
+ synthetic_divmod_0 synthetic_divmod_pCons
+
+text {* code generator setup for div and mod *}
+
+definition
+ pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
+where
+ [code del]: "pdivmod x y = (x div y, x mod y)"
+
+lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
+ unfolding pdivmod_def by simp
+
+lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
+ unfolding pdivmod_def by simp
+
+lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
+ unfolding pdivmod_def by simp
+
+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])
+done
+
+end
--- a/src/HOL/Polynomial.thy Wed Feb 18 19:51:39 2009 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1441 +0,0 @@
-(* Title: HOL/Polynomial.thy
- Author: Brian Huffman
- Based on an earlier development by Clemens Ballarin
-*)
-
-header {* Univariate Polynomials *}
-
-theory Polynomial
-imports Plain SetInterval Main
-begin
-
-subsection {* Definition of type @{text poly} *}
-
-typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
- morphisms coeff Abs_poly
- by auto
-
-lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
-by (simp add: coeff_inject [symmetric] expand_fun_eq)
-
-lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
-by (simp add: expand_poly_eq)
-
-
-subsection {* Degree of a polynomial *}
-
-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"
-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"
- unfolding degree_def by (rule LeastI_ex)
- moreover assume "degree p < n"
- ultimately show ?thesis by simp
-qed
-
-lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
- by (erule contrapos_np, rule coeff_eq_0, simp)
-
-lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
- unfolding degree_def by (erule Least_le)
-
-lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
- unfolding degree_def by (drule not_less_Least, simp)
-
-
-subsection {* The zero polynomial *}
-
-instantiation poly :: (zero) zero
-begin
-
-definition
- zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
-
-instance ..
-end
-
-lemma coeff_0 [simp]: "coeff 0 n = 0"
- unfolding zero_poly_def
- by (simp add: Abs_poly_inverse Poly_def)
-
-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"
-proof (cases "degree p")
- case 0
- from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
- by (simp add: expand_poly_eq)
- 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`
- show "coeff p (degree p) \<noteq> 0" by simp
-next
- case (Suc n)
- from `degree p = Suc n` have "n < degree p" by simp
- hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
- then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
- from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
- also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
- finally have "degree p = i" .
- 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"
- 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
- [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
-
-syntax
- "_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
-
-translations
- "[:x, xs:]" == "CONST pCons x [:xs:]"
- "[:x:]" == "CONST pCons x 0"
-
-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"
- 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_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
-
-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
-
-lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
-by (rule poly_ext, 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)
- 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
-next
- 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)
-qed
-
-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)
-qed
-
-lemma pCons_induct [case_names 0 pCons, induct type: poly]:
- assumes zero: "P 0"
- assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
- shows "P p"
-proof (induct p rule: measure_induct_rule [where f=degree])
- case (less p)
- obtain a q where "p = pCons a q" by (rule pCons_cases)
- have "P q"
- proof (cases "q = 0")
- case True
- then show "P q" by (simp add: zero)
- next
- case False
- then have "degree (pCons a q) = Suc (degree q)"
- by (rule degree_pCons_eq)
- then have "degree q < degree p"
- using `p = pCons a q` by simp
- then show "P q"
- by (rule less.hyps)
- qed
- then have "P (pCons a q)"
- by (rule pCons)
- then show ?case
- using `p = pCons a q` by simp
-qed
-
-
-subsection {* Recursion combinator for polynomials *}
-
-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, code 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)
-
-termination poly_rec
-by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
- (simp add: degree_pCons_eq)
-
-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
-
-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)
-
-
-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)"
-
-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_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_eq_0 [simp]: "monom 0 n = 0"
- by (rule poly_ext) simp
-
-lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
- by (simp add: expand_poly_eq)
-
-lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
- by (simp add: expand_poly_eq)
-
-lemma degree_monom_le: "degree (monom a n) \<le> n"
- by (rule degree_le, simp)
-
-lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
- apply (rule order_antisym [OF degree_monom_le])
- apply (rule le_degree, simp)
- done
-
-
-subsection {* Addition and subtraction *}
-
-instantiation poly :: (comm_monoid_add) comm_monoid_add
-begin
-
-definition
- plus_poly_def [code del]:
- "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
-
-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)
-
-instance proof
- fix p q r :: "'a poly"
- show "(p + q) + r = p + (q + r)"
- by (simp add: expand_poly_eq add_assoc)
- show "p + q = q + p"
- by (simp add: expand_poly_eq add_commute)
- show "0 + p = p"
- by (simp add: expand_poly_eq)
-qed
-
-end
-
-instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
-proof
- fix p q r :: "'a poly"
- assume "p + q = p + r" thus "q = r"
- by (simp add: expand_poly_eq)
-qed
-
-instantiation poly :: (ab_group_add) ab_group_add
-begin
-
-definition
- uminus_poly_def [code del]:
- "- p = Abs_poly (\<lambda>n. - coeff p n)"
-
-definition
- minus_poly_def [code del]:
- "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
-
-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)
-
-lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
- unfolding uminus_poly_def
- by (simp add: Abs_poly_inverse coeff Poly_minus)
-
-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)
-
-instance proof
- fix p q :: "'a poly"
- show "- p + p = 0"
- by (simp add: expand_poly_eq)
- show "p - q = p + - q"
- by (simp add: expand_poly_eq 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)
-
-lemma minus_pCons [simp]:
- "- pCons a p = pCons (- a) (- p)"
- by (rule poly_ext, 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)
-
-lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
- by (rule degree_le, auto simp add: coeff_eq_0)
-
-lemma degree_add_le:
- "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
- by (auto intro: order_trans degree_add_le_max)
-
-lemma degree_add_less:
- "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
- by (auto intro: le_less_trans degree_add_le_max)
-
-lemma degree_add_eq_right:
- "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
- apply (cases "q = 0", simp)
- apply (rule order_antisym)
- apply (simp add: degree_add_le)
- apply (rule le_degree)
- apply (simp add: coeff_eq_0)
- done
-
-lemma degree_add_eq_left:
- "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
- using degree_add_eq_right [of q p]
- by (simp add: add_commute)
-
-lemma degree_minus [simp]: "degree (- p) = degree p"
- unfolding degree_def by simp
-
-lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
- using degree_add_le [where p=p and q="-q"]
- by (simp add: diff_minus)
-
-lemma degree_diff_le:
- "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
- by (simp add: diff_minus degree_add_le)
-
-lemma degree_diff_less:
- "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
- 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
-
-lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
- by (rule poly_ext) simp
-
-lemma minus_monom: "- monom a n = monom (-a) n"
- by (rule poly_ext) 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)
-
-
-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)"
-
-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)
-
-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 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)
-
-lemma smult_0_right [simp]: "smult a 0 = 0"
- by (rule poly_ext, simp)
-
-lemma smult_0_left [simp]: "smult 0 p = 0"
- by (rule poly_ext, simp)
-
-lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
- by (rule poly_ext, simp)
-
-lemma smult_add_right:
- "smult a (p + q) = smult a p + smult a q"
- by (rule poly_ext, 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)
-
-lemma smult_minus_right [simp]:
- "smult (a::'a::comm_ring) (- p) = - smult a p"
- by (rule poly_ext, simp)
-
-lemma smult_minus_left [simp]:
- "smult (- a::'a::comm_ring) p = - smult a p"
- by (rule poly_ext, 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)
-
-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)
-
-lemmas smult_distribs =
- smult_add_left smult_add_right
- smult_diff_left smult_diff_right
-
-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)
-
-lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
- by (induct n, simp add: monom_0, simp add: monom_Suc)
-
-lemma degree_smult_eq [simp]:
- fixes a :: "'a::idom"
- shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
- by (cases "a = 0", simp, simp add: degree_def)
-
-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 *}
-
-text {* TODO: move to SetInterval.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
-
-instantiation poly :: (comm_semiring_0) comm_semiring_0
-begin
-
-definition
- times_poly_def [code del]:
- "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
-
-lemma mult_poly_0_left: "(0::'a poly) * q = 0"
- unfolding times_poly_def by (simp add: poly_rec_0)
-
-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)
-
-lemma mult_poly_0_right: "p * (0::'a poly) = 0"
- 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)
-
-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_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)
-
-instance proof
- fix p q r :: "'a poly"
- show 0: "0 * p = 0"
- by (rule mult_poly_0_left)
- show "p * 0 = 0"
- by (rule mult_poly_0_right)
- show "(p + q) * r = p * r + q * r"
- by (rule mult_poly_add_left)
- show "(p * q) * r = p * (q * r)"
- by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
- show "p * q = q * p"
- by (induct p, simp add: mult_poly_0, simp)
-qed
-
-end
-
-instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
-
-lemma coeff_mult:
- "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
-proof (induct p arbitrary: n)
- case 0 show ?case by simp
-next
- case (pCons a p n) thus ?case
- by (cases n, simp, simp add: setsum_atMost_Suc_shift
- del: setsum_atMost_Suc)
-qed
-
-lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
-apply (rule degree_le)
-apply (induct p)
-apply simp
-apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
-done
-
-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"
-
-instance proof
- fix p :: "'a poly" show "1 * p = p"
- unfolding one_poly_def
- by simp
-next
- show "0 \<noteq> (1::'a poly)"
- unfolding one_poly_def by simp
-qed
-
-end
-
-instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
-
-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)
-
-lemma degree_1 [simp]: "degree 1 = 0"
- unfolding one_poly_def
- by (rule degree_pCons_0)
-
-text {* Lemmas about divisibility *}
-
-lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
-proof -
- assume "p dvd q"
- then obtain k where "q = p * k" ..
- then have "smult a q = p * smult a k" by simp
- then show "p dvd smult a q" ..
-qed
-
-lemma dvd_smult_cancel:
- fixes a :: "'a::field"
- shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
- by (drule dvd_smult [where a="inverse a"]) simp
-
-lemma dvd_smult_iff:
- fixes a :: "'a::field"
- shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
- by (safe elim!: dvd_smult dvd_smult_cancel)
-
-instantiation poly :: (comm_semiring_1) recpower
-begin
-
-primrec power_poly where
- power_poly_0: "(p::'a poly) ^ 0 = 1"
-| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
-
-instance
- by default simp_all
-
-end
-
-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 ..
-
-instantiation poly :: (comm_ring_1) number_ring
-begin
-
-definition
- "number_of k = (of_int k :: 'a poly)"
-
-instance
- by default (rule number_of_poly_def)
-
-end
-
-
-subsection {* Polynomials form an integral domain *}
-
-lemma coeff_mult_degree_sum:
- "coeff (p * q) (degree p + degree q) =
- coeff p (degree p) * coeff q (degree q)"
- by (induct p, simp, simp add: coeff_eq_0)
-
-instance poly :: (idom) idom
-proof
- fix p q :: "'a poly"
- assume "p \<noteq> 0" and "q \<noteq> 0"
- have "coeff (p * q) (degree p + degree q) =
- coeff p (degree p) * coeff q (degree q)"
- by (rule coeff_mult_degree_sum)
- 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)
-qed
-
-lemma degree_mult_eq:
- fixes p q :: "'a::idom poly"
- shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
-apply (rule order_antisym [OF degree_mult_le le_degree])
-apply (simp add: coeff_mult_degree_sum)
-done
-
-lemma dvd_imp_degree_le:
- fixes p q :: "'a::idom poly"
- shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
- by (erule dvdE, simp add: degree_mult_eq)
-
-
-subsection {* Polynomials form an ordered integral domain *}
-
-definition
- pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool"
-where
- "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
-
-lemma pos_poly_pCons:
- "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
- unfolding pos_poly_def by simp
-
-lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
- unfolding pos_poly_def by simp
-
-lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
- apply (induct p arbitrary: q, simp)
- apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
- done
-
-lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
- unfolding pos_poly_def
- apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
- apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)
- apply auto
- done
-
-lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
-by (induct p) (auto simp add: pos_poly_pCons)
-
-instantiation poly :: (ordered_idom) ordered_idom
-begin
-
-definition
- [code del]:
- "x < y \<longleftrightarrow> pos_poly (y - x)"
-
-definition
- [code del]:
- "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
-
-definition
- [code del]:
- "abs (x::'a poly) = (if x < 0 then - x else x)"
-
-definition
- [code del]:
- "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
-
-instance proof
- fix x y :: "'a poly"
- show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
- unfolding less_eq_poly_def less_poly_def
- apply safe
- apply simp
- apply (drule (1) pos_poly_add)
- apply simp
- done
-next
- fix x :: "'a poly" show "x \<le> x"
- unfolding less_eq_poly_def by simp
-next
- fix x y z :: "'a poly"
- assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
- unfolding less_eq_poly_def
- apply safe
- apply (drule (1) pos_poly_add)
- apply (simp add: algebra_simps)
- done
-next
- fix x y :: "'a poly"
- assume "x \<le> y" and "y \<le> x" thus "x = y"
- unfolding less_eq_poly_def
- apply safe
- apply (drule (1) pos_poly_add)
- apply simp
- done
-next
- fix x y z :: "'a poly"
- assume "x \<le> y" thus "z + x \<le> z + y"
- unfolding less_eq_poly_def
- apply safe
- apply (simp add: algebra_simps)
- done
-next
- fix x y :: "'a poly"
- show "x \<le> y \<or> y \<le> x"
- unfolding less_eq_poly_def
- using pos_poly_total [of "x - y"]
- by auto
-next
- fix x y z :: "'a poly"
- assume "x < y" and "0 < z"
- thus "z * x < z * y"
- unfolding less_poly_def
- by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
-next
- fix x :: "'a poly"
- show "\<bar>x\<bar> = (if x < 0 then - x else x)"
- by (rule abs_poly_def)
-next
- fix x :: "'a poly"
- show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
- by (rule sgn_poly_def)
-qed
-
-end
-
-text {* TODO: Simplification rules for comparisons *}
-
-
-subsection {* Long division of polynomials *}
-
-definition
- pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
-where
- [code del]:
- "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)"
-
-lemma pdivmod_rel_0:
- "pdivmod_rel 0 y 0 0"
- unfolding pdivmod_rel_def by simp
-
-lemma pdivmod_rel_by_0:
- "pdivmod_rel x 0 0 x"
- unfolding pdivmod_rel_def by simp
-
-lemma eq_zero_or_degree_less:
- assumes "degree p \<le> n" and "coeff p n = 0"
- shows "p = 0 \<or> degree p < n"
-proof (cases n)
- case 0
- with `degree p \<le> n` and `coeff p n = 0`
- have "coeff p (degree p) = 0" by simp
- then have "p = 0" by simp
- then show ?thesis ..
-next
- case (Suc m)
- have "\<forall>i>n. coeff p i = 0"
- using `degree p \<le> n` by (simp add: coeff_eq_0)
- then have "\<forall>i\<ge>n. coeff p i = 0"
- using `coeff p n = 0` by (simp add: le_less)
- then have "\<forall>i>m. coeff p i = 0"
- using `n = Suc m` by (simp add: less_eq_Suc_le)
- then have "degree p \<le> m"
- by (rule degree_le)
- then have "degree p < n"
- using `n = Suc m` by (simp add: less_Suc_eq_le)
- then show ?thesis ..
-qed
-
-lemma pdivmod_rel_pCons:
- assumes rel: "pdivmod_rel x y q r"
- assumes y: "y \<noteq> 0"
- assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
- shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
- (is "pdivmod_rel ?x y ?q ?r")
-proof -
- have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
- using assms unfolding pdivmod_rel_def by simp_all
-
- have 1: "?x = ?q * y + ?r"
- using b x by simp
-
- have 2: "?r = 0 \<or> degree ?r < degree y"
- proof (rule eq_zero_or_degree_less)
- show "degree ?r \<le> degree y"
- proof (rule degree_diff_le)
- show "degree (pCons a r) \<le> degree y"
- using r by auto
- show "degree (smult b y) \<le> degree y"
- by (rule degree_smult_le)
- qed
- next
- show "coeff ?r (degree y) = 0"
- using `y \<noteq> 0` unfolding b by simp
- qed
-
- from 1 2 show ?thesis
- unfolding pdivmod_rel_def
- using `y \<noteq> 0` by simp
-qed
-
-lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
-apply (cases "y = 0")
-apply (fast intro!: pdivmod_rel_by_0)
-apply (induct x)
-apply (fast intro!: pdivmod_rel_0)
-apply (fast intro!: pdivmod_rel_pCons)
-done
-
-lemma pdivmod_rel_unique:
- assumes 1: "pdivmod_rel x y q1 r1"
- assumes 2: "pdivmod_rel x y q2 r2"
- shows "q1 = q2 \<and> r1 = r2"
-proof (cases "y = 0")
- assume "y = 0" with assms show ?thesis
- by (simp add: pdivmod_rel_def)
-next
- assume [simp]: "y \<noteq> 0"
- from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
- unfolding pdivmod_rel_def by simp_all
- from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
- unfolding pdivmod_rel_def by simp_all
- from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
- by (simp add: algebra_simps)
- from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
- by (auto intro: degree_diff_less)
-
- show "q1 = q2 \<and> r1 = r2"
- proof (rule ccontr)
- assume "\<not> (q1 = q2 \<and> r1 = r2)"
- with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
- with r3 have "degree (r2 - r1) < degree y" by simp
- also have "degree y \<le> degree (q1 - q2) + degree y" by simp
- also have "\<dots> = degree ((q1 - q2) * y)"
- using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
- also have "\<dots> = degree (r2 - r1)"
- using q3 by simp
- finally have "degree (r2 - r1) < degree (r2 - r1)" .
- then show "False" by simp
- qed
-qed
-
-lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
-by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
-
-lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
-by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
-
-lemmas pdivmod_rel_unique_div =
- pdivmod_rel_unique [THEN conjunct1, standard]
-
-lemmas pdivmod_rel_unique_mod =
- pdivmod_rel_unique [THEN conjunct2, standard]
-
-instantiation poly :: (field) ring_div
-begin
-
-definition div_poly where
- [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
-
-definition mod_poly where
- [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
-
-lemma div_poly_eq:
- "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
-unfolding div_poly_def
-by (fast elim: pdivmod_rel_unique_div)
-
-lemma mod_poly_eq:
- "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
-unfolding mod_poly_def
-by (fast elim: pdivmod_rel_unique_mod)
-
-lemma pdivmod_rel:
- "pdivmod_rel x y (x div y) (x mod y)"
-proof -
- from pdivmod_rel_exists
- obtain q r where "pdivmod_rel x y q r" by fast
- thus ?thesis
- by (simp add: div_poly_eq mod_poly_eq)
-qed
-
-instance proof
- fix x y :: "'a poly"
- show "x div y * y + x mod y = x"
- using pdivmod_rel [of x y]
- by (simp add: pdivmod_rel_def)
-next
- fix x :: "'a poly"
- have "pdivmod_rel x 0 0 x"
- by (rule pdivmod_rel_by_0)
- thus "x div 0 = 0"
- by (rule div_poly_eq)
-next
- fix y :: "'a poly"
- have "pdivmod_rel 0 y 0 0"
- by (rule pdivmod_rel_0)
- thus "0 div y = 0"
- by (rule div_poly_eq)
-next
- fix x y z :: "'a poly"
- assume "y \<noteq> 0"
- hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
- using pdivmod_rel [of x y]
- by (simp add: pdivmod_rel_def left_distrib)
- thus "(x + z * y) div y = z + x div y"
- by (rule div_poly_eq)
-qed
-
-end
-
-lemma degree_mod_less:
- "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
- using pdivmod_rel [of x y]
- unfolding pdivmod_rel_def by simp
-
-lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
-proof -
- assume "degree x < degree y"
- hence "pdivmod_rel x y 0 x"
- by (simp add: pdivmod_rel_def)
- thus "x div y = 0" by (rule div_poly_eq)
-qed
-
-lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
-proof -
- assume "degree x < degree y"
- hence "pdivmod_rel x y 0 x"
- by (simp add: pdivmod_rel_def)
- thus "x mod y = x" by (rule mod_poly_eq)
-qed
-
-lemma pdivmod_rel_smult_left:
- "pdivmod_rel x y q r
- \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
- unfolding pdivmod_rel_def by (simp add: smult_add_right)
-
-lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
- by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
-
-lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
- by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
-
-lemma pdivmod_rel_smult_right:
- "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
- \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
- unfolding pdivmod_rel_def by simp
-
-lemma div_smult_right:
- "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
- by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
-
-lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
- by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
-
-lemma pdivmod_rel_mult:
- "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
- \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
-apply (cases "z = 0", simp add: pdivmod_rel_def)
-apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
-apply (cases "r = 0")
-apply (cases "r' = 0")
-apply (simp add: pdivmod_rel_def)
-apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq)
-apply (cases "r' = 0")
-apply (simp add: pdivmod_rel_def degree_mult_eq)
-apply (simp add: pdivmod_rel_def ring_simps)
-apply (simp add: degree_mult_eq degree_add_less)
-done
-
-lemma poly_div_mult_right:
- fixes x y z :: "'a::field poly"
- shows "x div (y * z) = (x div y) div z"
- by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
-
-lemma poly_mod_mult_right:
- fixes x y z :: "'a::field poly"
- shows "x mod (y * z) = y * (x div y mod z) + x mod y"
- by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
-
-lemma mod_pCons:
- fixes a and x
- assumes y: "y \<noteq> 0"
- defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
- shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
-unfolding b
-apply (rule mod_poly_eq)
-apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
-done
-
-
-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,recpower}"
- 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)
- 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,recpower} 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 [code del]:
- "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
- with `k \<noteq> 0` have "finite {x. poly k x = 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: expand_fun_eq)
-
-
-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
-
-
-subsection {* Order of polynomial roots *}
-
-definition
- order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
-where
- [code del]:
- "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
-
-lemma coeff_linear_power:
- fixes a :: "'a::comm_semiring_1"
- shows "coeff ([:a, 1:] ^ n) n = 1"
-apply (induct n, simp_all)
-apply (subst coeff_eq_0)
-apply (auto intro: le_less_trans degree_power_le)
-done
-
-lemma degree_linear_power:
- fixes a :: "'a::comm_semiring_1"
- shows "degree ([:a, 1:] ^ n) = n"
-apply (rule order_antisym)
-apply (rule ord_le_eq_trans [OF degree_power_le], simp)
-apply (rule le_degree, simp add: coeff_linear_power)
-done
-
-lemma order_1: "[:-a, 1:] ^ order a p dvd p"
-apply (cases "p = 0", simp)
-apply (cases "order a p", simp)
-apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
-apply (drule not_less_Least, simp)
-apply (fold order_def, simp)
-done
-
-lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
-unfolding order_def
-apply (rule LeastI_ex)
-apply (rule_tac x="degree p" in exI)
-apply (rule notI)
-apply (drule (1) dvd_imp_degree_le)
-apply (simp only: degree_linear_power)
-done
-
-lemma order:
- "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
-by (rule conjI [OF order_1 order_2])
-
-lemma order_degree:
- assumes p: "p \<noteq> 0"
- shows "order a p \<le> degree p"
-proof -
- have "order a p = degree ([:-a, 1:] ^ order a p)"
- by (simp only: degree_linear_power)
- also have "\<dots> \<le> degree p"
- using order_1 p by (rule dvd_imp_degree_le)
- finally show ?thesis .
-qed
-
-lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
-apply (cases "p = 0", simp_all)
-apply (rule iffI)
-apply (rule ccontr, simp)
-apply (frule order_2 [where a=a], simp)
-apply (simp add: poly_eq_0_iff_dvd)
-apply (simp add: poly_eq_0_iff_dvd)
-apply (simp only: order_def)
-apply (drule not_less_Least, simp)
-done
-
-
-subsection {* Configuration of the code generator *}
-
-code_datatype "0::'a::zero poly" pCons
-
-declare pCons_0_0 [code post]
-
-instantiation poly :: ("{zero,eq}") eq
-begin
-
-definition [code del]:
- "eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q"
-
-instance
- by default (rule eq_poly_def)
-
-end
-
-lemma eq_poly_code [code]:
- "eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
- "eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q"
- "eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0"
- "eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q"
-unfolding eq by simp_all
-
-lemmas coeff_code [code] =
- coeff_0 coeff_pCons_0 coeff_pCons_Suc
-
-lemmas degree_code [code] =
- degree_0 degree_pCons_eq_if
-
-lemmas monom_poly_code [code] =
- monom_0 monom_Suc
-
-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 minus_poly_code [code]:
- "- 0 = (0 :: _ poly)"
- "- pCons a p = pCons (- a) (- p)"
-by simp_all
-
-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
-
-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
-
-lemmas poly_code [code] =
- poly_0 poly_pCons
-
-lemmas synthetic_divmod_code [code] =
- synthetic_divmod_0 synthetic_divmod_pCons
-
-text {* code generator setup for div and mod *}
-
-definition
- pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
-where
- [code del]: "pdivmod x y = (x div y, x mod y)"
-
-lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
- unfolding pdivmod_def by simp
-
-lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
- unfolding pdivmod_def by simp
-
-lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
- unfolding pdivmod_def by simp
-
-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])
-done
-
-end