src/HOL/Library/Polynomial.thy
 author paulson Thu Apr 03 17:26:04 2014 +0100 (2014-04-03) changeset 56383 8e7052e9fda4 parent 55642 63beb38e9258 child 56544 b60d5d119489 permissions -rw-r--r--
Cleaned up some messy proofs
1 (*  Title:      HOL/Library/Polynomial.thy
2     Author:     Brian Huffman
3     Author:     Clemens Ballarin
4     Author:     Florian Haftmann
5 *)
7 header {* Polynomials as type over a ring structure *}
9 theory Polynomial
10 imports Main GCD
11 begin
13 subsection {* Auxiliary: operations for lists (later) representing coefficients *}
15 definition strip_while :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
16 where
17   "strip_while P = rev \<circ> dropWhile P \<circ> rev"
19 lemma strip_while_Nil [simp]:
20   "strip_while P [] = []"
23 lemma strip_while_append [simp]:
24   "\<not> P x \<Longrightarrow> strip_while P (xs @ [x]) = xs @ [x]"
27 lemma strip_while_append_rec [simp]:
28   "P x \<Longrightarrow> strip_while P (xs @ [x]) = strip_while P xs"
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)"
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:
48   fixes xs :: "'a list"
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"
51 proof (rule that)
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)))"
56   then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
57     by (simp only: rev_is_rev_conv)
58 qed
61 definition nth_default :: "'a \<Rightarrow> 'a list \<Rightarrow> nat \<Rightarrow> 'a"
62 where
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"
69 lemma nth_default_Cons_0 [simp]:
70   "nth_default y (x # xs) 0 = x"
73 lemma nth_default_Cons_Suc [simp]:
74   "nth_default y (x # xs) (Suc n) = nth_default y xs n"
77 lemma nth_default_map_eq:
78   "f y = x \<Longrightarrow> nth_default x (map f xs) n = f (nth_default y xs n)"
81 lemma nth_default_strip_while_eq [simp]:
82   "nth_default x (strip_while (HOL.eq x) xs) n = nth_default x xs n"
83 proof -
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)
87 qed
90 definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
91 where
92   "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
94 lemma cCons_0_Nil_eq [simp]:
95   "0 ## [] = []"
98 lemma cCons_Cons_eq [simp]:
99   "x ## y # ys = x # y # ys"
102 lemma cCons_append_Cons_eq [simp]:
103   "x ## xs @ y # ys = x # xs @ y # ys"
106 lemma cCons_not_0_eq [simp]:
107   "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
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
114 next
115   case True show ?thesis
116   proof (induct xs rule: rev_induct)
117     case Nil with True show ?case by simp
118   next
119     case (snoc y ys) then show ?case
120       by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
121   qed
122 qed
124 lemma tl_cCons [simp]:
125   "tl (x ## xs) = xs"
129 subsection {* Almost everywhere zero functions *}
131 definition almost_everywhere_zero :: "(nat \<Rightarrow> 'a::zero) \<Rightarrow> bool"
132 where
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"
142 proof -
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 .
146 qed
148 lemma almost_everywhere_zero_case_nat:
149   assumes "almost_everywhere_zero f"
150   shows "almost_everywhere_zero (case_nat a f)"
151   using assms
152   by (auto intro!: almost_everywhere_zeroI elim!: almost_everywhere_zeroE split: nat.split)
153     blast
155 lemma almost_everywhere_zero_Suc:
156   assumes "almost_everywhere_zero f"
157   shows "almost_everywhere_zero (\<lambda>n. f (Suc n))"
158 proof -
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)
162 qed
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"
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"
187 where
188   "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
190 lemma coeff_eq_0:
191   assumes "degree p < n"
192   shows "coeff p n = 0"
193 proof -
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
199 qed
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
214 begin
216 lift_definition zero_poly :: "'a poly"
217   is "\<lambda>_. 0" by (rule almost_everywhere_zeroI) simp
219 instance ..
221 end
223 lemma coeff_0 [simp]:
224   "coeff 0 n = 0"
225   by transfer rule
227 lemma degree_0 [simp]:
228   "degree 0 = 0"
229   by (rule order_antisym [OF degree_le le0]) simp
232   assumes "p \<noteq> 0"
233   shows "coeff p (degree p) \<noteq> 0"
234 proof (cases "degree p")
235   case 0
236   from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
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
242 next
243   case (Suc n)
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
251 qed
254   "coeff p (degree p) = 0 \<longleftrightarrow> p = 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"
268   by transfer simp
270 lemma coeff_pCons_Suc [simp]:
271   "coeff (pCons a p) (Suc n) = coeff p n"
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)
282   done
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)
288   done
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)
297   done
299 lemma pCons_0_0 [simp]:
300   "pCons 0 0 = 0"
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"
305 proof safe
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
309 next
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)
314 qed
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"
322 proof
323   show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
324     by transfer
325       (simp add: Abs_poly_inverse almost_everywhere_zero_Suc fun_eq_iff split: nat.split)
326 qed
328 lemma pCons_induct [case_names 0 pCons, induct type: poly]:
329   assumes zero: "P 0"
330   assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"
331   shows "P p"
332 proof (induct p rule: measure_induct_rule [where f=degree])
333   case (less p)
334   obtain a q where "p = pCons a q" by (rule pCons_cases)
335   have "P q"
336   proof (cases "q = 0")
337     case True
338     then show "P q" by (simp add: zero)
339   next
340     case False
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
345     then show "P q"
346       by (rule less.hyps)
347   qed
348   have "P (pCons a q)"
349   proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")
350     case True
351     with `P q` show ?thesis by (auto intro: pCons)
352   next
353     case False
354     with zero show ?thesis by simp
355   qed
356   then show ?case
357     using `p = pCons a q` by simp
358 qed
361 subsection {* List-style syntax for polynomials *}
363 syntax
364   "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
366 translations
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"
375 where
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
383 lemma Poly_eq_0:
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"
388 where
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"
395 lemma not_0_coeffs_not_Nil:
396   "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
397   by simp
399 lemma coeffs_0_eq_Nil [simp]:
400   "coeffs 0 = []"
401   by simp
403 lemma coeffs_pCons_eq_cCons [simp]:
404   "coeffs (pCons a p) = a ## coeffs p"
405 proof -
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)) }
410   note * = this
411   show ?thesis
412     by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt One_nat_def del: upt_Suc)
413 qed
415 lemma not_0_cCons_eq [simp]:
416   "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
419 lemma Poly_coeffs [simp, code abstype]:
420   "Poly (coeffs p) = p"
421   by (induct p) auto
423 lemma coeffs_Poly [simp]:
424   "coeffs (Poly as) = strip_while (HOL.eq 0) as"
425 proof (induct as)
426   case Nil then show ?case by simp
427 next
428   case (Cons a as)
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
432 qed
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)
442 lemma coeffs_eq_iff:
443   "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
444 proof
445   assume ?P then show ?Q by simp
446 next
447   assume ?Q
448   then have "Poly (coeffs p) = Poly (coeffs q)" by simp
449   then show ?P by simp
450 qed
452 lemma coeff_Poly_eq:
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])
461 lemma [code]:
462   "coeff p = nth_default 0 (coeffs p)"
465 lemma coeffs_eqI:
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"
469 proof -
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)
472 qed
474 lemma degree_eq_length_coeffs [code]:
475   "degree p = length (coeffs p) - 1"
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]:
483   "coeffs 0 = []"
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
491 begin
493 definition
494   [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
496 instance proof
497 qed (simp add: equal equal_poly_def coeffs_eq_iff)
499 end
501 lemma [code nbe]:
502   "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
503   by (fact equal_refl)
505 definition is_zero :: "'a::zero poly \<Rightarrow> bool"
506 where
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"
517 where
518   "fold_coeffs f p = foldr f (coeffs p)"
520 lemma fold_coeffs_0_eq [simp]:
521   "fold_coeffs f 0 = id"
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"
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"
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"
541 subsection {* Canonical morphism on polynomials -- evaluation *}
543 definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
544 where
545   "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" -- {* The Horner Schema *}
547 lemma poly_0 [simp]:
548   "poly 0 x = 0"
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)"
564   by transfer rule
566 lemma monom_0:
567   "monom a 0 = pCons a 0"
568   by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
570 lemma monom_Suc:
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"
580 lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
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)
589   done
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)
599 lemma poly_monom:
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 *}
609 begin
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"
618 qed
621   "coeff (p + q) n = coeff p n + coeff q n"
624 instance proof
625   fix p q r :: "'a poly"
626   show "(p + q) + r = p + (q + r)"
628   show "p + q = q + p"
630   show "0 + p = p"
632 qed
634 end
637 proof
638   fix p q r :: "'a poly"
639   assume "p + q = p + r" thus "q = r"
641 qed
644 begin
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"
653 qed
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"
662 qed
664 lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
667 lemma coeff_diff [simp]:
668   "coeff (p - q) n = coeff p n - coeff q n"
671 instance proof
672   fix p q :: "'a poly"
673   show "- p + p = 0"
675   show "p - q = p + - q"
677 qed
679 end
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)
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)
705   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
706   apply (cases "q = 0", simp)
707   apply (rule order_antisym)
709   apply (rule le_degree)
711   done
714   "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
715   using degree_add_eq_right [of q p]
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"]
723   by simp
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"
749 where
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)"
756 proof -
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)
761     qed simp_all }
762   note * = this
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
769     qed simp_all }
770   note ** = this
771   show ?thesis
772     apply (rule coeffs_eqI)
773     apply (simp add: * nth_default_coeffs_eq)
774     apply (rule **)
775     apply (auto dest: last_coeffs_not_0)
776     done
777 qed
779 lemma coeffs_uminus [code abstract]:
780   "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
781   by (rule coeffs_eqI)
782     (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
784 lemma [code]:
785   fixes p q :: "'a::ab_group_add poly"
786   shows "p - q = p + - q"
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)
792   done
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"
817 qed
819 lemma coeff_smult [simp]:
820   "coeff (smult a p) n = a * coeff p n"
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)
839   "smult a (p + q) = smult a p + smult a q"
840   by (rule poly_eqI, simp add: algebra_simps)
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 =
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"
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"
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))"
886   by (rule coeffs_eqI)
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
890 begin
892 definition
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"
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)"
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)"
915 lemma mult_smult_right [simp]:
916   "p * smult a q = smult a (p * q)"
920   fixes p q r :: "'a poly"
921   shows "(p + q) * r = p * r + q * r"
924 instance proof
925   fix p q r :: "'a poly"
926   show 0: "0 * p = 0"
927     by (rule mult_poly_0_left)
928   show "p * 0 = 0"
929     by (rule mult_poly_0_right)
930   show "(p + q) * r = p * r + q * r"
932   show "(p * q) * r = p * (q * r)"
934   show "p * q = q * p"
935     by (induct p, simp add: mult_poly_0, simp)
936 qed
938 end
940 instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
942 lemma coeff_mult:
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
946 next
947   case (pCons a p n) thus ?case
948     by (cases n, simp, simp add: setsum_atMost_Suc_shift
949                             del: setsum_atMost_Suc)
950 qed
952 lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
953 apply (rule degree_le)
954 apply (induct p)
955 apply simp
956 apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
957 done
959 lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
962 instantiation poly :: (comm_semiring_1) comm_semiring_1
963 begin
965 definition one_poly_def:
966   "1 = pCons 1 0"
968 instance proof
969   fix p :: "'a poly" show "1 * p = p"
970     unfolding one_poly_def by simp
971 next
972   show "0 \<noteq> (1::'a poly)"
973     unfolding one_poly_def by simp
974 qed
976 end
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]:
993   "coeffs 1 = [1]"
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]:
1009   "poly 1 x = 1"
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"
1021 proof -
1022   assume "p dvd 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" ..
1026 qed
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"
1040 proof -
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" ..
1045 qed
1047 lemma smult_dvd:
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
1066 proof
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)
1076 qed
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])
1083 done
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"
1094 where
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)
1107   done
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 mult_pos_pos)
1113   apply auto
1114   done
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)"
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")
1125 proof
1126   assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
1127 next
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)
1131 qed
1133 instantiation poly :: (linordered_idom) linordered_idom
1134 begin
1136 definition
1137   "x < y \<longleftrightarrow> pos_poly (y - x)"
1139 definition
1140   "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
1142 definition
1143   "abs (x::'a poly) = (if x < 0 then - x else x)"
1145 definition
1146   "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
1148 instance proof
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
1152     apply safe
1153     apply simp
1155     apply simp
1156     done
1157 next
1158   fix x :: "'a poly" show "x \<le> x"
1159     unfolding less_eq_poly_def by simp
1160 next
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
1164     apply safe
1167     done
1168 next
1169   fix x y :: "'a poly"
1170   assume "x \<le> y" and "y \<le> x" thus "x = y"
1171     unfolding less_eq_poly_def
1172     apply safe
1174     apply simp
1175     done
1176 next
1177   fix x y z :: "'a poly"
1178   assume "x \<le> y" thus "z + x \<le> z + y"
1179     unfolding less_eq_poly_def
1180     apply safe
1182     done
1183 next
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"]
1188     by auto
1189 next
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)
1195 next
1196   fix x :: "'a poly"
1197   show "\<bar>x\<bar> = (if x < 0 then - x else x)"
1198     by (rule abs_poly_def)
1199 next
1200   fix x :: "'a poly"
1201   show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
1202     by (rule sgn_poly_def)
1203 qed
1205 end
1207 text {* TODO: Simplification rules for comparisons *}
1210 subsection {* Synthetic division and polynomial roots *}
1212 text {*
1213   Synthetic division is simply division by the linear polynomial @{term "x - c"}.
1214 *}
1216 definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
1217 where
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"
1221 where
1222   "synthetic_div p c = fst (synthetic_divmod p c)"
1224 lemma synthetic_divmod_0 [simp]:
1225   "synthetic_divmod 0 c = (0, 0)"
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)
1265 done
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]
1273 lemma poly_eq_0_iff_dvd:
1274   fixes c :: "'a::idom"
1275   shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
1276 proof
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" ..
1281 next
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
1285 qed
1287 lemma dvd_iff_poly_eq_0:
1288   fixes c :: "'a::idom"
1289   shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
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)
1296   case (0 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
1300 next
1301   case (Suc n p)
1302   show "finite {x. poly p x = 0}"
1303   proof (cases "\<exists>x. poly p x = 0")
1304     case False
1305     then show "finite {x. poly p x = 0}" by simp
1306   next
1307     case True
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}"
1319                del: mult_pCons_left)
1320   qed
1321 qed
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")
1326 proof
1327   assume ?Q then show ?P by simp
1328 next
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)
1334       done
1335   } note this [of "p - q"]
1336   moreover assume ?P
1337   ultimately show ?Q by auto
1338 qed
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"
1349 where
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"
1364 proof (cases n)
1365   case 0
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 ..
1370 next
1371   case (Suc m)
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"
1379     by (rule degree_le)
1380   then have "degree p < n"
1381     using `n = Suc m` by (simp add: less_Suc_eq_le)
1382   then show ?thesis ..
1383 qed
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")
1391 proof -
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"
1396     using b x by simp
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"
1403         using r by auto
1404       show "degree (smult b y) \<le> degree y"
1405         by (rule degree_smult_le)
1406     qed
1407   next
1408     show "coeff ?r (degree y) = 0"
1409       using `y \<noteq> 0` unfolding b by simp
1410   qed
1412   from 1 2 show ?thesis
1413     unfolding pdivmod_rel_def
1414     using `y \<noteq> 0` by simp
1415 qed
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)
1420 apply (induct x)
1421 apply (fast intro!: pdivmod_rel_0)
1422 apply (fast intro!: pdivmod_rel_pCons)
1423 done
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
1432 next
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"
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"
1444   proof (rule ccontr)
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)"
1452       using q3 by simp
1453     finally have "degree (r2 - r1) < degree (r2 - r1)" .
1454     then show "False" by simp
1455   qed
1456 qed
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
1469 begin
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)"
1477 lemma div_poly_eq:
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)
1482 lemma mod_poly_eq:
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)
1487 lemma pdivmod_rel:
1488   "pdivmod_rel x y (x div y) (x mod y)"
1489 proof -
1490   from pdivmod_rel_exists
1491     obtain q r where "pdivmod_rel x y q r" by fast
1492   thus ?thesis
1493     by (simp add: div_poly_eq mod_poly_eq)
1494 qed
1496 instance proof
1497   fix x y :: "'a poly"
1498   show "x div y * y + x mod y = x"
1499     using pdivmod_rel [of x y]
1501 next
1502   fix x :: "'a poly"
1503   have "pdivmod_rel x 0 0 x"
1504     by (rule pdivmod_rel_by_0)
1505   thus "x div 0 = 0"
1506     by (rule div_poly_eq)
1507 next
1508   fix y :: "'a poly"
1509   have "pdivmod_rel 0 y 0 0"
1510     by (rule pdivmod_rel_0)
1511   thus "0 div y = 0"
1512     by (rule div_poly_eq)
1513 next
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)
1521 next
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
1535   next
1536     case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
1537     with `x \<noteq> 0`
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)
1544   qed
1545 qed
1547 end
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"
1555 proof -
1556   assume "degree x < degree y"
1557   hence "pdivmod_rel x y 0 x"
1559   thus "x div y = 0" by (rule div_poly_eq)
1560 qed
1562 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
1563 proof -
1564   assume "degree x < degree y"
1565   hence "pdivmod_rel x y 0 x"
1567   thus "x mod y = x" by (rule mod_poly_eq)
1568 qed
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)"
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_smult_right:
1592   "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
1593     \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
1594   unfolding pdivmod_rel_def by simp
1596 lemma div_smult_right:
1597   "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
1598   by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
1600 lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
1601   by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
1603 lemma poly_div_minus_right [simp]:
1604   fixes x y :: "'a::field poly"
1605   shows "x div (- y) = - (x div y)"
1606   using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)
1608 lemma poly_mod_minus_right [simp]:
1609   fixes x y :: "'a::field poly"
1610   shows "x mod (- y) = x mod y"
1611   using mod_smult_right [of "- 1::'a"] by simp
1613 lemma pdivmod_rel_mult:
1614   "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
1615     \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
1616 apply (cases "z = 0", simp add: pdivmod_rel_def)
1617 apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
1618 apply (cases "r = 0")
1619 apply (cases "r' = 0")
1621 apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)
1622 apply (cases "r' = 0")
1623 apply (simp add: pdivmod_rel_def degree_mult_eq)
1624 apply (simp add: pdivmod_rel_def field_simps)
1626 done
1628 lemma poly_div_mult_right:
1629   fixes x y z :: "'a::field poly"
1630   shows "x div (y * z) = (x div y) div z"
1631   by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
1633 lemma poly_mod_mult_right:
1634   fixes x y z :: "'a::field poly"
1635   shows "x mod (y * z) = y * (x div y mod z) + x mod y"
1636   by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
1638 lemma mod_pCons:
1639   fixes a and x
1640   assumes y: "y \<noteq> 0"
1641   defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
1642   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
1643 unfolding b
1644 apply (rule mod_poly_eq)
1645 apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
1646 done
1648 definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
1649 where
1650   "pdivmod p q = (p div q, p mod q)"
1652 lemma div_poly_code [code]:
1653   "p div q = fst (pdivmod p q)"
1656 lemma mod_poly_code [code]:
1657   "p mod q = snd (pdivmod p q)"
1660 lemma pdivmod_0:
1661   "pdivmod 0 q = (0, 0)"
1664 lemma pdivmod_pCons:
1665   "pdivmod (pCons a p) q =
1666     (if q = 0 then (0, pCons a p) else
1667       (let (s, r) = pdivmod p q;
1668            b = coeff (pCons a r) (degree q) / coeff q (degree q)
1669         in (pCons b s, pCons a r - smult b q)))"
1670   apply (simp add: pdivmod_def Let_def, safe)
1671   apply (rule div_poly_eq)
1672   apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
1673   apply (rule mod_poly_eq)
1674   apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
1675   done
1677 lemma pdivmod_fold_coeffs [code]:
1678   "pdivmod p q = (if q = 0 then (0, p)
1679     else fold_coeffs (\<lambda>a (s, r).
1680       let b = coeff (pCons a r) (degree q) / coeff q (degree q)
1681       in (pCons b s, pCons a r - smult b q)
1682    ) p (0, 0))"
1683   apply (cases "q = 0")
1685   apply (rule sym)
1686   apply (induct p)
1687   apply (simp_all add: pdivmod_0 pdivmod_pCons)
1688   apply (case_tac "a = 0 \<and> p = 0")
1689   apply (auto simp add: pdivmod_def)
1690   done
1693 subsection {* Order of polynomial roots *}
1695 definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
1696 where
1697   "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
1699 lemma coeff_linear_power:
1700   fixes a :: "'a::comm_semiring_1"
1701   shows "coeff ([:a, 1:] ^ n) n = 1"
1702 apply (induct n, simp_all)
1703 apply (subst coeff_eq_0)
1704 apply (auto intro: le_less_trans degree_power_le)
1705 done
1707 lemma degree_linear_power:
1708   fixes a :: "'a::comm_semiring_1"
1709   shows "degree ([:a, 1:] ^ n) = n"
1710 apply (rule order_antisym)
1711 apply (rule ord_le_eq_trans [OF degree_power_le], simp)
1712 apply (rule le_degree, simp add: coeff_linear_power)
1713 done
1715 lemma order_1: "[:-a, 1:] ^ order a p dvd p"
1716 apply (cases "p = 0", simp)
1717 apply (cases "order a p", simp)
1718 apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
1719 apply (drule not_less_Least, simp)
1720 apply (fold order_def, simp)
1721 done
1723 lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
1724 unfolding order_def
1725 apply (rule LeastI_ex)
1726 apply (rule_tac x="degree p" in exI)
1727 apply (rule notI)
1728 apply (drule (1) dvd_imp_degree_le)
1729 apply (simp only: degree_linear_power)
1730 done
1732 lemma order:
1733   "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
1734 by (rule conjI [OF order_1 order_2])
1736 lemma order_degree:
1737   assumes p: "p \<noteq> 0"
1738   shows "order a p \<le> degree p"
1739 proof -
1740   have "order a p = degree ([:-a, 1:] ^ order a p)"
1741     by (simp only: degree_linear_power)
1742   also have "\<dots> \<le> degree p"
1743     using order_1 p by (rule dvd_imp_degree_le)
1744   finally show ?thesis .
1745 qed
1747 lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
1748 apply (cases "p = 0", simp_all)
1749 apply (rule iffI)
1750 apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)
1751 unfolding poly_eq_0_iff_dvd
1752 apply (metis dvd_power dvd_trans order_1)
1753 done
1756 subsection {* GCD of polynomials *}
1758 instantiation poly :: (field) gcd
1759 begin
1761 function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
1762 where
1763   "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
1764 | "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
1765 by auto
1767 termination "gcd :: _ poly \<Rightarrow> _"
1768 by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
1769    (auto dest: degree_mod_less)
1771 declare gcd_poly.simps [simp del]
1773 instance ..
1775 end
1777 lemma
1778   fixes x y :: "_ poly"
1779   shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
1780     and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
1781   apply (induct x y rule: gcd_poly.induct)
1783   apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
1784   apply (blast dest: dvd_mod_imp_dvd)
1785   done
1787 lemma poly_gcd_greatest:
1788   fixes k x y :: "_ poly"
1789   shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
1790   by (induct x y rule: gcd_poly.induct)
1791      (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
1793 lemma dvd_poly_gcd_iff [iff]:
1794   fixes k x y :: "_ poly"
1795   shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
1796   by (blast intro!: poly_gcd_greatest intro: dvd_trans)
1798 lemma poly_gcd_monic:
1799   fixes x y :: "_ poly"
1800   shows "coeff (gcd x y) (degree (gcd x y)) =
1801     (if x = 0 \<and> y = 0 then 0 else 1)"
1802   by (induct x y rule: gcd_poly.induct)
1805 lemma poly_gcd_zero_iff [simp]:
1806   fixes x y :: "_ poly"
1807   shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
1808   by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
1810 lemma poly_gcd_0_0 [simp]:
1811   "gcd (0::_ poly) 0 = 0"
1812   by simp
1814 lemma poly_dvd_antisym:
1815   fixes p q :: "'a::idom poly"
1816   assumes coeff: "coeff p (degree p) = coeff q (degree q)"
1817   assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
1818 proof (cases "p = 0")
1819   case True with coeff show "p = q" by simp
1820 next
1821   case False with coeff have "q \<noteq> 0" by auto
1822   have degree: "degree p = degree q"
1823     using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
1824     by (intro order_antisym dvd_imp_degree_le)
1826   from `p dvd q` obtain a where a: "q = p * a" ..
1827   with `q \<noteq> 0` have "a \<noteq> 0" by auto
1828   with degree a `p \<noteq> 0` have "degree a = 0"
1830   with coeff a show "p = q"
1831     by (cases a, auto split: if_splits)
1832 qed
1834 lemma poly_gcd_unique:
1835   fixes d x y :: "_ poly"
1836   assumes dvd1: "d dvd x" and dvd2: "d dvd y"
1837     and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
1838     and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
1839   shows "gcd x y = d"
1840 proof -
1841   have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
1842     by (simp_all add: poly_gcd_monic monic)
1843   moreover have "gcd x y dvd d"
1844     using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
1845   moreover have "d dvd gcd x y"
1846     using dvd1 dvd2 by (rule poly_gcd_greatest)
1847   ultimately show ?thesis
1848     by (rule poly_dvd_antisym)
1849 qed
1851 interpretation gcd_poly!: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
1852 proof
1853   fix x y z :: "'a poly"
1854   show "gcd (gcd x y) z = gcd x (gcd y z)"
1855     by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
1856   show "gcd x y = gcd y x"
1857     by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1858 qed
1860 lemmas poly_gcd_assoc = gcd_poly.assoc
1861 lemmas poly_gcd_commute = gcd_poly.commute
1862 lemmas poly_gcd_left_commute = gcd_poly.left_commute
1864 lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
1866 lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
1867 by (rule poly_gcd_unique) simp_all
1869 lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
1870 by (rule poly_gcd_unique) simp_all
1872 lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
1873 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1875 lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
1876 by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
1878 lemma poly_gcd_code [code]:
1879   "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
1883 subsection {* Composition of polynomials *}
1885 definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
1886 where
1887   "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
1889 lemma pcompose_0 [simp]:
1890   "pcompose 0 q = 0"
1893 lemma pcompose_pCons:
1894   "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
1895   by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
1897 lemma poly_pcompose:
1898   "poly (pcompose p q) x = poly p (poly q x)"
1899   by (induct p) (simp_all add: pcompose_pCons)
1901 lemma degree_pcompose_le:
1902   "degree (pcompose p q) \<le> degree p * degree q"
1903 apply (induct p, simp)
1904 apply (simp add: pcompose_pCons, clarify)