1 (* Title: HOL/Library/Polynomial.thy
3 Author: Clemens Ballarin
4 Author: Florian Haftmann
7 header {* Polynomials as type over a ring structure *}
13 subsection {* Auxiliary: operations for lists (later) representing coefficients *}
15 definition strip_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
17 "strip_while P = rev \<circ> dropWhile P \<circ> rev"
19 lemma strip_while_Nil [simp]:
20 "strip_while P [] = []"
21 by (simp add: strip_while_def)
23 lemma strip_while_append [simp]:
24 "\<not> P x \<Longrightarrow> strip_while P (xs @ [x]) = xs @ [x]"
25 by (simp add: strip_while_def)
27 lemma strip_while_append_rec [simp]:
28 "P x \<Longrightarrow> strip_while P (xs @ [x]) = strip_while P xs"
29 by (simp add: strip_while_def)
31 lemma strip_while_Cons [simp]:
32 "\<not> P x \<Longrightarrow> strip_while P (x # xs) = x # strip_while P xs"
33 by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
35 lemma strip_while_eq_Nil [simp]:
36 "strip_while P xs = [] \<longleftrightarrow> (\<forall>x\<in>set xs. P x)"
37 by (simp add: strip_while_def)
39 lemma strip_while_eq_Cons_rec:
40 "strip_while P (x # xs) = x # strip_while P xs \<longleftrightarrow> \<not> (P x \<and> (\<forall>x\<in>set xs. P x))"
41 by (induct xs rule: rev_induct) (simp_all add: strip_while_def)
43 lemma strip_while_not_last [simp]:
44 "\<not> P (last xs) \<Longrightarrow> strip_while P xs = xs"
45 by (cases xs rule: rev_cases) simp_all
47 lemma split_strip_while_append:
49 obtains ys zs :: "'a list"
50 where "strip_while P xs = ys" and "\<forall>x\<in>set zs. P x" and "xs = ys @ zs"
52 show "strip_while P xs = strip_while P xs" ..
53 show "\<forall>x\<in>set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
54 have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
55 by (simp add: strip_while_def)
56 then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
57 by (simp only: rev_is_rev_conv)
61 definition nth_default :: "'a \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a"
63 "nth_default x xs n = (if n < length xs then xs ! n else x)"
65 lemma nth_default_Nil [simp]:
66 "nth_default y [] n = y"
67 by (simp add: nth_default_def)
69 lemma nth_default_Cons_0 [simp]:
70 "nth_default y (x # xs) 0 = x"
71 by (simp add: nth_default_def)
73 lemma nth_default_Cons_Suc [simp]:
74 "nth_default y (x # xs) (Suc n) = nth_default y xs n"
75 by (simp add: nth_default_def)
77 lemma nth_default_map_eq:
78 "f y = x \<Longrightarrow> nth_default x (map f xs) n = f (nth_default y xs n)"
79 by (simp add: nth_default_def)
81 lemma nth_default_strip_while_eq [simp]:
82 "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
84 from split_strip_while_append obtain ys zs
85 where "strip_while (HOL.eq x) xs = ys" and "\<forall>z\<in>set zs. x = z" and "xs = ys @ zs" by blast
86 then show ?thesis by (simp add: nth_default_def not_less nth_append)
90 definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "##" 65)
92 "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
94 lemma cCons_0_Nil_eq [simp]:
96 by (simp add: cCons_def)
98 lemma cCons_Cons_eq [simp]:
99 "x ## y # ys = x # y # ys"
100 by (simp add: cCons_def)
102 lemma cCons_append_Cons_eq [simp]:
103 "x ## xs @ y # ys = x # xs @ y # ys"
104 by (simp add: cCons_def)
106 lemma cCons_not_0_eq [simp]:
107 "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
108 by (simp add: cCons_def)
110 lemma strip_while_not_0_Cons_eq [simp]:
111 "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
112 proof (cases "x = 0")
113 case False then show ?thesis by simp
115 case True show ?thesis
116 proof (induct xs rule: rev_induct)
117 case Nil with True show ?case by simp
119 case (snoc y ys) then show ?case
120 by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
124 lemma tl_cCons [simp]:
126 by (simp add: cCons_def)
129 subsection {* Almost everywhere zero functions *}
131 definition almost_everywhere_zero :: "(nat \<Rightarrow> 'a::zero) \<Rightarrow> bool"
133 "almost_everywhere_zero f \<longleftrightarrow> (\<exists>n. \<forall>i>n. f i = 0)"
135 lemma almost_everywhere_zeroI:
136 "(\<And>i. i > n \<Longrightarrow> f i = 0) \<Longrightarrow> almost_everywhere_zero f"
137 by (auto simp add: almost_everywhere_zero_def)
139 lemma almost_everywhere_zeroE:
140 assumes "almost_everywhere_zero f"
141 obtains n where "\<And>i. i > n \<Longrightarrow> f i = 0"
143 from assms have "\<exists>n. \<forall>i>n. f i = 0" by (simp add: almost_everywhere_zero_def)
144 then obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by blast
145 with that show thesis .
148 lemma almost_everywhere_zero_case_nat:
149 assumes "almost_everywhere_zero f"
150 shows "almost_everywhere_zero (case_nat a f)"
152 by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
155 lemma almost_everywhere_zero_Suc:
156 assumes "almost_everywhere_zero f"
157 shows "almost_everywhere_zero (\<lambda>n. f (Suc n))"
159 from assms obtain n where "\<And>i. i > n \<Longrightarrow> f i = 0" by (erule almost_everywhere_zeroE)
160 then have "\<And>i. i > n \<Longrightarrow> f (Suc i) = 0" by auto
161 then show ?thesis by (rule almost_everywhere_zeroI)
165 subsection {* Definition of type @{text poly} *}
167 typedef 'a poly = "{f :: nat \<Rightarrow> 'a::zero. almost_everywhere_zero f}"
168 morphisms coeff Abs_poly
169 unfolding almost_everywhere_zero_def by auto
171 setup_lifting (no_code) type_definition_poly
173 lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
174 by (simp add: coeff_inject [symmetric] fun_eq_iff)
176 lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
177 by (simp add: poly_eq_iff)
179 lemma coeff_almost_everywhere_zero:
180 "almost_everywhere_zero (coeff p)"
181 using coeff [of p] by simp
184 subsection {* Degree of a polynomial *}
186 definition degree :: "'a::zero poly \<Rightarrow> nat"
188 "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
191 assumes "degree p < n"
192 shows "coeff p n = 0"
194 from coeff_almost_everywhere_zero
195 have "\<exists>n. \<forall>i>n. coeff p i = 0" by (blast intro: almost_everywhere_zeroE)
196 then have "\<forall>i>degree p. coeff p i = 0"
197 unfolding degree_def by (rule LeastI_ex)
198 with assms show ?thesis by simp
201 lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
202 by (erule contrapos_np, rule coeff_eq_0, simp)
204 lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
205 unfolding degree_def by (erule Least_le)
207 lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
208 unfolding degree_def by (drule not_less_Least, simp)
211 subsection {* The zero polynomial *}
213 instantiation poly :: (zero) zero
216 lift_definition zero_poly :: "'a poly"
217 is "\<lambda>_. 0" by (rule almost_everywhere_zeroI) simp
223 lemma coeff_0 [simp]:
227 lemma degree_0 [simp]:
229 by (rule order_antisym [OF degree_le le0]) simp
231 lemma leading_coeff_neq_0:
232 assumes "p \<noteq> 0"
233 shows "coeff p (degree p) \<noteq> 0"
234 proof (cases "degree p")
236 from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
237 by (simp add: poly_eq_iff)
238 then obtain n where "coeff p n \<noteq> 0" ..
239 hence "n \<le> degree p" by (rule le_degree)
240 with `coeff p n \<noteq> 0` and `degree p = 0`
241 show "coeff p (degree p) \<noteq> 0" by simp
244 from `degree p = Suc n` have "n < degree p" by simp
245 hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
246 then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
247 from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
248 also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
249 finally have "degree p = i" .
250 with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
253 lemma leading_coeff_0_iff [simp]:
254 "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
255 by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
258 subsection {* List-style constructor for polynomials *}
260 lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
261 is "\<lambda>a p. case_nat a (coeff p)"
262 using coeff_almost_everywhere_zero by (rule almost_everywhere_zero_case_nat)
264 lemmas coeff_pCons = pCons.rep_eq
266 lemma coeff_pCons_0 [simp]:
267 "coeff (pCons a p) 0 = a"
270 lemma coeff_pCons_Suc [simp]:
271 "coeff (pCons a p) (Suc n) = coeff p n"
272 by (simp add: coeff_pCons)
274 lemma degree_pCons_le:
275 "degree (pCons a p) \<le> Suc (degree p)"
276 by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
278 lemma degree_pCons_eq:
279 "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
280 apply (rule order_antisym [OF degree_pCons_le])
281 apply (rule le_degree, simp)
284 lemma degree_pCons_0:
285 "degree (pCons a 0) = 0"
286 apply (rule order_antisym [OF _ le0])
287 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
290 lemma degree_pCons_eq_if [simp]:
291 "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
292 apply (cases "p = 0", simp_all)
293 apply (rule order_antisym [OF _ le0])
294 apply (rule degree_le, simp add: coeff_pCons split: nat.split)
295 apply (rule order_antisym [OF degree_pCons_le])
296 apply (rule le_degree, simp)
299 lemma pCons_0_0 [simp]:
301 by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
303 lemma pCons_eq_iff [simp]:
304 "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
306 assume "pCons a p = pCons b q"
307 then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
308 then show "a = b" by simp
310 assume "pCons a p = pCons b q"
311 then have "\<forall>n. coeff (pCons a p) (Suc n) =
312 coeff (pCons b q) (Suc n)" by simp
313 then show "p = q" by (simp add: poly_eq_iff)
316 lemma pCons_eq_0_iff [simp]:
317 "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
318 using pCons_eq_iff [of a p 0 0] by simp
320 lemma pCons_cases [cases type: poly]:
321 obtains (pCons) a q where "p = pCons a q"
323 show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
325 (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
328 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
330 assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"
332 proof (induct p rule: measure_induct_rule [where f=degree])
334 obtain a q where "p = pCons a q" by (rule pCons_cases)
336 proof (cases "q = 0")
338 then show "P q" by (simp add: zero)
341 then have "degree (pCons a q) = Suc (degree q)"
342 by (rule degree_pCons_eq)
343 then have "degree q < degree p"
344 using `p = pCons a q` by simp
349 proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")
351 with `P q` show ?thesis by (auto intro: pCons)
354 with zero show ?thesis by simp
357 using `p = pCons a q` by simp
361 subsection {* List-style syntax for polynomials *}
364 "_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
367 "[:x, xs:]" == "CONST pCons x [:xs:]"
368 "[:x:]" == "CONST pCons x 0"
369 "[:x:]" <= "CONST pCons x (_constrain 0 t)"
372 subsection {* Representation of polynomials by lists of coefficients *}
374 primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
376 [code_post]: "Poly [] = 0"
377 | [code_post]: "Poly (a # as) = pCons a (Poly as)"
379 lemma Poly_replicate_0 [simp]:
380 "Poly (replicate n 0) = 0"
381 by (induct n) simp_all
384 "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
385 by (induct as) (auto simp add: Cons_replicate_eq)
387 definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
389 "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
391 lemma coeffs_eq_Nil [simp]:
392 "coeffs p = [] \<longleftrightarrow> p = 0"
393 by (simp add: coeffs_def)
395 lemma not_0_coeffs_not_Nil:
396 "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
399 lemma coeffs_0_eq_Nil [simp]:
403 lemma coeffs_pCons_eq_cCons [simp]:
404 "coeffs (pCons a p) = a ## coeffs p"
406 { fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
407 assume "\<forall>m\<in>set ms. m > 0"
408 then have "map (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)"
409 by (induct ms) (auto, metis Suc_pred' nat.case(2)) }
412 by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
415 lemma not_0_cCons_eq [simp]:
416 "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
417 by (simp add: cCons_def)
419 lemma Poly_coeffs [simp, code abstype]:
420 "Poly (coeffs p) = p"
423 lemma coeffs_Poly [simp]:
424 "coeffs (Poly as) = strip_while (HOL.eq 0) as"
426 case Nil then show ?case by simp
429 have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
430 using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
431 with Cons show ?case by auto
434 lemma last_coeffs_not_0:
435 "p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
436 by (induct p) (auto simp add: cCons_def)
438 lemma strip_while_coeffs [simp]:
439 "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
440 by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)
443 "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
445 assume ?P then show ?Q by simp
448 then have "Poly (coeffs p) = Poly (coeffs q)" by simp
453 "coeff (Poly xs) n = nth_default 0 xs n"
454 apply (induct xs arbitrary: n) apply simp_all
455 by (metis nat.case not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)
457 lemma nth_default_coeffs_eq:
458 "nth_default 0 (coeffs p) = coeff p"
459 by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
462 "coeff p = nth_default 0 (coeffs p)"
463 by (simp add: nth_default_coeffs_eq)
466 assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
467 assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
468 shows "coeffs p = xs"
470 from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
471 with zero show ?thesis by simp (cases xs, simp_all)
474 lemma degree_eq_length_coeffs [code]:
475 "degree p = length (coeffs p) - 1"
476 by (simp add: coeffs_def)
478 lemma length_coeffs_degree:
479 "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
480 by (induct p) (auto simp add: cCons_def)
482 lemma [code abstract]:
484 by (fact coeffs_0_eq_Nil)
486 lemma [code abstract]:
487 "coeffs (pCons a p) = a ## coeffs p"
488 by (fact coeffs_pCons_eq_cCons)
490 instantiation poly :: ("{zero, equal}") equal
494 [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
497 qed (simp add: equal equal_poly_def coeffs_eq_iff)
502 "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
505 definition is_zero :: "'a::zero poly \<Rightarrow> bool"
507 [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
509 lemma is_zero_null [code_abbrev]:
510 "is_zero p \<longleftrightarrow> p = 0"
511 by (simp add: is_zero_def null_def)
514 subsection {* Fold combinator for polynomials *}
516 definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
518 "fold_coeffs f p = foldr f (coeffs p)"
520 lemma fold_coeffs_0_eq [simp]:
521 "fold_coeffs f 0 = id"
522 by (simp add: fold_coeffs_def)
524 lemma fold_coeffs_pCons_eq [simp]:
525 "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
526 by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
528 lemma fold_coeffs_pCons_0_0_eq [simp]:
529 "fold_coeffs f (pCons 0 0) = id"
530 by (simp add: fold_coeffs_def)
532 lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
533 "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
534 by (simp add: fold_coeffs_def)
536 lemma fold_coeffs_pCons_not_0_0_eq [simp]:
537 "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
538 by (simp add: fold_coeffs_def)
541 subsection {* Canonical morphism on polynomials -- evaluation *}
543 definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
545 "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" -- {* The Horner Schema *}
549 by (simp add: poly_def)
551 lemma poly_pCons [simp]:
552 "poly (pCons a p) x = a + x * poly p x"
553 by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
556 subsection {* Monomials *}
558 lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
559 is "\<lambda>a m n. if m = n then a else 0"
560 by (auto intro!: almost_everywhere_zeroI)
562 lemma coeff_monom [simp]:
563 "coeff (monom a m) n = (if m = n then a else 0)"
567 "monom a 0 = pCons a 0"
568 by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
571 "monom a (Suc n) = pCons 0 (monom a n)"
572 by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
574 lemma monom_eq_0 [simp]: "monom 0 n = 0"
575 by (rule poly_eqI) simp
577 lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
578 by (simp add: poly_eq_iff)
580 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
581 by (simp add: poly_eq_iff)
583 lemma degree_monom_le: "degree (monom a n) \<le> n"
584 by (rule degree_le, simp)
586 lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
587 apply (rule order_antisym [OF degree_monom_le])
588 apply (rule le_degree, simp)
591 lemma coeffs_monom [code abstract]:
592 "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
593 by (induct n) (simp_all add: monom_0 monom_Suc)
595 lemma fold_coeffs_monom [simp]:
596 "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
597 by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
600 fixes a x :: "'a::{comm_semiring_1}"
601 shows "poly (monom a n) x = a * x ^ n"
602 by (cases "a = 0", simp_all)
603 (induct n, simp_all add: mult.left_commute poly_def)
606 subsection {* Addition and subtraction *}
608 instantiation poly :: (comm_monoid_add) comm_monoid_add
611 lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
612 is "\<lambda>p q n. coeff p n + coeff q n"
613 proof (rule almost_everywhere_zeroI)
614 fix q p :: "'a poly" and i
615 assume "max (degree q) (degree p) < i"
616 then show "coeff p i + coeff q i = 0"
617 by (simp add: coeff_eq_0)
620 lemma coeff_add [simp]:
621 "coeff (p + q) n = coeff p n + coeff q n"
622 by (simp add: plus_poly.rep_eq)
625 fix p q r :: "'a poly"
626 show "(p + q) + r = p + (q + r)"
627 by (simp add: poly_eq_iff add_assoc)
629 by (simp add: poly_eq_iff add_commute)
631 by (simp add: poly_eq_iff)
636 instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
638 fix p q r :: "'a poly"
639 assume "p + q = p + r" thus "q = r"
640 by (simp add: poly_eq_iff)
643 instantiation poly :: (ab_group_add) ab_group_add
646 lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
647 is "\<lambda>p n. - coeff p n"
648 proof (rule almost_everywhere_zeroI)
649 fix p :: "'a poly" and i
650 assume "degree p < i"
651 then show "- coeff p i = 0"
652 by (simp add: coeff_eq_0)
655 lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
656 is "\<lambda>p q n. coeff p n - coeff q n"
657 proof (rule almost_everywhere_zeroI)
658 fix q p :: "'a poly" and i
659 assume "max (degree q) (degree p) < i"
660 then show "coeff p i - coeff q i = 0"
661 by (simp add: coeff_eq_0)
664 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
665 by (simp add: uminus_poly.rep_eq)
667 lemma coeff_diff [simp]:
668 "coeff (p - q) n = coeff p n - coeff q n"
669 by (simp add: minus_poly.rep_eq)
674 by (simp add: poly_eq_iff)
675 show "p - q = p + - q"
676 by (simp add: poly_eq_iff)
681 lemma add_pCons [simp]:
682 "pCons a p + pCons b q = pCons (a + b) (p + q)"
683 by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
685 lemma minus_pCons [simp]:
686 "- pCons a p = pCons (- a) (- p)"
687 by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
689 lemma diff_pCons [simp]:
690 "pCons a p - pCons b q = pCons (a - b) (p - q)"
691 by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
693 lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
694 by (rule degree_le, auto simp add: coeff_eq_0)
697 "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
698 by (auto intro: order_trans degree_add_le_max)
700 lemma degree_add_less:
701 "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
702 by (auto intro: le_less_trans degree_add_le_max)
704 lemma degree_add_eq_right:
705 "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
706 apply (cases "q = 0", simp)
707 apply (rule order_antisym)
708 apply (simp add: degree_add_le)
709 apply (rule le_degree)
710 apply (simp add: coeff_eq_0)
713 lemma degree_add_eq_left:
714 "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
715 using degree_add_eq_right [of q p]
716 by (simp add: add_commute)
718 lemma degree_minus [simp]: "degree (- p) = degree p"
719 unfolding degree_def by simp
721 lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
722 using degree_add_le [where p=p and q="-q"]
725 lemma degree_diff_le:
726 "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
727 using degree_add_le [of p n "- q"] by simp
729 lemma degree_diff_less:
730 "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
731 using degree_add_less [of p n "- q"] by simp
733 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
734 by (rule poly_eqI) simp
736 lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
737 by (rule poly_eqI) simp
739 lemma minus_monom: "- monom a n = monom (-a) n"
740 by (rule poly_eqI) simp
742 lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
743 by (cases "finite A", induct set: finite, simp_all)
745 lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
746 by (rule poly_eqI) (simp add: coeff_setsum)
748 fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
750 "plus_coeffs xs [] = xs"
751 | "plus_coeffs [] ys = ys"
752 | "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
754 lemma coeffs_plus_eq_plus_coeffs [code abstract]:
755 "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
757 { fix xs ys :: "'a list" and n
758 have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
759 proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
760 case (3 x xs y ys n) then show ?case by (cases n) (auto simp add: cCons_def)
763 { fix xs ys :: "'a list"
764 assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
765 moreover assume "plus_coeffs xs ys \<noteq> []"
766 ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
767 proof (induct xs ys rule: plus_coeffs.induct)
768 case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
772 apply (rule coeffs_eqI)
773 apply (simp add: * nth_default_coeffs_eq)
775 apply (auto dest: last_coeffs_not_0)
779 lemma coeffs_uminus [code abstract]:
780 "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
782 (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
785 fixes p q :: "'a::ab_group_add poly"
786 shows "p - q = p + - q"
787 by (fact ab_add_uminus_conv_diff)
789 lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
790 apply (induct p arbitrary: q, simp)
791 apply (case_tac q, simp, simp add: algebra_simps)
794 lemma poly_minus [simp]:
795 fixes x :: "'a::comm_ring"
796 shows "poly (- p) x = - poly p x"
797 by (induct p) simp_all
799 lemma poly_diff [simp]:
800 fixes x :: "'a::comm_ring"
801 shows "poly (p - q) x = poly p x - poly q x"
802 using poly_add [of p "- q" x] by simp
804 lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
805 by (induct A rule: infinite_finite_induct) simp_all
808 subsection {* Multiplication by a constant, polynomial multiplication and the unit polynomial *}
810 lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
811 is "\<lambda>a p n. a * coeff p n"
812 proof (rule almost_everywhere_zeroI)
813 fix a :: 'a and p :: "'a poly" and i
814 assume "degree p < i"
815 then show "a * coeff p i = 0"
816 by (simp add: coeff_eq_0)
819 lemma coeff_smult [simp]:
820 "coeff (smult a p) n = a * coeff p n"
821 by (simp add: smult.rep_eq)
823 lemma degree_smult_le: "degree (smult a p) \<le> degree p"
824 by (rule degree_le, simp add: coeff_eq_0)
826 lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
827 by (rule poly_eqI, simp add: mult_assoc)
829 lemma smult_0_right [simp]: "smult a 0 = 0"
830 by (rule poly_eqI, simp)
832 lemma smult_0_left [simp]: "smult 0 p = 0"
833 by (rule poly_eqI, simp)
835 lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
836 by (rule poly_eqI, simp)
838 lemma smult_add_right:
839 "smult a (p + q) = smult a p + smult a q"
840 by (rule poly_eqI, simp add: algebra_simps)
842 lemma smult_add_left:
843 "smult (a + b) p = smult a p + smult b p"
844 by (rule poly_eqI, simp add: algebra_simps)
846 lemma smult_minus_right [simp]:
847 "smult (a::'a::comm_ring) (- p) = - smult a p"
848 by (rule poly_eqI, simp)
850 lemma smult_minus_left [simp]:
851 "smult (- a::'a::comm_ring) p = - smult a p"
852 by (rule poly_eqI, simp)
854 lemma smult_diff_right:
855 "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
856 by (rule poly_eqI, simp add: algebra_simps)
858 lemma smult_diff_left:
859 "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
860 by (rule poly_eqI, simp add: algebra_simps)
862 lemmas smult_distribs =
863 smult_add_left smult_add_right
864 smult_diff_left smult_diff_right
866 lemma smult_pCons [simp]:
867 "smult a (pCons b p) = pCons (a * b) (smult a p)"
868 by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
870 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
871 by (induct n, simp add: monom_0, simp add: monom_Suc)
873 lemma degree_smult_eq [simp]:
874 fixes a :: "'a::idom"
875 shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
876 by (cases "a = 0", simp, simp add: degree_def)
878 lemma smult_eq_0_iff [simp]:
879 fixes a :: "'a::idom"
880 shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
881 by (simp add: poly_eq_iff)
883 lemma coeffs_smult [code abstract]:
884 fixes p :: "'a::idom poly"
885 shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
887 (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
889 instantiation poly :: (comm_semiring_0) comm_semiring_0
893 "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
895 lemma mult_poly_0_left: "(0::'a poly) * q = 0"
896 by (simp add: times_poly_def)
898 lemma mult_pCons_left [simp]:
899 "pCons a p * q = smult a q + pCons 0 (p * q)"
900 by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
902 lemma mult_poly_0_right: "p * (0::'a poly) = 0"
903 by (induct p) (simp add: mult_poly_0_left, simp)
905 lemma mult_pCons_right [simp]:
906 "p * pCons a q = smult a p + pCons 0 (p * q)"
907 by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps)
909 lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
911 lemma mult_smult_left [simp]:
912 "smult a p * q = smult a (p * q)"
913 by (induct p) (simp add: mult_poly_0, simp add: smult_add_right)
915 lemma mult_smult_right [simp]:
916 "p * smult a q = smult a (p * q)"
917 by (induct q) (simp add: mult_poly_0, simp add: smult_add_right)
919 lemma mult_poly_add_left:
920 fixes p q r :: "'a poly"
921 shows "(p + q) * r = p * r + q * r"
922 by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps)
925 fix p q r :: "'a poly"
927 by (rule mult_poly_0_left)
929 by (rule mult_poly_0_right)
930 show "(p + q) * r = p * r + q * r"
931 by (rule mult_poly_add_left)
932 show "(p * q) * r = p * (q * r)"
933 by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
935 by (induct p, simp add: mult_poly_0, simp)
940 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
943 "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
944 proof (induct p arbitrary: n)
945 case 0 show ?case by simp
947 case (pCons a p n) thus ?case
948 by (cases n, simp, simp add: setsum_atMost_Suc_shift
949 del: setsum_atMost_Suc)
952 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
953 apply (rule degree_le)
956 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
959 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
960 by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
962 instantiation poly :: (comm_semiring_1) comm_semiring_1
965 definition one_poly_def:
969 fix p :: "'a poly" show "1 * p = p"
970 unfolding one_poly_def by simp
972 show "0 \<noteq> (1::'a poly)"
973 unfolding one_poly_def by simp
978 instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
980 instance poly :: (comm_ring) comm_ring ..
982 instance poly :: (comm_ring_1) comm_ring_1 ..
984 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
985 unfolding one_poly_def
986 by (simp add: coeff_pCons split: nat.split)
988 lemma degree_1 [simp]: "degree 1 = 0"
989 unfolding one_poly_def
990 by (rule degree_pCons_0)
992 lemma coeffs_1_eq [simp, code abstract]:
994 by (simp add: one_poly_def)
996 lemma degree_power_le:
997 "degree (p ^ n) \<le> degree p * n"
998 by (induct n) (auto intro: order_trans degree_mult_le)
1000 lemma poly_smult [simp]:
1001 "poly (smult a p) x = a * poly p x"
1002 by (induct p, simp, simp add: algebra_simps)
1004 lemma poly_mult [simp]:
1005 "poly (p * q) x = poly p x * poly q x"
1006 by (induct p, simp_all, simp add: algebra_simps)
1008 lemma poly_1 [simp]:
1010 by (simp add: one_poly_def)
1012 lemma poly_power [simp]:
1013 fixes p :: "'a::{comm_semiring_1} poly"
1014 shows "poly (p ^ n) x = poly p x ^ n"
1015 by (induct n) simp_all
1018 subsection {* Lemmas about divisibility *}
1020 lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
1023 then obtain k where "q = p * k" ..
1024 then have "smult a q = p * smult a k" by simp
1025 then show "p dvd smult a q" ..
1028 lemma dvd_smult_cancel:
1029 fixes a :: "'a::field"
1030 shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
1031 by (drule dvd_smult [where a="inverse a"]) simp
1033 lemma dvd_smult_iff:
1034 fixes a :: "'a::field"
1035 shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
1036 by (safe elim!: dvd_smult dvd_smult_cancel)
1038 lemma smult_dvd_cancel:
1039 "smult a p dvd q \<Longrightarrow> p dvd q"
1041 assume "smult a p dvd q"
1042 then obtain k where "q = smult a p * k" ..
1043 then have "q = p * smult a k" by simp
1044 then show "p dvd q" ..
1048 fixes a :: "'a::field"
1049 shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
1050 by (rule smult_dvd_cancel [where a="inverse a"]) simp
1052 lemma smult_dvd_iff:
1053 fixes a :: "'a::field"
1054 shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
1055 by (auto elim: smult_dvd smult_dvd_cancel)
1058 subsection {* Polynomials form an integral domain *}
1060 lemma coeff_mult_degree_sum:
1061 "coeff (p * q) (degree p + degree q) =
1062 coeff p (degree p) * coeff q (degree q)"
1063 by (induct p, simp, simp add: coeff_eq_0)
1065 instance poly :: (idom) idom
1067 fix p q :: "'a poly"
1068 assume "p \<noteq> 0" and "q \<noteq> 0"
1069 have "coeff (p * q) (degree p + degree q) =
1070 coeff p (degree p) * coeff q (degree q)"
1071 by (rule coeff_mult_degree_sum)
1072 also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
1073 using `p \<noteq> 0` and `q \<noteq> 0` by simp
1074 finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
1075 thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
1078 lemma degree_mult_eq:
1079 fixes p q :: "'a::idom poly"
1080 shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
1081 apply (rule order_antisym [OF degree_mult_le le_degree])
1082 apply (simp add: coeff_mult_degree_sum)
1085 lemma dvd_imp_degree_le:
1086 fixes p q :: "'a::idom poly"
1087 shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
1088 by (erule dvdE, simp add: degree_mult_eq)
1091 subsection {* Polynomials form an ordered integral domain *}
1093 definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
1095 "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
1097 lemma pos_poly_pCons:
1098 "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
1099 unfolding pos_poly_def by simp
1101 lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
1102 unfolding pos_poly_def by simp
1104 lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
1105 apply (induct p arbitrary: q, simp)
1106 apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
1109 lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
1110 unfolding pos_poly_def
1111 apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
1112 apply (simp add: degree_mult_eq coeff_mult_degree_sum)
1116 lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
1117 by (induct p) (auto simp add: pos_poly_pCons)
1119 lemma last_coeffs_eq_coeff_degree:
1120 "p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"
1121 by (simp add: coeffs_def)
1123 lemma pos_poly_coeffs [code]:
1124 "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q")
1126 assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
1128 assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def)
1129 then have "p \<noteq> 0" by auto
1130 with * show ?Q by (simp add: last_coeffs_eq_coeff_degree)
1133 instantiation poly :: (linordered_idom) linordered_idom
1137 "x < y \<longleftrightarrow> pos_poly (y - x)"
1140 "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
1143 "abs (x::'a poly) = (if x < 0 then - x else x)"
1146 "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
1149 fix x y :: "'a poly"
1150 show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
1151 unfolding less_eq_poly_def less_poly_def
1154 apply (drule (1) pos_poly_add)
1158 fix x :: "'a poly" show "x \<le> x"
1159 unfolding less_eq_poly_def by simp
1161 fix x y z :: "'a poly"
1162 assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
1163 unfolding less_eq_poly_def
1165 apply (drule (1) pos_poly_add)
1166 apply (simp add: algebra_simps)
1169 fix x y :: "'a poly"
1170 assume "x \<le> y" and "y \<le> x" thus "x = y"
1171 unfolding less_eq_poly_def
1173 apply (drule (1) pos_poly_add)
1177 fix x y z :: "'a poly"
1178 assume "x \<le> y" thus "z + x \<le> z + y"
1179 unfolding less_eq_poly_def
1181 apply (simp add: algebra_simps)
1184 fix x y :: "'a poly"
1185 show "x \<le> y \<or> y \<le> x"
1186 unfolding less_eq_poly_def
1187 using pos_poly_total [of "x - y"]
1190 fix x y z :: "'a poly"
1191 assume "x < y" and "0 < z"
1192 thus "z * x < z * y"
1193 unfolding less_poly_def
1194 by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
1197 show "\<bar>x\<bar> = (if x < 0 then - x else x)"
1198 by (rule abs_poly_def)
1201 show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
1202 by (rule sgn_poly_def)
1207 text {* TODO: Simplification rules for comparisons *}
1210 subsection {* Synthetic division and polynomial roots *}
1213 Synthetic division is simply division by the linear polynomial @{term "x - c"}.
1216 definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
1218 "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
1220 definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
1222 "synthetic_div p c = fst (synthetic_divmod p c)"
1224 lemma synthetic_divmod_0 [simp]:
1225 "synthetic_divmod 0 c = (0, 0)"
1226 by (simp add: synthetic_divmod_def)
1228 lemma synthetic_divmod_pCons [simp]:
1229 "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
1230 by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
1232 lemma synthetic_div_0 [simp]:
1233 "synthetic_div 0 c = 0"
1234 unfolding synthetic_div_def by simp
1236 lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
1237 by (induct p arbitrary: a) simp_all
1239 lemma snd_synthetic_divmod:
1240 "snd (synthetic_divmod p c) = poly p c"
1241 by (induct p, simp, simp add: split_def)
1243 lemma synthetic_div_pCons [simp]:
1244 "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
1245 unfolding synthetic_div_def
1246 by (simp add: split_def snd_synthetic_divmod)
1248 lemma synthetic_div_eq_0_iff:
1249 "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
1250 by (induct p, simp, case_tac p, simp)
1252 lemma degree_synthetic_div:
1253 "degree (synthetic_div p c) = degree p - 1"
1254 by (induct p, simp, simp add: synthetic_div_eq_0_iff)
1256 lemma synthetic_div_correct:
1257 "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
1258 by (induct p) simp_all
1260 lemma synthetic_div_unique:
1261 "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
1262 apply (induct p arbitrary: q r)
1263 apply (simp, frule synthetic_div_unique_lemma, simp)
1264 apply (case_tac q, force)
1267 lemma synthetic_div_correct':
1268 fixes c :: "'a::comm_ring_1"
1269 shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
1270 using synthetic_div_correct [of p c]
1271 by (simp add: algebra_simps)
1273 lemma poly_eq_0_iff_dvd:
1274 fixes c :: "'a::idom"
1275 shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
1277 assume "poly p c = 0"
1278 with synthetic_div_correct' [of c p]
1279 have "p = [:-c, 1:] * synthetic_div p c" by simp
1280 then show "[:-c, 1:] dvd p" ..
1282 assume "[:-c, 1:] dvd p"
1283 then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
1284 then show "poly p c = 0" by simp
1287 lemma dvd_iff_poly_eq_0:
1288 fixes c :: "'a::idom"
1289 shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
1290 by (simp add: poly_eq_0_iff_dvd)
1292 lemma poly_roots_finite:
1293 fixes p :: "'a::idom poly"
1294 shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
1295 proof (induct n \<equiv> "degree p" arbitrary: p)
1297 then obtain a where "a \<noteq> 0" and "p = [:a:]"
1298 by (cases p, simp split: if_splits)
1299 then show "finite {x. poly p x = 0}" by simp
1302 show "finite {x. poly p x = 0}"
1303 proof (cases "\<exists>x. poly p x = 0")
1305 then show "finite {x. poly p x = 0}" by simp
1308 then obtain a where "poly p a = 0" ..
1309 then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
1310 then obtain k where k: "p = [:-a, 1:] * k" ..
1311 with `p \<noteq> 0` have "k \<noteq> 0" by auto
1312 with k have "degree p = Suc (degree k)"
1313 by (simp add: degree_mult_eq del: mult_pCons_left)
1314 with `Suc n = degree p` have "n = degree k" by simp
1315 then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
1316 then have "finite (insert a {x. poly k x = 0})" by simp
1317 then show "finite {x. poly p x = 0}"
1318 by (simp add: k uminus_add_conv_diff Collect_disj_eq
1319 del: mult_pCons_left)
1323 lemma poly_eq_poly_eq_iff:
1324 fixes p q :: "'a::{idom,ring_char_0} poly"
1325 shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
1327 assume ?Q then show ?P by simp
1329 { fix p :: "'a::{idom,ring_char_0} poly"
1330 have "poly p = poly 0 \<longleftrightarrow> p = 0"
1331 apply (cases "p = 0", simp_all)
1332 apply (drule poly_roots_finite)
1333 apply (auto simp add: infinite_UNIV_char_0)
1335 } note this [of "p - q"]
1337 ultimately show ?Q by auto
1340 lemma poly_all_0_iff_0:
1341 fixes p :: "'a::{ring_char_0, idom} poly"
1342 shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
1343 by (auto simp add: poly_eq_poly_eq_iff [symmetric])
1346 subsection {* Long division of polynomials *}
1348 definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
1350 "pdivmod_rel x y q r \<longleftrightarrow>
1351 x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
1353 lemma pdivmod_rel_0:
1354 "pdivmod_rel 0 y 0 0"
1355 unfolding pdivmod_rel_def by simp
1357 lemma pdivmod_rel_by_0:
1358 "pdivmod_rel x 0 0 x"
1359 unfolding pdivmod_rel_def by simp
1361 lemma eq_zero_or_degree_less:
1362 assumes "degree p \<le> n" and "coeff p n = 0"
1363 shows "p = 0 \<or> degree p < n"
1366 with `degree p \<le> n` and `coeff p n = 0`
1367 have "coeff p (degree p) = 0" by simp
1368 then have "p = 0" by simp
1369 then show ?thesis ..
1372 have "\<forall>i>n. coeff p i = 0"
1373 using `degree p \<le> n` by (simp add: coeff_eq_0)
1374 then have "\<forall>i\<ge>n. coeff p i = 0"
1375 using `coeff p n = 0` by (simp add: le_less)
1376 then have "\<forall>i>m. coeff p i = 0"
1377 using `n = Suc m` by (simp add: less_eq_Suc_le)
1378 then have "degree p \<le> m"
1380 then have "degree p < n"
1381 using `n = Suc m` by (simp add: less_Suc_eq_le)
1382 then show ?thesis ..
1385 lemma pdivmod_rel_pCons:
1386 assumes rel: "pdivmod_rel x y q r"
1387 assumes y: "y \<noteq> 0"
1388 assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
1389 shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
1390 (is "pdivmod_rel ?x y ?q ?r")
1392 have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
1393 using assms unfolding pdivmod_rel_def by simp_all
1395 have 1: "?x = ?q * y + ?r"
1398 have 2: "?r = 0 \<or> degree ?r < degree y"
1399 proof (rule eq_zero_or_degree_less)
1400 show "degree ?r \<le> degree y"
1401 proof (rule degree_diff_le)
1402 show "degree (pCons a r) \<le> degree y"
1404 show "degree (smult b y) \<le> degree y"
1405 by (rule degree_smult_le)
1408 show "coeff ?r (degree y) = 0"
1409 using `y \<noteq> 0` unfolding b by simp
1412 from 1 2 show ?thesis
1413 unfolding pdivmod_rel_def
1414 using `y \<noteq> 0` by simp
1417 lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
1418 apply (cases "y = 0")
1419 apply (fast intro!: pdivmod_rel_by_0)
1421 apply (fast intro!: pdivmod_rel_0)
1422 apply (fast intro!: pdivmod_rel_pCons)
1425 lemma pdivmod_rel_unique:
1426 assumes 1: "pdivmod_rel x y q1 r1"
1427 assumes 2: "pdivmod_rel x y q2 r2"
1428 shows "q1 = q2 \<and> r1 = r2"
1429 proof (cases "y = 0")
1430 assume "y = 0" with assms show ?thesis
1431 by (simp add: pdivmod_rel_def)
1433 assume [simp]: "y \<noteq> 0"
1434 from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
1435 unfolding pdivmod_rel_def by simp_all
1436 from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
1437 unfolding pdivmod_rel_def by simp_all
1438 from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
1439 by (simp add: algebra_simps)
1440 from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
1441 by (auto intro: degree_diff_less)
1443 show "q1 = q2 \<and> r1 = r2"
1445 assume "\<not> (q1 = q2 \<and> r1 = r2)"
1446 with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
1447 with r3 have "degree (r2 - r1) < degree y" by simp
1448 also have "degree y \<le> degree (q1 - q2) + degree y" by simp
1449 also have "\<dots> = degree ((q1 - q2) * y)"
1450 using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
1451 also have "\<dots> = degree (r2 - r1)"
1453 finally have "degree (r2 - r1) < degree (r2 - r1)" .
1454 then show "False" by simp
1458 lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
1459 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
1461 lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
1462 by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
1464 lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]
1466 lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]
1468 instantiation poly :: (field) ring_div
1471 definition div_poly where
1472 "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
1474 definition mod_poly where
1475 "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
1478 "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
1479 unfolding div_poly_def
1480 by (fast elim: pdivmod_rel_unique_div)
1483 "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
1484 unfolding mod_poly_def
1485 by (fast elim: pdivmod_rel_unique_mod)
1488 "pdivmod_rel x y (x div y) (x mod y)"
1490 from pdivmod_rel_exists
1491 obtain q r where "pdivmod_rel x y q r" by fast
1493 by (simp add: div_poly_eq mod_poly_eq)
1497 fix x y :: "'a poly"
1498 show "x div y * y + x mod y = x"
1499 using pdivmod_rel [of x y]
1500 by (simp add: pdivmod_rel_def)
1503 have "pdivmod_rel x 0 0 x"
1504 by (rule pdivmod_rel_by_0)
1506 by (rule div_poly_eq)
1509 have "pdivmod_rel 0 y 0 0"
1510 by (rule pdivmod_rel_0)
1512 by (rule div_poly_eq)
1514 fix x y z :: "'a poly"
1515 assume "y \<noteq> 0"
1516 hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
1517 using pdivmod_rel [of x y]
1518 by (simp add: pdivmod_rel_def distrib_right)
1519 thus "(x + z * y) div y = z + x div y"
1520 by (rule div_poly_eq)
1522 fix x y z :: "'a poly"
1523 assume "x \<noteq> 0"
1524 show "(x * y) div (x * z) = y div z"
1525 proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
1526 have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
1527 by (rule pdivmod_rel_by_0)
1528 then have [simp]: "\<And>x::'a poly. x div 0 = 0"
1529 by (rule div_poly_eq)
1530 have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
1531 by (rule pdivmod_rel_0)
1532 then have [simp]: "\<And>x::'a poly. 0 div x = 0"
1533 by (rule div_poly_eq)
1534 case False then show ?thesis by auto
1536 case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
1538 have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
1539 by (auto simp add: pdivmod_rel_def algebra_simps)
1540 (rule classical, simp add: degree_mult_eq)
1541 moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
1542 ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
1543 then show ?thesis by (simp add: div_poly_eq)
1549 lemma degree_mod_less:
1550 "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
1551 using pdivmod_rel [of x y]
1552 unfolding pdivmod_rel_def by simp
1554 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
1556 assume "degree x < degree y"
1557 hence "pdivmod_rel x y 0 x"
1558 by (simp add: pdivmod_rel_def)
1559 thus "x div y = 0" by (rule div_poly_eq)
1562 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
1564 assume "degree x < degree y"
1565 hence "pdivmod_rel x y 0 x"
1566 by (simp add: pdivmod_rel_def)
1567 thus "x mod y = x" by (rule mod_poly_eq)
1570 lemma pdivmod_rel_smult_left:
1571 "pdivmod_rel x y q r
1572 \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
1573 unfolding pdivmod_rel_def by (simp add: smult_add_right)
1575 lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
1576 by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
1578 lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
1579 by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
1581 lemma poly_div_minus_left [simp]:
1582 fixes x y :: "'a::field poly"
1583 shows "(- x) div y = - (x div y)"
1584 using div_smult_left [of "- 1::'a"] by simp
1586 lemma poly_mod_minus_left [simp]:
1587 fixes x y :: "'a::field poly"
1588 shows "(- x) mod y = - (x mod y)"
1589 using mod_smult_left [of "- 1::'a"] by simp
1591 lemma pdivmod_rel_add_left:
1592 assumes "pdivmod_rel x y q r"
1593 assumes "pdivmod_rel x' y q' r'"
1594 shows "pdivmod_rel (x + x') y (q + q') (r + r')"
1595 using assms unfolding pdivmod_rel_def
1596 by (auto simp add: distrib degree_add_less)
1598 lemma poly_div_add_left:
1599 fixes x y z :: "'a::field poly"
1600 shows "(x + y) div z = x div z + y div z"
1601 using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
1602 by (rule div_poly_eq)
1604 lemma poly_mod_add_left:
1605 fixes x y z :: "'a::field poly"
1606 shows "(x + y) mod z = x mod z + y mod z"
1607 using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
1608 by (rule mod_poly_eq)
1610 lemma poly_div_diff_left:
1611 fixes x y z :: "'a::field poly"
1612 shows "(x - y) div z = x div z - y div z"
1613 by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)
1615 lemma poly_mod_diff_left:
1616 fixes x y z :: "'a::field poly"
1617 shows "(x - y) mod z = x mod z - y mod z"
1618 by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)
1620 lemma pdivmod_rel_smult_right:
1621 "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
1622 \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
1623 unfolding pdivmod_rel_def by simp
1625 lemma div_smult_right:
1626 "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
1627 by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
1629 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
1630 by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
1632 lemma poly_div_minus_right [simp]:
1633 fixes x y :: "'a::field poly"
1634 shows "x div (- y) = - (x div y)"
1635 using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)
1637 lemma poly_mod_minus_right [simp]:
1638 fixes x y :: "'a::field poly"
1639 shows "x mod (- y) = x mod y"
1640 using mod_smult_right [of "- 1::'a"] by simp
1642 lemma pdivmod_rel_mult:
1643 "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
1644 \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
1645 apply (cases "z = 0", simp add: pdivmod_rel_def)
1646 apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
1647 apply (cases "r = 0")
1648 apply (cases "r' = 0")
1649 apply (simp add: pdivmod_rel_def)
1650 apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)
1651 apply (cases "r' = 0")
1652 apply (simp add: pdivmod_rel_def degree_mult_eq)
1653 apply (simp add: pdivmod_rel_def field_simps)
1654 apply (simp add: degree_mult_eq degree_add_less)
1657 lemma poly_div_mult_right:
1658 fixes x y z :: "'a::field poly"
1659 shows "x div (y * z) = (x div y) div z"
1660 by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
1662 lemma poly_mod_mult_right:
1663 fixes x y z :: "'a::field poly"
1664 shows "x mod (y * z) = y * (x div y mod z) + x mod y"
1665 by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
1669 assumes y: "y \<noteq> 0"
1670 defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
1671 shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
1673 apply (rule mod_poly_eq)
1674 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
1677 definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
1679 "pdivmod p q = (p div q, p mod q)"
1681 lemma div_poly_code [code]:
1682 "p div q = fst (pdivmod p q)"
1683 by (simp add: pdivmod_def)
1685 lemma mod_poly_code [code]:
1686 "p mod q = snd (pdivmod p q)"
1687 by (simp add: pdivmod_def)
1690 "pdivmod 0 q = (0, 0)"
1691 by (simp add: pdivmod_def)
1693 lemma pdivmod_pCons:
1694 "pdivmod (pCons a p) q =
1695 (if q = 0 then (0, pCons a p) else
1696 (let (s, r) = pdivmod p q;
1697 b = coeff (pCons a r) (degree q) / coeff q (degree q)
1698 in (pCons b s, pCons a r - smult b q)))"
1699 apply (simp add: pdivmod_def Let_def, safe)
1700 apply (rule div_poly_eq)
1701 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
1702 apply (rule mod_poly_eq)
1703 apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
1706 lemma pdivmod_fold_coeffs [code]:
1707 "pdivmod p q = (if q = 0 then (0, p)
1708 else fold_coeffs (\<lambda>a (s, r).
1709 let b = coeff (pCons a r) (degree q) / coeff q (degree q)
1710 in (pCons b s, pCons a r - smult b q)
1712 apply (cases "q = 0")
1713 apply (simp add: pdivmod_def)
1716 apply (simp_all add: pdivmod_0 pdivmod_pCons)
1717 apply (case_tac "a = 0 \<and> p = 0")
1718 apply (auto simp add: pdivmod_def)
1722 subsection {* Order of polynomial roots *}
1724 definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
1726 "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
1728 lemma coeff_linear_power:
1729 fixes a :: "'a::comm_semiring_1"
1730 shows "coeff ([:a, 1:] ^ n) n = 1"
1731 apply (induct n, simp_all)
1732 apply (subst coeff_eq_0)
1733 apply (auto intro: le_less_trans degree_power_le)
1736 lemma degree_linear_power:
1737 fixes a :: "'a::comm_semiring_1"
1738 shows "degree ([:a, 1:] ^ n) = n"
1739 apply (rule order_antisym)
1740 apply (rule ord_le_eq_trans [OF degree_power_le], simp)
1741 apply (rule le_degree, simp add: coeff_linear_power)
1744 lemma order_1: "[:-a, 1:] ^ order a p dvd p"
1745 apply (cases "p = 0", simp)
1746 apply (cases "order a p", simp)
1747 apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
1748 apply (drule not_less_Least, simp)
1749 apply (fold order_def, simp)
1752 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
1754 apply (rule LeastI_ex)
1755 apply (rule_tac x="degree p" in exI)
1757 apply (drule (1) dvd_imp_degree_le)
1758 apply (simp only: degree_linear_power)
1762 "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
1763 by (rule conjI [OF order_1 order_2])
1766 assumes p: "p \<noteq> 0"
1767 shows "order a p \<le> degree p"
1769 have "order a p = degree ([:-a, 1:] ^ order a p)"
1770 by (simp only: degree_linear_power)
1771 also have "\<dots> \<le> degree p"
1772 using order_1 p by (rule dvd_imp_degree_le)
1773 finally show ?thesis .
1776 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
1777 apply (cases "p = 0", simp_all)
1779 apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)
1780 unfolding poly_eq_0_iff_dvd
1781 apply (metis dvd_power dvd_trans order_1)
1785 subsection {* GCD of polynomials *}
1787 instantiation poly :: (field) gcd
1790 function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
1792 "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
1793 | "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
1796 termination "gcd :: _ poly \<Rightarrow> _"
1797 by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
1798 (auto dest: degree_mod_less)
1800 declare gcd_poly.simps [simp del]
1807 fixes x y :: "_ poly"
1808 shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
1809 and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
1810 apply (induct x y rule: gcd_poly.induct)
1811 apply (simp_all add: gcd_poly.simps)
1812 apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
1813 apply (blast dest: dvd_mod_imp_dvd)
1816 lemma poly_gcd_greatest:
1817 fixes k x y :: "_ poly"
1818 shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
1819 by (induct x y rule: gcd_poly.induct)
1820 (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
1822 lemma dvd_poly_gcd_iff [iff]:
1823 fixes k x y :: "_ poly"
1824 shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
1825 by (blast intro!: poly_gcd_greatest intro: dvd_trans)
1827 lemma poly_gcd_monic:
1828 fixes x y :: "_ poly"
1829 shows "coeff (gcd x y) (degree (gcd x y)) =
1830 (if x = 0 \<and> y = 0 then 0 else 1)"
1831 by (induct x y rule: gcd_poly.induct)
1832 (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
1834 lemma poly_gcd_zero_iff [simp]:
1835 fixes x y :: "_ poly"
1836 shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
1837 by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
1839 lemma poly_gcd_0_0 [simp]:
1840 "gcd (0::_ poly) 0 = 0"
1843 lemma poly_dvd_antisym:
1844 fixes p q :: "'a::idom poly"
1845 assumes coeff: "coeff p (degree p) = coeff q (degree q)"
1846 assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
1847 proof (cases "p = 0")
1848 case True with coeff show "p = q" by simp
1850 case False with coeff have "q \<noteq> 0" by auto
1851 have degree: "degree p = degree q"
1852 using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
1853 by (intro order_antisym dvd_imp_degree_le)
1855 from `p dvd q` obtain a where a: "q = p * a" ..
1856 with `q \<noteq> 0` have "a \<noteq> 0" by auto
1857 with degree a `p \<noteq> 0` have "degree a = 0"
1858 by (simp add: degree_mult_eq)
1859 with coeff a show "p = q"
1860 by (cases a, auto split: if_splits)
1863 lemma poly_gcd_unique:
1864 fixes d x y :: "_ poly"
1865 assumes dvd1: "d dvd x" and dvd2: "d dvd y"
1866 and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
1867 and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
1870 have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
1871 by (simp_all add: poly_gcd_monic monic)
1872 moreover have "gcd x y dvd d"
1873 using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
1874 moreover have "d dvd gcd x y"
1875 using dvd1 dvd2 by (rule poly_gcd_greatest)
1876 ultimately show ?thesis
1877 by (rule poly_dvd_antisym)
1880 interpretation gcd_poly!: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
1882 fix x y z :: "'a poly"
1883 show "gcd (gcd x y) z = gcd x (gcd y z)"
1884 by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
1885 show "gcd x y = gcd y x"
1886 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1889 lemmas poly_gcd_assoc = gcd_poly.assoc
1890 lemmas poly_gcd_commute = gcd_poly.commute
1891 lemmas poly_gcd_left_commute = gcd_poly.left_commute
1893 lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
1895 lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
1896 by (rule poly_gcd_unique) simp_all
1898 lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
1899 by (rule poly_gcd_unique) simp_all
1901 lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
1902 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1904 lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
1905 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1907 lemma poly_gcd_code [code]:
1908 "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
1909 by (simp add: gcd_poly.simps)
1912 subsection {* Composition of polynomials *}
1914 definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
1916 "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
1918 lemma pcompose_0 [simp]:
1920 by (simp add: pcompose_def)
1922 lemma pcompose_pCons:
1923 "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
1924 by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
1926 lemma poly_pcompose:
1927 "poly (pcompose p q) x = poly p (poly q x)"
1928 by (induct p) (simp_all add: pcompose_pCons)
1930 lemma degree_pcompose_le:
1931 "degree (pcompose p q) \<le> degree p * degree q"
1932 apply (induct p, simp)
1933 apply (simp add: pcompose_pCons, clarify)
1934 apply (rule degree_add_le, simp)
1935 apply (rule order_trans [OF degree_mult_le], simp)
1939 no_notation cCons (infixr "##" 65)