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Quellcode-Bibliothek
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Datei:
median_overlap.prf
Sprache: Lisp
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(* Title: HOL/Algebra/Polynomials.thy
Author: Paulo Emílio de Vilhena
*)
theory Polynomials
imports Ring Ring_Divisibility Subrings
begin
section \<open>Polynomials\<close>
subsection \<open>Definitions\<close>
abbreviation lead_coeff :: "'a list \ 'a"
where "lead_coeff \ hd"
abbreviation degree :: "'a list \ nat"
where "degree p \ length p - 1"
definition polynomial :: "_ \ 'a set \ 'a list \ bool" ("polynomial\")
where "polynomial\<^bsub>R\<^esub> K p \ p = [] \ (set p \ K \ lead_coeff p \ \\<^bsub>R\<^esub>)"
definition (in ring) monom :: "'a \ nat \ 'a list"
where "monom a n = a # (replicate n \\<^bsub>R\<^esub>)"
fun (in ring) eval :: "'a list \ 'a \ 'a"
where
"eval [] = (\_. \)"
| "eval p = (\x. ((lead_coeff p) \ (x [^] (degree p))) \ (eval (tl p) x))"
fun (in ring) coeff :: "'a list \ nat \ 'a"
where
"coeff [] = (\_. \)"
| "coeff p = (\i. if i = degree p then lead_coeff p else (coeff (tl p)) i)"
fun (in ring) normalize :: "'a list \ 'a list"
where
"normalize [] = []"
| "normalize p = (if lead_coeff p \ \ then p else normalize (tl p))"
fun (in ring) poly_add :: "'a list \ 'a list \ 'a list"
where "poly_add p1 p2 =
(if length p1 \<ge> length p2
then normalize (map2 (\<oplus>) p1 ((replicate (length p1 - length p2) \<zero>) @ p2))
else poly_add p2 p1)"
fun (in ring) poly_mult :: "'a list \ 'a list \ 'a list"
where
"poly_mult [] p2 = []"
| "poly_mult p1 p2 =
poly_add ((map (\<lambda>a. lead_coeff p1 \<otimes> a) p2) @ (replicate (degree p1) \<zero>)) (poly_mult (tl p1) p2)"
fun (in ring) dense_repr :: "'a list \ ('a \ nat) list"
where
"dense_repr [] = []"
| "dense_repr p = (if lead_coeff p \ \
then (lead_coeff p, degree p) # (dense_repr (tl p))
else (dense_repr (tl p)))"
fun (in ring) poly_of_dense :: "('a \ nat) list \ 'a list"
where "poly_of_dense dl = foldr (\(a, n) l. poly_add (monom a n) l) dl []"
definition (in ring) poly_of_const :: "'a \ 'a list"
where "poly_of_const = (\k. normalize [ k ])"
subsection \<open>Basic Properties\<close>
context ring
begin
lemma polynomialI [intro]: "\ set p \ K; lead_coeff p \ \ \ \ polynomial K p"
unfolding polynomial_def by auto
lemma polynomial_incl: "polynomial K p \ set p \ K"
unfolding polynomial_def by auto
lemma monom_in_carrier [intro]: "a \ carrier R \ set (monom a n) \ carrier R"
unfolding monom_def by auto
lemma lead_coeff_not_zero: "polynomial K (a # p) \ a \ K - { \ }"
unfolding polynomial_def by simp
lemma zero_is_polynomial [intro]: "polynomial K []"
unfolding polynomial_def by simp
lemma const_is_polynomial [intro]: "a \ K - { \ } \ polynomial K [ a ]"
unfolding polynomial_def by auto
lemma normalize_gives_polynomial: "set p \ K \ polynomial K (normalize p)"
by (induction p) (auto simp add: polynomial_def)
lemma normalize_in_carrier: "set p \ carrier R \ set (normalize p) \ carrier R"
by (induction p) (auto)
lemma normalize_polynomial: "polynomial K p \ normalize p = p"
unfolding polynomial_def by (cases p) (auto)
lemma normalize_idem: "normalize ((normalize p) @ q) = normalize (p @ q)"
by (induct p) (auto)
lemma normalize_length_le: "length (normalize p) \ length p"
by (induction p) (auto)
lemma eval_in_carrier: "\ set p \ carrier R; x \ carrier R \ \ (eval p) x \ carrier R"
by (induction p) (auto)
lemma coeff_in_carrier [simp]: "set p \ carrier R \ (coeff p) i \ carrier R"
by (induction p) (auto)
lemma lead_coeff_simp [simp]: "p \ [] \ (coeff p) (degree p) = lead_coeff p"
by (metis coeff.simps(2) list.exhaust_sel)
lemma coeff_list: "map (coeff p) (rev [0..< length p]) = p"
proof (induction p)
case Nil thus ?case by simp
next
case (Cons a p)
have "map (coeff (a # p)) (rev [0..
a # (map (coeff p) (rev [0..<length p]))"
by auto
also have " ... = a # p"
using Cons by simp
finally show ?case .
qed
lemma coeff_nth: "i < length p \ (coeff p) i = p ! (length p - 1 - i)"
proof -
assume i_lt: "i < length p"
hence "(coeff p) i = (map (coeff p) [0..< length p]) ! i"
by simp
also have " ... = (rev (map (coeff p) (rev [0..< length p]))) ! i"
by (simp add: rev_map)
also have " ... = (map (coeff p) (rev [0..< length p])) ! (length p - 1 - i)"
using coeff_list i_lt rev_nth by auto
also have " ... = p ! (length p - 1 - i)"
using coeff_list[of p] by simp
finally show "(coeff p) i = p ! (length p - 1 - i)" .
qed
lemma coeff_iff_length_cond:
assumes "length p1 = length p2"
shows "p1 = p2 \ coeff p1 = coeff p2"
proof
show "p1 = p2 \ coeff p1 = coeff p2"
by simp
next
assume A: "coeff p1 = coeff p2"
have "p1 = map (coeff p1) (rev [0..< length p1])"
using coeff_list[of p1] by simp
also have " ... = map (coeff p2) (rev [0..< length p2])"
using A assms by simp
also have " ... = p2"
using coeff_list[of p2] by simp
finally show "p1 = p2" .
qed
lemma coeff_img_restrict: "(coeff p) ` {..< length p} = set p"
using coeff_list[of p] by (metis atLeast_upt image_set set_rev)
lemma coeff_length: "\i. i \ length p \ (coeff p) i = \"
by (induction p) (auto)
lemma coeff_degree: "\i. i > degree p \ (coeff p) i = \"
using coeff_length by (simp)
lemma replicate_zero_coeff [simp]: "coeff (replicate n \) = (\_. \)"
by (induction n) (auto)
lemma scalar_coeff: "a \ carrier R \ coeff (map (\b. a \ b) p) = (\i. a \ (coeff p) i)"
by (induction p) (auto)
lemma monom_coeff: "coeff (monom a n) = (\i. if i = n then a else \)"
unfolding monom_def by (induction n) (auto)
lemma coeff_img:
"(coeff p) ` {..< length p} = set p"
"(coeff p) ` { length p ..} = { \ }"
"(coeff p) ` UNIV = (set p) \ { \ }"
using coeff_img_restrict
proof (simp)
show coeff_img_up: "(coeff p) ` { length p ..} = { \ }"
using coeff_length[of p] by force
from coeff_img_up and coeff_img_restrict[of p]
show "(coeff p) ` UNIV = (set p) \ { \ }"
by force
qed
lemma degree_def':
assumes "polynomial K p"
shows "degree p = (LEAST n. \i. i > n \ (coeff p) i = \)"
proof (cases p)
case Nil thus ?thesis by auto
next
define P where "P = (\n. \i. i > n \ (coeff p) i = \)"
case (Cons a ps)
hence "(coeff p) (degree p) \ \"
using assms unfolding polynomial_def by auto
hence "\n. n < degree p \ \ P n"
unfolding P_def by auto
moreover have "P (degree p)"
unfolding P_def using coeff_degree[of p] by simp
ultimately have "degree p = (LEAST n. P n)"
by (meson LeastI nat_neq_iff not_less_Least)
thus ?thesis unfolding P_def .
qed
lemma coeff_iff_polynomial_cond:
assumes "polynomial K p1" and "polynomial K p2"
shows "p1 = p2 \ coeff p1 = coeff p2"
proof
show "p1 = p2 \ coeff p1 = coeff p2"
by simp
next
assume coeff_eq: "coeff p1 = coeff p2"
hence deg_eq: "degree p1 = degree p2"
using degree_def'[OF assms(1)] degree_def'[OF assms(2)] by auto
thus "p1 = p2"
proof (cases)
assume "p1 \ [] \ p2 \ []"
hence "length p1 = length p2"
using deg_eq by (simp add: Nitpick.size_list_simp(2))
thus ?thesis
using coeff_iff_length_cond[of p1 p2] coeff_eq by simp
next
{ fix p1 p2 assume A: "p1 = []" "coeff p1 = coeff p2" "polynomial K p2"
have "p2 = []"
proof (rule ccontr)
assume "p2 \ []"
hence "(coeff p2) (degree p2) \ \"
using A(3) unfolding polynomial_def
by (metis coeff.simps(2) list.collapse)
moreover have "(coeff p1) ` UNIV = { \ }"
using A(1) by auto
hence "(coeff p2) ` UNIV = { \ }"
using A(2) by simp
ultimately show False
by blast
qed } note aux_lemma = this
assume "\ (p1 \ [] \ p2 \ [])"
hence "p1 = [] \ p2 = []" by simp
thus ?thesis
using assms coeff_eq aux_lemma[of p1 p2] aux_lemma[of p2 p1] by auto
qed
qed
lemma normalize_lead_coeff:
assumes "length (normalize p) < length p"
shows "lead_coeff p = \"
proof (cases p)
case Nil thus ?thesis
using assms by simp
next
case (Cons a ps) thus ?thesis
using assms by (cases "a = \") (auto)
qed
lemma normalize_length_lt:
assumes "lead_coeff p = \" and "length p > 0"
shows "length (normalize p) < length p"
proof (cases p)
case Nil thus ?thesis
using assms by simp
next
case (Cons a ps) thus ?thesis
using normalize_length_le[of ps] assms by simp
qed
lemma normalize_length_eq:
assumes "lead_coeff p \ \"
shows "length (normalize p) = length p"
using normalize_length_le[of p] assms nat_less_le normalize_lead_coeff by auto
lemma normalize_replicate_zero: "normalize ((replicate n \) @ p) = normalize p"
by (induction n) (auto)
lemma normalize_def':
shows "p = (replicate (length p - length (normalize p)) \) @
(drop (length p - length (normalize p)) p)" (is ?statement1)
and "normalize p = drop (length p - length (normalize p)) p" (is ?statement2)
proof -
show ?statement1
proof (induction p)
case Nil thus ?case by simp
next
case (Cons a p) thus ?case
proof (cases "a = \")
assume "a \ \" thus ?case
using Cons by simp
next
assume eq_zero: "a = \"
hence len_eq:
"Suc (length p - length (normalize p)) = length (a # p) - length (normalize (a # p))"
by (simp add: Suc_diff_le normalize_length_le)
have "a # p = \ # (replicate (length p - length (normalize p)) \ @
drop (length p - length (normalize p)) p)"
using eq_zero Cons by simp
also have " ... = (replicate (Suc (length p - length (normalize p))) \ @
drop (Suc (length p - length (normalize p))) (a # p))"
by simp
also have " ... = (replicate (length (a # p) - length (normalize (a # p))) \ @
drop (length (a # p) - length (normalize (a # p))) (a # p))"
using len_eq by simp
finally show ?case .
qed
qed
next
show ?statement2
proof -
have "\m. normalize p = drop m p"
proof (induction p)
case Nil thus ?case by simp
next
case (Cons a p) thus ?case
apply (cases "a = \")
apply (auto)
apply (metis drop_Suc_Cons)
apply (metis drop0)
done
qed
then obtain m where m: "normalize p = drop m p" by auto
hence "length (normalize p) = length p - m" by simp
thus ?thesis
using m by (metis rev_drop rev_rev_ident take_rev)
qed
qed
corollary normalize_trick:
shows "p = (replicate (length p - length (normalize p)) \) @ (normalize p)"
using normalize_def'(1)[of p] unfolding sym[OF normalize_def'(2)] .
lemma normalize_coeff: "coeff p = coeff (normalize p)"
proof (induction p)
case Nil thus ?case by simp
next
case (Cons a p)
have "coeff (normalize p) (length p) = \"
using normalize_length_le[of p] coeff_degree[of "normalize p"] coeff_length by blast
then show ?case
using Cons by (cases "a = \") (auto)
qed
lemma append_coeff:
"coeff (p @ q) = (\i. if i < length q then (coeff q) i else (coeff p) (i - length q))"
proof (induction p)
case Nil thus ?case
using coeff_length[of q] by auto
next
case (Cons a p)
have "coeff ((a # p) @ q) = (\i. if i = length p + length q then a else (coeff (p @ q)) i)"
by auto
also have " ... = (\i. if i = length p + length q then a
else if i < length q then (coeff q) i
else (coeff p) (i - length q))"
using Cons by auto
also have " ... = (\i. if i < length q then (coeff q) i
else if i = length p + length q then a else (coeff p) (i - length q))"
by auto
also have " ... = (\i. if i < length q then (coeff q) i
else if i - length q = length p then a else (coeff p) (i - length q))"
by fastforce
also have " ... = (\i. if i < length q then (coeff q) i else (coeff (a # p)) (i - length q))"
by auto
finally show ?case .
qed
lemma prefix_replicate_zero_coeff: "coeff p = coeff ((replicate n \) @ p)"
using append_coeff[of "replicate n \" p] replicate_zero_coeff[of n] coeff_length[of p] by auto
(* ========================================================================== *)
context
fixes K :: "'a set" assumes K: "subring K R"
begin
lemma polynomial_in_carrier [intro]: "polynomial K p \ set p \ carrier R"
unfolding polynomial_def using subringE(1)[OF K] by auto
lemma carrier_polynomial [intro]: "polynomial K p \ polynomial (carrier R) p"
unfolding polynomial_def using subringE(1)[OF K] by auto
lemma append_is_polynomial: "\ polynomial K p; p \ [] \ \ polynomial K (p @ (replicate n \))"
unfolding polynomial_def using subringE(2)[OF K] by auto
lemma lead_coeff_in_carrier: "polynomial K (a # p) \ a \ carrier R - { \ }"
unfolding polynomial_def using subringE(1)[OF K] by auto
lemma monom_is_polynomial [intro]: "a \ K - { \ } \ polynomial K (monom a n)"
unfolding polynomial_def monom_def using subringE(2)[OF K] by auto
lemma eval_poly_in_carrier: "\ polynomial K p; x \ carrier R \ \ (eval p) x \ carrier R"
using eval_in_carrier[OF polynomial_in_carrier] .
lemma poly_coeff_in_carrier [simp]: "polynomial K p \ coeff p i \ carrier R"
using coeff_in_carrier[OF polynomial_in_carrier] .
end (* of fixed K context. *)
(* ========================================================================== *)
subsection \<open>Polynomial Addition\<close>
(* ========================================================================== *)
context
fixes K :: "'a set" assumes K: "subring K R"
begin
lemma poly_add_is_polynomial:
assumes "set p1 \ K" and "set p2 \ K"
shows "polynomial K (poly_add p1 p2)"
proof -
{ fix p1 p2 assume A: "set p1 \ K" "set p2 \ K" "length p1 \ length p2"
hence "polynomial K (poly_add p1 p2)"
proof -
define p2' where "p2' = (replicate (length p1 - length p2) \<zero>) @ p2"
hence "set p2' \ K" and "length p1 = length p2'"
using A(2-3) subringE(2)[OF K] by auto
hence "set (map2 (\) p1 p2') \ K"
using A(1) subringE(7)[OF K]
by (induct p1) (auto, metis set_ConsD subsetD set_zip_leftD set_zip_rightD)
thus ?thesis
unfolding p2'_def using normalize_gives_polynomial A(3) by simp
qed }
thus ?thesis
using assms by auto
qed
lemma poly_add_closed: "\ polynomial K p1; polynomial K p2 \ \ polynomial K (poly_add p1 p2)"
using poly_add_is_polynomial polynomial_incl by simp
lemma poly_add_length_eq:
assumes "polynomial K p1" "polynomial K p2" and "length p1 \ length p2"
shows "length (poly_add p1 p2) = max (length p1) (length p2)"
proof -
{ fix p1 p2 assume A: "polynomial K p1" "polynomial K p2" "length p1 > length p2"
hence "length (poly_add p1 p2) = max (length p1) (length p2)"
proof -
let ?p2 = "(replicate (length p1 - length p2) \) @ p2"
have p1: "p1 \ []" and p2: "?p2 \ []"
using A(3) by auto
then have "zip p1 (replicate (length p1 - length p2) \ @ p2) = zip (lead_coeff p1 # tl p1) (lead_coeff (replicate (length p1 - length p2) \ @ p2) # tl (replicate (length p1 - length p2) \ @ p2))"
by auto
hence "lead_coeff (map2 (\) p1 ?p2) = lead_coeff p1 \ lead_coeff ?p2"
by simp
moreover have "lead_coeff p1 \ carrier R"
using p1 A(1) lead_coeff_in_carrier[OF K, of "hd p1" "tl p1"] by auto
ultimately have "lead_coeff (map2 (\) p1 ?p2) = lead_coeff p1"
using A(3) by auto
moreover have "lead_coeff p1 \ \"
using p1 A(1) unfolding polynomial_def by simp
ultimately have "length (normalize (map2 (\) p1 ?p2)) = length p1"
using normalize_length_eq by auto
thus ?thesis
using A(3) by auto
qed }
thus ?thesis
using assms by auto
qed
lemma poly_add_degree_eq:
assumes "polynomial K p1" "polynomial K p2" and "degree p1 \ degree p2"
shows "degree (poly_add p1 p2) = max (degree p1) (degree p2)"
using poly_add_length_eq[OF assms(1-2)] assms(3) by simp
end (* of fixed K context. *)
(* ========================================================================== *)
lemma poly_add_in_carrier:
"\ set p1 \ carrier R; set p2 \ carrier R \ \ set (poly_add p1 p2) \ carrier R"
using polynomial_incl[OF poly_add_is_polynomial[OF carrier_is_subring]] by simp
lemma poly_add_length_le: "length (poly_add p1 p2) \ max (length p1) (length p2)"
proof -
{ fix p1 p2 :: "'a list" assume A: "length p1 \ length p2"
let ?p2 = "(replicate (length p1 - length p2) \) @ p2"
have "length (poly_add p1 p2) \ max (length p1) (length p2)"
using normalize_length_le[of "map2 (\) p1 ?p2"] A by auto }
thus ?thesis
by (metis le_cases max.commute poly_add.simps)
qed
lemma poly_add_degree: "degree (poly_add p1 p2) \ max (degree p1) (degree p2)"
using poly_add_length_le by (meson diff_le_mono le_max_iff_disj)
lemma poly_add_coeff_aux:
assumes "length p1 \ length p2"
shows "coeff (poly_add p1 p2) = (\i. ((coeff p1) i) \ ((coeff p2) i))"
proof
fix i
have "i < length p1 \ (coeff (poly_add p1 p2)) i = ((coeff p1) i) \ ((coeff p2) i)"
proof -
let ?p2 = "(replicate (length p1 - length p2) \) @ p2"
have len_eqs: "length p1 = length ?p2" "length (map2 (\) p1 ?p2) = length p1"
using assms by auto
assume i_lt: "i < length p1"
have "(coeff (poly_add p1 p2)) i = (coeff (map2 (\) p1 ?p2)) i"
using normalize_coeff[of "map2 (\) p1 ?p2"] assms by auto
also have " ... = (map2 (\) p1 ?p2) ! (length p1 - 1 - i)"
using coeff_nth[of i "map2 (\) p1 ?p2"] len_eqs(2) i_lt by auto
also have " ... = (p1 ! (length p1 - 1 - i)) \ (?p2 ! (length ?p2 - 1 - i))"
using len_eqs i_lt by auto
also have " ... = ((coeff p1) i) \ ((coeff ?p2) i)"
using coeff_nth[of i p1] coeff_nth[of i ?p2] i_lt len_eqs(1) by auto
also have " ... = ((coeff p1) i) \ ((coeff p2) i)"
using prefix_replicate_zero_coeff by simp
finally show "(coeff (poly_add p1 p2)) i = ((coeff p1) i) \ ((coeff p2) i)" .
qed
moreover
have "i \ length p1 \ (coeff (poly_add p1 p2)) i = ((coeff p1) i) \ ((coeff p2) i)"
using coeff_length[of "poly_add p1 p2"] coeff_length[of p1] coeff_length[of p2]
poly_add_length_le[of p1 p2] assms by auto
ultimately show "(coeff (poly_add p1 p2)) i = ((coeff p1) i) \ ((coeff p2) i)"
using not_le by blast
qed
lemma poly_add_coeff:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "coeff (poly_add p1 p2) = (\i. ((coeff p1) i) \ ((coeff p2) i))"
proof -
have "length p1 \ length p2 \ length p2 > length p1"
by auto
thus ?thesis
proof
assume "length p1 \ length p2" thus ?thesis
using poly_add_coeff_aux by simp
next
assume "length p2 > length p1"
hence "coeff (poly_add p1 p2) = (\i. ((coeff p2) i) \ ((coeff p1) i))"
using poly_add_coeff_aux by simp
thus ?thesis
using assms by (simp add: add.m_comm)
qed
qed
lemma poly_add_comm:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_add p1 p2 = poly_add p2 p1"
proof -
have "coeff (poly_add p1 p2) = coeff (poly_add p2 p1)"
using poly_add_coeff[OF assms] poly_add_coeff[OF assms(2) assms(1)]
coeff_in_carrier[OF assms(1)] coeff_in_carrier[OF assms(2)] add.m_comm by auto
thus ?thesis
using coeff_iff_polynomial_cond[OF
poly_add_is_polynomial[OF carrier_is_subring assms]
poly_add_is_polynomial[OF carrier_is_subring assms(2,1)]] by simp
qed
lemma poly_add_monom:
assumes "set p \ carrier R" and "a \ carrier R - { \ }"
shows "poly_add (monom a (length p)) p = a # p"
unfolding monom_def using assms by (induction p) (auto)
lemma poly_add_append_replicate:
assumes "set p \ carrier R" "set q \ carrier R"
shows "poly_add (p @ (replicate (length q) \)) q = normalize (p @ q)"
proof -
have "map2 (\) (p @ (replicate (length q) \)) ((replicate (length p) \) @ q) = p @ q"
using assms by (induct p) (induct q, auto)
thus ?thesis by simp
qed
lemma poly_add_append_zero:
assumes "set p \ carrier R" "set q \ carrier R"
shows "poly_add (p @ [ \ ]) (q @ [ \ ]) = normalize ((poly_add p q) @ [ \ ])"
proof -
have in_carrier: "set (p @ [ \ ]) \ carrier R" "set (q @ [ \ ]) \ carrier R"
using assms by auto
have "coeff (poly_add (p @ [ \ ]) (q @ [ \ ])) = coeff ((poly_add p q) @ [ \ ])"
using append_coeff[of p "[ \ ]"] poly_add_coeff[OF in_carrier]
append_coeff[of q "[ \ ]"] append_coeff[of "poly_add p q" "[ \ ]"]
poly_add_coeff[OF assms] assms[THEN coeff_in_carrier] by auto
hence "coeff (poly_add (p @ [ \ ]) (q @ [ \ ])) = coeff (normalize ((poly_add p q) @ [ \ ]))"
using normalize_coeff by simp
moreover have "set ((poly_add p q) @ [ \ ]) \ carrier R"
using poly_add_in_carrier[OF assms] by simp
ultimately show ?thesis
using coeff_iff_polynomial_cond[OF poly_add_is_polynomial[OF carrier_is_subring in_carrier]
normalize_gives_polynomial] by simp
qed
lemma poly_add_normalize_aux:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_add p1 p2 = poly_add (normalize p1) p2"
proof -
{ fix n p1 p2 assume "set p1 \ carrier R" "set p2 \ carrier R"
hence "poly_add p1 p2 = poly_add ((replicate n \) @ p1) p2"
proof (induction n)
case 0 thus ?case by simp
next
{ fix p1 p2 :: "'a list"
assume in_carrier: "set p1 \ carrier R" "set p2 \ carrier R"
have "poly_add p1 p2 = poly_add (\ # p1) p2"
proof -
have "length p1 \ length p2 \ ?thesis"
proof -
assume A: "length p1 \ length p2"
let ?p2 = "\n. (replicate n \) @ p2"
have "poly_add p1 p2 = normalize (map2 (\) (\ # p1) (\ # ?p2 (length p1 - length p2)))"
using A by simp
also have " ... = normalize (map2 (\) (\ # p1) (?p2 (length (\ # p1) - length p2)))"
by (simp add: A Suc_diff_le)
also have " ... = poly_add (\ # p1) p2"
using A by simp
finally show ?thesis .
qed
moreover have "length p2 > length p1 \ ?thesis"
proof -
assume A: "length p2 > length p1"
let ?f = "\n p. (replicate n \) @ p"
have "poly_add p1 p2 = poly_add p2 p1"
using A by simp
also have " ... = normalize (map2 (\) p2 (?f (length p2 - length p1) p1))"
using A by simp
also have " ... = normalize (map2 (\) p2 (?f (length p2 - Suc (length p1)) (\ # p1)))"
by (metis A Suc_diff_Suc append_Cons replicate_Suc replicate_app_Cons_same)
also have " ... = poly_add p2 (\ # p1)"
using A by simp
also have " ... = poly_add (\ # p1) p2"
using poly_add_comm[of p2 "\ # p1"] in_carrier by auto
finally show ?thesis .
qed
ultimately show ?thesis by auto
qed } note aux_lemma = this
case (Suc n)
hence in_carrier: "set (replicate n \ @ p1) \ carrier R"
by auto
have "poly_add p1 p2 = poly_add (replicate n \ @ p1) p2"
using Suc by simp
also have " ... = poly_add (replicate (Suc n) \ @ p1) p2"
using aux_lemma[OF in_carrier Suc(3)] by simp
finally show ?case .
qed } note aux_lemma = this
have "poly_add p1 p2 =
poly_add ((replicate (length p1 - length (normalize p1)) \<zero>) @ normalize p1) p2"
using normalize_def'[of p1] by simp
also have " ... = poly_add (normalize p1) p2"
using aux_lemma[OF normalize_in_carrier[OF assms(1)] assms(2)] by simp
finally show ?thesis .
qed
lemma poly_add_normalize:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_add p1 p2 = poly_add (normalize p1) p2"
and "poly_add p1 p2 = poly_add p1 (normalize p2)"
and "poly_add p1 p2 = poly_add (normalize p1) (normalize p2)"
proof -
show "poly_add p1 p2 = poly_add p1 (normalize p2)"
unfolding poly_add_comm[OF assms] poly_add_normalize_aux[OF assms(2) assms(1)]
poly_add_comm[OF normalize_in_carrier[OF assms(2)] assms(1)] by simp
next
show "poly_add p1 p2 = poly_add (normalize p1) p2"
using poly_add_normalize_aux[OF assms] .
also have " ... = poly_add (normalize p2) (normalize p1)"
unfolding poly_add_comm[OF normalize_in_carrier[OF assms(1)] assms(2)]
poly_add_normalize_aux[OF assms(2) normalize_in_carrier[OF assms(1)]] by simp
finally show "poly_add p1 p2 = poly_add (normalize p1) (normalize p2)"
unfolding poly_add_comm[OF assms[THEN normalize_in_carrier]] .
qed
lemma poly_add_zero':
assumes "set p \ carrier R"
shows "poly_add p [] = normalize p" and "poly_add [] p = normalize p"
proof -
have "map2 (\) p (replicate (length p) \) = p"
using assms by (induct p) (auto)
thus "poly_add p [] = normalize p" and "poly_add [] p = normalize p"
using poly_add_comm[OF assms, of "[]"] by simp+
qed
lemma poly_add_zero:
assumes "subring K R" "polynomial K p"
shows "poly_add p [] = p" and "poly_add [] p = p"
using poly_add_zero' normalize_polynomial polynomial_in_carrier assms by auto
lemma poly_add_replicate_zero':
assumes "set p \ carrier R"
shows "poly_add p (replicate n \) = normalize p" and "poly_add (replicate n \) p = normalize p"
proof -
have "poly_add p (replicate n \) = poly_add p []"
using poly_add_normalize(2)[OF assms, of "replicate n \"]
normalize_replicate_zero[of n "[]"] by force
also have " ... = normalize p"
using poly_add_zero'[OF assms] by simp
finally show "poly_add p (replicate n \) = normalize p" .
thus "poly_add (replicate n \) p = normalize p"
using poly_add_comm[OF assms, of "replicate n \"] by force
qed
lemma poly_add_replicate_zero:
assumes "subring K R" "polynomial K p"
shows "poly_add p (replicate n \) = p" and "poly_add (replicate n \) p = p"
using poly_add_replicate_zero' normalize_polynomial polynomial_in_carrier assms by auto
subsection \<open>Dense Representation\<close>
lemma dense_repr_replicate_zero: "dense_repr ((replicate n \) @ p) = dense_repr p"
by (induction n) (auto)
lemma dense_repr_normalize: "dense_repr (normalize p) = dense_repr p"
by (induct p) (auto)
lemma polynomial_dense_repr:
assumes "polynomial K p" and "p \ []"
shows "dense_repr p = (lead_coeff p, degree p) # dense_repr (normalize (tl p))"
proof -
let ?len = length and ?norm = normalize
obtain a p' where p: "p = a # p'"
using assms(2) list.exhaust_sel by blast
hence a: "a \ K - { \ }" and p': "set p' \ K"
using assms(1) unfolding p by (auto simp add: polynomial_def)
hence "dense_repr p = (lead_coeff p, degree p) # dense_repr p'"
unfolding p by simp
also have " ... =
(lead_coeff p, degree p) # dense_repr ((replicate (?len p' - ?len (?norm p')) \<zero>) @ ?norm p')"
using normalize_def' dense_repr_replicate_zero by simp
also have " ... = (lead_coeff p, degree p) # dense_repr (?norm p')"
using dense_repr_replicate_zero by simp
finally show ?thesis
unfolding p by simp
qed
lemma monom_decomp:
assumes "subring K R" "polynomial K p"
shows "p = poly_of_dense (dense_repr p)"
using assms(2)
proof (induct "length p" arbitrary: p rule: less_induct)
case less thus ?case
proof (cases p)
case Nil thus ?thesis by simp
next
case (Cons a l)
hence a: "a \ carrier R - { \ }" and l: "set l \ carrier R" "set l \ K"
using less(2) subringE(1)[OF assms(1)] by (auto simp add: polynomial_def)
hence "a # l = poly_add (monom a (degree (a # l))) l"
using poly_add_monom[of l a] by simp
also have " ... = poly_add (monom a (degree (a # l))) (normalize l)"
using poly_add_normalize(2)[of "monom a (degree (a # l))", OF _ l(1)] a
unfolding monom_def by force
also have " ... = poly_add (monom a (degree (a # l))) (poly_of_dense (dense_repr (normalize l)))"
using less(1)[OF _ normalize_gives_polynomial[OF l(2)]] normalize_length_le[of l]
unfolding Cons by simp
also have " ... = poly_of_dense ((a, degree (a # l)) # dense_repr (normalize l))"
by simp
also have " ... = poly_of_dense (dense_repr (a # l))"
using polynomial_dense_repr[OF less(2)] unfolding Cons by simp
finally show ?thesis
unfolding Cons by simp
qed
qed
subsection \<open>Polynomial Multiplication\<close>
lemma poly_mult_is_polynomial:
assumes "subring K R" "set p1 \ K" and "set p2 \ K"
shows "polynomial K (poly_mult p1 p2)"
using assms(2-3)
proof (induction p1)
case Nil thus ?case
by (simp add: polynomial_def)
next
case (Cons a p1)
let ?a_p2 = "(map (\b. a \ b) p2) @ (replicate (degree (a # p1)) \)"
have "set (poly_mult p1 p2) \ K"
using Cons unfolding polynomial_def by auto
moreover have "set ?a_p2 \ K"
using assms(3) Cons(2) subringE(1-2,6)[OF assms(1)] by(induct p2) (auto)
ultimately have "polynomial K (poly_add ?a_p2 (poly_mult p1 p2))"
using poly_add_is_polynomial[OF assms(1)] by blast
thus ?case by simp
qed
lemma poly_mult_closed:
assumes "subring K R"
shows "\ polynomial K p1; polynomial K p2 \ \ polynomial K (poly_mult p1 p2)"
using poly_mult_is_polynomial polynomial_incl assms by simp
lemma poly_mult_in_carrier:
"\ set p1 \ carrier R; set p2 \ carrier R \ \ set (poly_mult p1 p2) \ carrier R"
using poly_mult_is_polynomial polynomial_in_carrier carrier_is_subring by simp
lemma poly_mult_coeff:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "coeff (poly_mult p1 p2) = (\i. \ k \ {..i}. (coeff p1) k \ (coeff p2) (i - k))"
using assms(1)
proof (induction p1)
case Nil thus ?case using assms(2) by auto
next
case (Cons a p1)
hence in_carrier:
"a \ carrier R" "\i. (coeff p1) i \ carrier R" "\i. (coeff p2) i \ carrier R"
using coeff_in_carrier assms(2) by auto
let ?a_p2 = "(map (\b. a \ b) p2) @ (replicate (degree (a # p1)) \)"
have "coeff (replicate (degree (a # p1)) \) = (\_. \)"
and "length (replicate (degree (a # p1)) \) = length p1"
using prefix_replicate_zero_coeff[of "[]" "length p1"] by auto
hence "coeff ?a_p2 = (\i. if i < length p1 then \ else (coeff (map (\b. a \ b) p2)) (i - length p1))"
using append_coeff[of "map (\b. a \ b) p2" "replicate (length p1) \"] by auto
also have " ... = (\i. if i < length p1 then \ else a \ ((coeff p2) (i - length p1)))"
proof -
have "\i. i < length p2 \ (coeff (map (\b. a \ b) p2)) i = a \ ((coeff p2) i)"
proof -
fix i assume i_lt: "i < length p2"
hence "(coeff (map (\b. a \ b) p2)) i = (map (\b. a \ b) p2) ! (length p2 - 1 - i)"
using coeff_nth[of i "map (\b. a \ b) p2"] by auto
also have " ... = a \ (p2 ! (length p2 - 1 - i))"
using i_lt by auto
also have " ... = a \ ((coeff p2) i)"
using coeff_nth[OF i_lt] by simp
finally show "(coeff (map (\b. a \ b) p2)) i = a \ ((coeff p2) i)" .
qed
moreover have "\i. i \ length p2 \ (coeff (map (\b. a \ b) p2)) i = a \ ((coeff p2) i)"
using coeff_length[of p2] coeff_length[of "map (\b. a \ b) p2"] in_carrier by auto
ultimately show ?thesis by (meson not_le)
qed
also have " ... = (\i. \ k \ {..i}. (if k = length p1 then a else \) \ (coeff p2) (i - k))"
(is "?f1 = (\i. (\ k \ {..i}. ?f2 k \ ?f3 (i - k)))")
proof
fix i
have "\k. k \ {..i} \ ?f2 k \ ?f3 (i - k) = \" if "i < length p1"
using in_carrier that by auto
hence "(\ k \ {..i}. ?f2 k \ ?f3 (i - k)) = \" if "i < length p1"
using that in_carrier
add.finprod_cong'[of "{..i}" "{..i}" "\k. ?f2 k \ ?f3 (i - k)" "\i. \"]
by auto
hence eq_lt: "?f1 i = (\i. (\ k \ {..i}. ?f2 k \ ?f3 (i - k))) i" if "i < length p1"
using that by auto
have "\k. k \ {..i} \
?f2 k \<otimes>\<^bsub>R\<^esub> ?f3 (i - k) = (if length p1 = k then a \<otimes> coeff p2 (i - k) else \<zero>)"
using in_carrier by auto
hence "(\ k \ {..i}. ?f2 k \ ?f3 (i - k)) =
(\<Oplus> k \<in> {..i}. (if length p1 = k then a \<otimes> coeff p2 (i - k) else \<zero>))"
using in_carrier
add.finprod_cong'[of "{..i}" "{..i}" "\k. ?f2 k \ ?f3 (i - k)"
"\k. (if length p1 = k then a \ coeff p2 (i - k) else \)"]
by fastforce
also have " ... = a \ (coeff p2) (i - length p1)" if "i \ length p1"
using add.finprod_singleton[of "length p1" "{..i}" "\j. a \ (coeff p2) (i - j)"]
in_carrier that by auto
finally
have "(\ k \ {..i}. ?f2 k \ ?f3 (i - k)) = a \ (coeff p2) (i - length p1)" if "i \ length p1"
using that by simp
hence eq_ge: "?f1 i = (\i. (\ k \ {..i}. ?f2 k \ ?f3 (i - k))) i" if "i \ length p1"
using that by auto
from eq_lt eq_ge show "?f1 i = (\i. (\ k \ {..i}. ?f2 k \ ?f3 (i - k))) i" by auto
qed
finally have coeff_a_p2:
"coeff ?a_p2 = (\i. \ k \ {..i}. (if k = length p1 then a else \) \ (coeff p2) (i - k))" .
have "set ?a_p2 \ carrier R"
using in_carrier(1) assms(2) by auto
moreover have "set (poly_mult p1 p2) \ carrier R"
using poly_mult_in_carrier[OF _ assms(2)] Cons(2) by simp
ultimately
have "coeff (poly_mult (a # p1) p2) = (\i. ((coeff ?a_p2) i) \ ((coeff (poly_mult p1 p2)) i))"
using poly_add_coeff[of ?a_p2 "poly_mult p1 p2"] by simp
also have " ... = (\i. (\ k \ {..i}. (if k = length p1 then a else \) \ (coeff p2) (i - k)) \
(\<Oplus> k \<in> {..i}. (coeff p1) k \<otimes> (coeff p2) (i - k)))"
using Cons coeff_a_p2 by simp
also have " ... = (\i. (\ k \ {..i}. ((if k = length p1 then a else \) \ (coeff p2) (i - k)) \
((coeff p1) k \<otimes> (coeff p2) (i - k))))"
using add.finprod_multf in_carrier by auto
also have " ... = (\i. (\ k \ {..i}. (coeff (a # p1) k) \ (coeff p2) (i - k)))"
(is "(\i. (\ k \ {..i}. ?f i k)) = (\i. (\ k \ {..i}. ?g i k))")
proof
fix i
have "\k. ?f i k = ?g i k"
using in_carrier coeff_length[of p1] by auto
thus "(\ k \ {..i}. ?f i k) = (\ k \ {..i}. ?g i k)" by simp
qed
finally show ?case .
qed
lemma poly_mult_zero:
assumes "set p \ carrier R"
shows "poly_mult [] p = []" and "poly_mult p [] = []"
proof (simp)
have "coeff (poly_mult p []) = (\_. \)"
using poly_mult_coeff[OF assms, of "[]"] coeff_in_carrier[OF assms] by auto
thus "poly_mult p [] = []"
using coeff_iff_polynomial_cond[OF
poly_mult_is_polynomial[OF carrier_is_subring assms] zero_is_polynomial] by simp
qed
lemma poly_mult_l_distr':
assumes "set p1 \ carrier R" "set p2 \ carrier R" "set p3 \ carrier R"
shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)"
proof -
let ?c1 = "coeff p1" and ?c2 = "coeff p2" and ?c3 = "coeff p3"
have in_carrier:
"\i. ?c1 i \ carrier R" "\i. ?c2 i \ carrier R" "\i. ?c3 i \ carrier R"
using assms coeff_in_carrier by auto
have "coeff (poly_mult (poly_add p1 p2) p3) = (\n. \i \ {..n}. (?c1 i \ ?c2 i) \ ?c3 (n - i))"
using poly_mult_coeff[of "poly_add p1 p2" p3] poly_add_coeff[OF assms(1-2)]
poly_add_in_carrier[OF assms(1-2)] assms by auto
also have " ... = (\n. \i \ {..n}. (?c1 i \ ?c3 (n - i)) \ (?c2 i \ ?c3 (n - i)))"
using in_carrier l_distr by auto
also
have " ... = (\n. (\i \ {..n}. (?c1 i \ ?c3 (n - i))) \ (\i \ {..n}. (?c2 i \ ?c3 (n - i))))"
using add.finprod_multf in_carrier by auto
also have " ... = coeff (poly_add (poly_mult p1 p3) (poly_mult p2 p3))"
using poly_mult_coeff[OF assms(1) assms(3)] poly_mult_coeff[OF assms(2-3)]
poly_add_coeff[OF poly_mult_in_carrier[OF assms(1) assms(3)]]
poly_mult_in_carrier[OF assms(2-3)] by simp
finally have "coeff (poly_mult (poly_add p1 p2) p3) =
coeff (poly_add (poly_mult p1 p3) (poly_mult p2 p3))" .
moreover have "polynomial (carrier R) (poly_mult (poly_add p1 p2) p3)"
and "polynomial (carrier R) (poly_add (poly_mult p1 p3) (poly_mult p2 p3))"
using assms poly_add_is_polynomial poly_mult_is_polynomial polynomial_in_carrier
carrier_is_subring by auto
ultimately show ?thesis
using coeff_iff_polynomial_cond by auto
qed
lemma poly_mult_l_distr:
assumes "subring K R" "polynomial K p1" "polynomial K p2" "polynomial K p3"
shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)"
using poly_mult_l_distr' polynomial_in_carrier assms by auto
lemma poly_mult_prepend_replicate_zero:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_mult p1 p2 = poly_mult ((replicate n \) @ p1) p2"
proof -
{ fix p1 p2 assume A: "set p1 \ carrier R" "set p2 \ carrier R"
hence "poly_mult p1 p2 = poly_mult (\ # p1) p2"
proof -
let ?a_p2 = "(map ((\) \) p2) @ (replicate (length p1) \)"
have "?a_p2 = replicate (length p2 + length p1) \"
using A(2) by (induction p2) (auto)
hence "poly_mult (\ # p1) p2 = poly_add (replicate (length p2 + length p1) \) (poly_mult p1 p2)"
by simp
also have " ... = poly_add (normalize (replicate (length p2 + length p1) \)) (poly_mult p1 p2)"
using poly_add_normalize(1)[of "replicate (length p2 + length p1) \" "poly_mult p1 p2"]
poly_mult_in_carrier[OF A] by force
also have " ... = poly_mult p1 p2"
using poly_add_zero(2)[OF _ poly_mult_is_polynomial[OF _ A]] carrier_is_subring
normalize_replicate_zero[of "length p2 + length p1" "[]"] by simp
finally show ?thesis by auto
qed } note aux_lemma = this
from assms show ?thesis
proof (induction n)
case 0 thus ?case by simp
next
case (Suc n) thus ?case
using aux_lemma[of "replicate n \ @ p1" p2] by force
qed
qed
lemma poly_mult_normalize:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_mult p1 p2 = poly_mult (normalize p1) p2"
proof -
let ?replicate = "replicate (length p1 - length (normalize p1)) \"
have "poly_mult p1 p2 = poly_mult (?replicate @ (normalize p1)) p2"
using normalize_def'[of p1] by simp
thus ?thesis
using poly_mult_prepend_replicate_zero normalize_in_carrier assms by auto
qed
lemma poly_mult_append_zero:
assumes "set p \ carrier R" "set q \ carrier R"
shows "poly_mult (p @ [ \ ]) q = normalize ((poly_mult p q) @ [ \ ])"
using assms(1)
proof (induct p)
case Nil thus ?case
using poly_mult_normalize[OF _ assms(2), of "[] @ [ \ ]"]
poly_mult_zero(1) poly_mult_zero(1)[of "q @ [ \ ]"] assms(2) by auto
next
case (Cons a p)
let ?q_a = "\n. (map ((\) a) q) @ (replicate n \)"
have set_q_a: "\n. set (?q_a n) \ carrier R"
using Cons(2) assms(2) by (induct q) (auto)
have set_poly_mult: "set ((poly_mult p q) @ [ \ ]) \ carrier R"
using poly_mult_in_carrier[OF _ assms(2)] Cons(2) by auto
have "poly_mult ((a # p) @ [\]) q = poly_add (?q_a (Suc (length p))) (poly_mult (p @ [\]) q)"
by auto
also have " ... = poly_add (?q_a (Suc (length p))) (normalize ((poly_mult p q) @ [ \ ]))"
using Cons by simp
also have " ... = poly_add ((?q_a (length p)) @ [ \ ]) ((poly_mult p q) @ [ \ ])"
using poly_add_normalize(2)[OF set_q_a[of "Suc (length p)"] set_poly_mult]
by (simp add: replicate_append_same)
also have " ... = normalize ((poly_add (?q_a (length p)) (poly_mult p q)) @ [ \ ])"
using poly_add_append_zero[OF set_q_a[of "length p"] poly_mult_in_carrier[OF _ assms(2)]] Cons(2) by auto
also have " ... = normalize ((poly_mult (a # p) q) @ [ \ ])"
by auto
finally show ?case .
qed
end (* of ring context. *)
subsection \<open>Properties Within a Domain\<close>
context domain
begin
lemma one_is_polynomial [intro]: "subring K R \ polynomial K [ \ ]"
unfolding polynomial_def using subringE(3) by auto
lemma poly_mult_comm:
assumes "set p1 \ carrier R" "set p2 \ carrier R"
shows "poly_mult p1 p2 = poly_mult p2 p1"
proof -
let ?c1 = "coeff p1" and ?c2 = "coeff p2"
have "\i. (\k \ {..i}. ?c1 k \ ?c2 (i - k)) = (\k \ {..i}. ?c2 k \ ?c1 (i - k))"
proof -
fix i :: nat
let ?f = "\k. ?c1 k \ ?c2 (i - k)"
have in_carrier: "\i. ?c1 i \ carrier R" "\i. ?c2 i \ carrier R"
using coeff_in_carrier[OF assms(1)] coeff_in_carrier[OF assms(2)] by auto
have reindex_inj: "inj_on (\k. i - k) {..i}"
using inj_on_def by force
moreover have "(\k. i - k) ` {..i} \ {..i}" by auto
hence "(\k. i - k) ` {..i} = {..i}"
using reindex_inj endo_inj_surj[of "{..i}" "\k. i - k"] by simp
ultimately have "(\k \ {..i}. ?f k) = (\k \ {..i}. ?f (i - k))"
using add.finprod_reindex[of ?f "\k. i - k" "{..i}"] in_carrier by auto
moreover have "\k. k \ {..i} \ ?f (i - k) = ?c2 k \ ?c1 (i - k)"
using in_carrier m_comm by auto
hence "(\k \ {..i}. ?f (i - k)) = (\k \ {..i}. ?c2 k \ ?c1 (i - k))"
using add.finprod_cong'[of "{..i}" "{..i}"] in_carrier by auto
ultimately show "(\k \ {..i}. ?f k) = (\k \ {..i}. ?c2 k \ ?c1 (i - k))"
by simp
qed
hence "coeff (poly_mult p1 p2) = coeff (poly_mult p2 p1)"
using poly_mult_coeff[OF assms] poly_mult_coeff[OF assms(2,1)] by simp
thus ?thesis
using coeff_iff_polynomial_cond[OF poly_mult_is_polynomial[OF _ assms]
poly_mult_is_polynomial[OF _ assms(2,1)]]
carrier_is_subring by simp
qed
lemma poly_mult_r_distr':
assumes "set p1 \ carrier R" "set p2 \ carrier R" "set p3 \ carrier R"
shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)"
unfolding poly_mult_comm[OF assms(1) poly_add_in_carrier[OF assms(2-3)]]
poly_mult_l_distr'[OF assms(2-3,1)] assms(2-3)[THEN poly_mult_comm[OF _ assms(1)]] ..
lemma poly_mult_r_distr:
assumes "subring K R" "polynomial K p1" "polynomial K p2" "polynomial K p3"
shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)"
using poly_mult_r_distr' polynomial_in_carrier assms by auto
lemma poly_mult_replicate_zero:
assumes "set p \ carrier R"
shows "poly_mult (replicate n \) p = []"
and "poly_mult p (replicate n \) = []"
proof -
have in_carrier: "\n. set (replicate n \) \ carrier R" by auto
show "poly_mult (replicate n \) p = []" using assms
proof (induction n)
case 0 thus ?case by simp
next
case (Suc n)
hence "poly_mult (replicate (Suc n) \) p = poly_mult (\ # (replicate n \)) p"
by simp
also have " ... = poly_add ((map (\a. \ \ a) p) @ (replicate n \)) []"
using Suc by simp
also have " ... = poly_add ((map (\a. \) p) @ (replicate n \)) []"
proof -
have "map ((\) \) p = map (\a. \) p"
using Suc.prems by auto
then show ?thesis
by presburger
qed
also have " ... = poly_add (replicate (length p + n) \) []"
by (simp add: map_replicate_const replicate_add)
also have " ... = poly_add [] []"
using poly_add_normalize(1)[of "replicate (length p + n) \" "[]"]
normalize_replicate_zero[of "length p + n" "[]"] by auto
also have " ... = []" by simp
finally show ?case .
qed
thus "poly_mult p (replicate n \) = []"
using poly_mult_comm[OF assms in_carrier] by simp
qed
lemma poly_mult_const':
assumes "set p \ carrier R" "a \ carrier R"
shows "poly_mult [ a ] p = normalize (map (\b. a \ b) p)"
and "poly_mult p [ a ] = normalize (map (\b. a \ b) p)"
proof -
have "map2 (\) (map ((\) a) p) (replicate (length p) \) = map ((\) a) p"
using assms by (induction p) (auto)
thus "poly_mult [ a ] p = normalize (map (\b. a \ b) p)" by simp
thus "poly_mult p [ a ] = normalize (map (\b. a \ b) p)"
using poly_mult_comm[OF assms(1), of "[ a ]"] assms(2) by auto
qed
lemma poly_mult_const:
assumes "subring K R" "polynomial K p" "a \ K - { \ }"
shows "poly_mult [ a ] p = map (\b. a \ b) p"
and "poly_mult p [ a ] = map (\b. a \ b) p"
proof -
have in_carrier: "set p \ carrier R" "a \ carrier R"
using polynomial_in_carrier[OF assms(1-2)] assms(3) subringE(1)[OF assms(1)] by auto
show "poly_mult [ a ] p = map (\b. a \ b) p"
proof (cases p)
case Nil thus ?thesis
using poly_mult_const'(1) in_carrier by auto
next
case (Cons b q)
have "lead_coeff (map (\b. a \ b) p) \ \"
using assms subringE(1)[OF assms(1)] integral[of a b] Cons lead_coeff_in_carrier by auto
hence "normalize (map (\b. a \ b) p) = (map (\b. a \ b) p)"
unfolding Cons by simp
thus ?thesis
using poly_mult_const'(1) in_carrier by auto
qed
thus "poly_mult p [ a ] = map (\b. a \ b) p"
using poly_mult_comm[OF in_carrier(1)] in_carrier(2) by auto
qed
lemma poly_mult_semiassoc:
assumes "set p \ carrier R" "set q \ carrier R" and "a \ carrier R"
shows "poly_mult (poly_mult [ a ] p) q = poly_mult [ a ] (poly_mult p q)"
proof -
let ?cp = "coeff p" and ?cq = "coeff q"
have "coeff (poly_mult [ a ] p) = (\i. (a \ ?cp i))"
using poly_mult_const'(1)[OF assms(1,3)] normalize_coeff scalar_coeff[OF assms(3)] by simp
hence "coeff (poly_mult (poly_mult [ a ] p) q) = (\i. (\j \ {..i}. (a \ ?cp j) \ ?cq (i - j)))"
using poly_mult_coeff[OF poly_mult_in_carrier[OF _ assms(1)] assms(2), of "[ a ]"] assms(3) by auto
also have " ... = (\i. a \ (\j \ {..i}. ?cp j \ ?cq (i - j)))"
proof
fix i show "(\j \ {..i}. (a \ ?cp j) \ ?cq (i - j)) = a \ (\j \ {..i}. ?cp j \ ?cq (i - j))"
using finsum_rdistr[OF _ assms(3), of _ "\j. ?cp j \ ?cq (i - j)"]
assms(1-2)[THEN coeff_in_carrier] by (simp add: assms(3) m_assoc)
qed
also have " ... = coeff (poly_mult [ a ] (poly_mult p q))"
unfolding poly_mult_const'(1)[OF poly_mult_in_carrier[OF assms(1-2)] assms(3)]
using scalar_coeff[OF assms(3), of "poly_mult p q"]
poly_mult_coeff[OF assms(1-2)] normalize_coeff by simp
finally have "coeff (poly_mult (poly_mult [ a ] p) q) = coeff (poly_mult [ a ] (poly_mult p q))" .
moreover have "polynomial (carrier R) (poly_mult (poly_mult [ a ] p) q)"
and "polynomial (carrier R) (poly_mult [ a ] (poly_mult p q))"
using poly_mult_is_polynomial[OF _ poly_mult_in_carrier[OF _ assms(1)] assms(2)]
poly_mult_is_polynomial[OF _ _ poly_mult_in_carrier[OF assms(1-2)]]
carrier_is_subring assms(3) by (auto simp del: poly_mult.simps)
ultimately show ?thesis
using coeff_iff_polynomial_cond by simp
qed
text \<open>Note that "polynomial (carrier R) p" and "subring K p; polynomial K p" are "equivalent"
assumptions for any lemma in ring which the result doesn't depend on K, because carrier
is a subring and a polynomial for a subset of the carrier is a carrier polynomial. The
decision between one of them should be based on how the lemma is going to be used and
proved. These are some tips:
(a) Lemmas about the algebraic structure of polynomials should use the latter option.
(b) Also, if the lemma deals with lots of polynomials, then the latter option is preferred.
(c) If the proof is going to be much easier with the first option, do not hesitate. \<close>
lemma poly_mult_monom':
assumes "set p \ carrier R" "a \ carrier R"
shows "poly_mult (monom a n) p = normalize ((map ((\) a) p) @ (replicate n \))"
proof -
have set_map: "set ((map ((\) a) p) @ (replicate n \)) \ carrier R"
using assms by (induct p) (auto)
show ?thesis
using poly_mult_replicate_zero(1)[OF assms(1), of n]
poly_add_zero'(1)[OF set_map]
unfolding monom_def by simp
qed
lemma poly_mult_monom:
assumes "polynomial (carrier R) p" "a \ carrier R - { \ }"
shows "poly_mult (monom a n) p =
(if p = [] then [] else (poly_mult [ a ] p) @ (replicate n \<zero>))"
proof (cases p)
case Nil thus ?thesis
using poly_mult_zero(2)[of "monom a n"] assms(2) monom_def by fastforce
next
case (Cons b ps)
hence "lead_coeff ((map (\b. a \ b) p) @ (replicate n \)) \ \"
using Cons assms integral[of a b] unfolding polynomial_def by auto
thus ?thesis
using poly_mult_monom'[OF polynomial_incl[OF assms(1)], of a n] assms(2) Cons
unfolding poly_mult_const(1)[OF carrier_is_subring assms] by simp
qed
lemma poly_mult_one':
assumes "set p \ carrier R"
shows "poly_mult [ \ ] p = normalize p" and "poly_mult p [ \ ] = normalize p"
proof -
have "map2 (\) (map ((\) \) p) (replicate (length p) \) = p"
using assms by (induct p) (auto)
thus "poly_mult [ \ ] p = normalize p" and "poly_mult p [ \ ] = normalize p"
using poly_mult_comm[OF assms, of "[ \ ]"] by auto
qed
lemma poly_mult_one:
assumes "subring K R" "polynomial K p"
shows "poly_mult [ \ ] p = p" and "poly_mult p [ \ ] = p"
using poly_mult_one'[OF polynomial_in_carrier[OF assms]] normalize_polynomial[OF assms(2)] by auto
lemma poly_mult_lead_coeff_aux:
assumes "subring K R" "polynomial K p1" "polynomial K p2" and "p1 \ []" and "p2 \ []"
shows "(coeff (poly_mult p1 p2)) (degree p1 + degree p2) = (lead_coeff p1) \ (lead_coeff p2)"
proof -
have p1: "lead_coeff p1 \ carrier R - { \ }" and p2: "lead_coeff p2 \ carrier R - { \ }"
using assms(2-5) lead_coeff_in_carrier[OF assms(1)] by (metis list.collapse)+
have "(coeff (poly_mult p1 p2)) (degree p1 + degree p2) =
(\<Oplus> k \<in> {..((degree p1) + (degree p2))}.
(coeff p1) k \<otimes> (coeff p2) ((degree p1) + (degree p2) - k))"
using poly_mult_coeff[OF assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]] by simp
also have " ... = (lead_coeff p1) \ (lead_coeff p2)"
proof -
let ?f = "\i. (coeff p1) i \ (coeff p2) ((degree p1) + (degree p2) - i)"
have in_carrier: "\i. (coeff p1) i \ carrier R" "\i. (coeff p2) i \ carrier R"
using coeff_in_carrier assms by auto
have "\i. i < degree p1 \ ?f i = \"
using coeff_degree[of p2] in_carrier by auto
moreover have "\i. i > degree p1 \ ?f i = \"
using coeff_degree[of p1] in_carrier by auto
moreover have "?f (degree p1) = (lead_coeff p1) \ (lead_coeff p2)"
using assms(4-5) lead_coeff_simp by simp
ultimately have "?f = (\i. if degree p1 = i then (lead_coeff p1) \ (lead_coeff p2) else \)"
using nat_neq_iff by auto
thus ?thesis
using add.finprod_singleton[of "degree p1" "{..((degree p1) + (degree p2))}"
"\i. (lead_coeff p1) \ (lead_coeff p2)"] p1 p2 by auto
qed
finally show ?thesis .
qed
lemma poly_mult_degree_eq:
assumes "subring K R" "polynomial K p1" "polynomial K p2"
shows "degree (poly_mult p1 p2) = (if p1 = [] \ p2 = [] then 0 else (degree p1) + (degree p2))"
proof (cases p1)
case Nil thus ?thesis by simp
next
case (Cons a p1') note p1 = Cons
show ?thesis
proof (cases p2)
case Nil thus ?thesis
using poly_mult_zero(2)[OF polynomial_in_carrier[OF assms(1-2)]] by simp
next
case (Cons b p2') note p2 = Cons
have a: "a \ carrier R" and b: "b \ carrier R"
using p1 p2 polynomial_in_carrier[OF assms(1-2)] polynomial_in_carrier[OF assms(1,3)] by auto
have "(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) = a \ b"
using poly_mult_lead_coeff_aux[OF assms] p1 p2 by simp
hence neq0: "(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) \ \"
using assms(2-3) integral[of a b] lead_coeff_in_carrier[OF assms(1)] p1 p2 by auto
moreover have eq0: "\i. i > (degree p1) + (degree p2) \ (coeff (poly_mult p1 p2)) i = \"
proof -
have aux_lemma: "degree (poly_mult p1 p2) \ (degree p1) + (degree p2)"
proof (induct p1)
case Nil
then show ?case by simp
next
case (Cons a p1)
let ?a_p2 = "(map (\b. a \ b) p2) @ (replicate (degree (a # p1)) \)"
have "poly_mult (a # p1) p2 = poly_add ?a_p2 (poly_mult p1 p2)" by simp
hence "degree (poly_mult (a # p1) p2) \ max (degree ?a_p2) (degree (poly_mult p1 p2))"
using poly_add_degree[of ?a_p2 "poly_mult p1 p2"] by simp
also have " ... \ max ((degree (a # p1)) + (degree p2)) (degree (poly_mult p1 p2))"
by auto
also have " ... \ max ((degree (a # p1)) + (degree p2)) ((degree p1) + (degree p2))"
using Cons by simp
also have " ... \ (degree (a # p1)) + (degree p2)"
by auto
finally show ?case .
qed
fix i show "i > (degree p1) + (degree p2) \ (coeff (poly_mult p1 p2)) i = \"
using coeff_degree aux_lemma by simp
qed
moreover have "polynomial K (poly_mult p1 p2)"
by (simp add: assms poly_mult_closed)
ultimately have "degree (poly_mult p1 p2) = degree p1 + degree p2"
by (metis (no_types) assms(1) coeff.simps(1) coeff_degree domain.poly_mult_one(1) domain_axioms eq0 lead_coeff_simp length_greater_0_conv neq0 normalize_length_lt not_less_iff_gr_or_eq poly_mult_one'(1) polynomial_in_carrier)
thus ?thesis
using p1 p2 by auto
qed
qed
lemma poly_mult_integral:
assumes "subring K R" "polynomial K p1" "polynomial K p2"
shows "poly_mult p1 p2 = [] \ p1 = [] \ p2 = []"
proof (rule ccontr)
assume A: "poly_mult p1 p2 = []" "\ (p1 = [] \ p2 = [])"
hence "degree (poly_mult p1 p2) = degree p1 + degree p2"
using poly_mult_degree_eq[OF assms] by simp
hence "length p1 = 1 \ length p2 = 1"
using A Suc_diff_Suc by fastforce
then obtain a b where p1: "p1 = [ a ]" and p2: "p2 = [ b ]"
by (metis One_nat_def length_0_conv length_Suc_conv)
hence "a \ carrier R - { \ }" and "b \ carrier R - { \ }"
using assms lead_coeff_in_carrier by auto
hence "poly_mult [ a ] [ b ] = [ a \ b ]"
using integral by auto
thus False using A(1) p1 p2 by simp
qed
lemma poly_mult_lead_coeff:
assumes "subring K R" "polynomial K p1" "polynomial K p2" and "p1 \ []" and "p2 \ []"
shows "lead_coeff (poly_mult p1 p2) = (lead_coeff p1) \ (lead_coeff p2)"
proof -
have "poly_mult p1 p2 \ []"
using poly_mult_integral[OF assms(1-3)] assms(4-5) by auto
hence "lead_coeff (poly_mult p1 p2) = (coeff (poly_mult p1 p2)) (degree p1 + degree p2)"
using poly_mult_degree_eq[OF assms(1-3)] assms(4-5) by (metis coeff.simps(2) list.collapse)
thus ?thesis
using poly_mult_lead_coeff_aux[OF assms] by simp
qed
lemma poly_mult_append_zero_lcancel:
assumes "subring K R" and "polynomial K p" "polynomial K q"
shows "poly_mult (p @ [ \ ]) q = r @ [ \ ] \ poly_mult p q = r"
proof -
note in_carrier = assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]
assume pmult: "poly_mult (p @ [ \ ]) q = r @ [ \ ]"
have "poly_mult (p @ [ \ ]) q = []" if "q = []"
using poly_mult_zero(2)[of "p @ [ \ ]"] that in_carrier(1) by auto
moreover have "poly_mult (p @ [ \ ]) q = []" if "p = []"
using poly_mult_normalize[OF _ in_carrier(2), of "p @ [ \ ]"] poly_mult_zero[OF in_carrier(2)]
unfolding that by auto
ultimately have "p \ []" and "q \ []"
using pmult by auto
hence "poly_mult p q \ []"
using poly_mult_integral[OF assms] by auto
hence "normalize ((poly_mult p q) @ [ \ ]) = (poly_mult p q) @ [ \ ]"
using normalize_polynomial[OF append_is_polynomial[OF assms(1) poly_mult_closed[OF assms], of "Suc 0"]] by auto
thus "poly_mult p q = r"
using poly_mult_append_zero[OF assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]] pmult by simp
qed
lemma poly_mult_append_zero_rcancel:
assumes "subring K R" and "polynomial K p" "polynomial K q"
shows "poly_mult p (q @ [ \ ]) = r @ [ \ ] \ poly_mult p q = r"
using poly_mult_append_zero_lcancel[OF assms(1,3,2)]
poly_mult_comm[of p "q @ [ \ ]"] poly_mult_comm[of p q]
assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]
by auto
end (* of domain context. *)
subsection \<open>Algebraic Structure of Polynomials\<close>
definition univ_poly :: "('a, 'b) ring_scheme \'a set \ ('a list) ring" ("_ [X]\" 80)
where "univ_poly R K =
\<lparr> carrier = { p. polynomial\<^bsub>R\<^esub> K p },
mult = ring.poly_mult R,
one = [ \<one>\<^bsub>R\<^esub> ],
zero = [],
add = ring.poly_add R \<rparr>"
text \<open>These lemmas allow you to unfold one field of the record at a time. \<close>
lemma univ_poly_carrier: "polynomial\<^bsub>R\<^esub> K p \ p \ carrier (K[X]\<^bsub>R\<^esub>)"
unfolding univ_poly_def by simp
lemma univ_poly_mult: "mult (K[X]\<^bsub>R\<^esub>) = ring.poly_mult R"
unfolding univ_poly_def by simp
lemma univ_poly_one: "one (K[X]\<^bsub>R\<^esub>) = [ \\<^bsub>R\<^esub> ]"
unfolding univ_poly_def by simp
lemma univ_poly_zero: "zero (K[X]\<^bsub>R\<^esub>) = []"
unfolding univ_poly_def by simp
lemma univ_poly_add: "add (K[X]\<^bsub>R\<^esub>) = ring.poly_add R"
unfolding univ_poly_def by simp
(* NEW ========== *)
lemma univ_poly_zero_closed [intro]: "[] \ carrier (K[X]\<^bsub>R\<^esub>)"
unfolding sym[OF univ_poly_carrier] polynomial_def by simp
context domain
begin
lemma poly_mult_monom_assoc:
assumes "set p \ carrier R" "set q \ carrier R" and "a \ carrier R"
shows "poly_mult (poly_mult (monom a n) p) q =
poly_mult (monom a n) (poly_mult p q)"
proof (induct n)
case 0 thus ?case
unfolding monom_def using poly_mult_semiassoc[OF assms] by (auto simp del: poly_mult.simps)
next
case (Suc n)
have "poly_mult (poly_mult (monom a (Suc n)) p) q =
poly_mult (normalize ((poly_mult (monom a n) p) @ [ \<zero> ])) q"
using poly_mult_append_zero[OF monom_in_carrier[OF assms(3), of n] assms(1)]
unfolding monom_def by (auto simp del: poly_mult.simps simp add: replicate_append_same)
also have " ... = normalize ((poly_mult (poly_mult (monom a n) p) q) @ [ \ ])"
using poly_mult_normalize[OF _ assms(2)] poly_mult_append_zero[OF _ assms(2)]
poly_mult_in_carrier[OF monom_in_carrier[OF assms(3), of n] assms(1)] by auto
also have " ... = normalize ((poly_mult (monom a n) (poly_mult p q)) @ [ \ ])"
using Suc by simp
also have " ... = poly_mult (monom a (Suc n)) (poly_mult p q)"
using poly_mult_append_zero[OF monom_in_carrier[OF assms(3), of n]
poly_mult_in_carrier[OF assms(1-2)]]
unfolding monom_def by (simp add: replicate_append_same)
finally show ?case .
qed
context
fixes K :: "'a set" assumes K: "subring K R"
begin
lemma univ_poly_is_monoid: "monoid (K[X])"
unfolding univ_poly_def using poly_mult_one[OF K]
proof (auto simp add: K poly_add_closed poly_mult_closed one_is_polynomial monoid_def)
fix p1 p2 p3
let ?P = "poly_mult (poly_mult p1 p2) p3 = poly_mult p1 (poly_mult p2 p3)"
assume A: "polynomial K p1" "polynomial K p2" "polynomial K p3"
show ?P using polynomial_in_carrier[OF K A(1)]
proof (induction p1)
case Nil thus ?case by simp
next
next
case (Cons a p1) thus ?case
proof (cases "a = \")
assume eq_zero: "a = \"
have p1: "set p1 \ carrier R"
using Cons(2) by simp
have "poly_mult (poly_mult (a # p1) p2) p3 = poly_mult (poly_mult p1 p2) p3"
using poly_mult_prepend_replicate_zero[OF p1 polynomial_in_carrier[OF K A(2)], of "Suc 0"]
eq_zero by simp
also have " ... = poly_mult p1 (poly_mult p2 p3)"
using p1[THEN Cons(1)] by simp
also have " ... = poly_mult (a # p1) (poly_mult p2 p3)"
using poly_mult_prepend_replicate_zero[OF p1
poly_mult_in_carrier[OF A(2-3)[THEN polynomial_in_carrier[OF K]]], of "Suc 0"] eq_zero
by simp
finally show ?thesis .
next
assume "a \ \" hence in_carrier:
--> --------------------
--> maximum size reached
--> --------------------
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