src/HOL/Library/Polynomial_Factorial.thy
 author haftmann Wed Jan 04 21:28:29 2017 +0100 (2017-01-04) changeset 64786 340db65fd2c1 parent 64784 5cb5e7ecb284 child 64794 6f7391f28197 permissions -rw-r--r--
reworked to provide auxiliary operations Euclidean_Algorithm.* to instantiate gcd etc. for euclidean rings
1 (*  Title:      HOL/Library/Polynomial_Factorial.thy
2     Author:     Brian Huffman
3     Author:     Clemens Ballarin
4     Author:     Amine Chaieb
5     Author:     Florian Haftmann
6     Author:     Manuel Eberl
7 *)
9 theory Polynomial_Factorial
10 imports
11   Complex_Main
12   "~~/src/HOL/Library/Polynomial"
13   "~~/src/HOL/Library/Normalized_Fraction"
14   "~~/src/HOL/Library/Field_as_Ring"
15 begin
17 subsection \<open>Various facts about polynomials\<close>
19 lemma prod_mset_const_poly: "prod_mset (image_mset (\<lambda>x. [:f x:]) A) = [:prod_mset (image_mset f A):]"
20   by (induction A) (simp_all add: one_poly_def mult_ac)
22 lemma is_unit_smult_iff: "smult c p dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1"
23 proof -
24   have "smult c p = [:c:] * p" by simp
25   also have "\<dots> dvd 1 \<longleftrightarrow> c dvd 1 \<and> p dvd 1"
26   proof safe
27     assume A: "[:c:] * p dvd 1"
28     thus "p dvd 1" by (rule dvd_mult_right)
29     from A obtain q where B: "1 = [:c:] * p * q" by (erule dvdE)
30     have "c dvd c * (coeff p 0 * coeff q 0)" by simp
31     also have "\<dots> = coeff ([:c:] * p * q) 0" by (simp add: mult.assoc coeff_mult)
32     also note B [symmetric]
33     finally show "c dvd 1" by simp
34   next
35     assume "c dvd 1" "p dvd 1"
36     from \<open>c dvd 1\<close> obtain d where "1 = c * d" by (erule dvdE)
37     hence "1 = [:c:] * [:d:]" by (simp add: one_poly_def mult_ac)
38     hence "[:c:] dvd 1" by (rule dvdI)
39     from mult_dvd_mono[OF this \<open>p dvd 1\<close>] show "[:c:] * p dvd 1" by simp
40   qed
41   finally show ?thesis .
42 qed
44 lemma degree_mod_less': "b \<noteq> 0 \<Longrightarrow> a mod b \<noteq> 0 \<Longrightarrow> degree (a mod b) < degree b"
45   using degree_mod_less[of b a] by auto
47 lemma smult_eq_iff:
48   assumes "(b :: 'a :: field) \<noteq> 0"
49   shows   "smult a p = smult b q \<longleftrightarrow> smult (a / b) p = q"
50 proof
51   assume "smult a p = smult b q"
52   also from assms have "smult (inverse b) \<dots> = q" by simp
53   finally show "smult (a / b) p = q" by (simp add: field_simps)
54 qed (insert assms, auto)
56 lemma irreducible_const_poly_iff:
57   fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}"
58   shows "irreducible [:c:] \<longleftrightarrow> irreducible c"
59 proof
60   assume A: "irreducible c"
61   show "irreducible [:c:]"
62   proof (rule irreducibleI)
63     fix a b assume ab: "[:c:] = a * b"
64     hence "degree [:c:] = degree (a * b)" by (simp only: )
65     also from A ab have "a \<noteq> 0" "b \<noteq> 0" by auto
66     hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq)
67     finally have "degree a = 0" "degree b = 0" by auto
68     then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE)
69     from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: )
70     hence "c = a' * b'" by (simp add: ab' mult_ac)
71     from A and this have "a' dvd 1 \<or> b' dvd 1" by (rule irreducibleD)
72     with ab' show "a dvd 1 \<or> b dvd 1" by (auto simp: one_poly_def)
73   qed (insert A, auto simp: irreducible_def is_unit_poly_iff)
74 next
75   assume A: "irreducible [:c:]"
76   show "irreducible c"
77   proof (rule irreducibleI)
78     fix a b assume ab: "c = a * b"
79     hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac)
80     from A and this have "[:a:] dvd 1 \<or> [:b:] dvd 1" by (rule irreducibleD)
81     thus "a dvd 1 \<or> b dvd 1" by (simp add: one_poly_def)
82   qed (insert A, auto simp: irreducible_def one_poly_def)
83 qed
86 subsection \<open>Lifting elements into the field of fractions\<close>
88 definition to_fract :: "'a :: idom \<Rightarrow> 'a fract" where "to_fract x = Fract x 1"
89   -- \<open>FIXME: name \<open>of_idom\<close>, abbreviation\<close>
91 lemma to_fract_0 [simp]: "to_fract 0 = 0"
92   by (simp add: to_fract_def eq_fract Zero_fract_def)
94 lemma to_fract_1 [simp]: "to_fract 1 = 1"
95   by (simp add: to_fract_def eq_fract One_fract_def)
97 lemma to_fract_add [simp]: "to_fract (x + y) = to_fract x + to_fract y"
98   by (simp add: to_fract_def)
100 lemma to_fract_diff [simp]: "to_fract (x - y) = to_fract x - to_fract y"
101   by (simp add: to_fract_def)
103 lemma to_fract_uminus [simp]: "to_fract (-x) = -to_fract x"
104   by (simp add: to_fract_def)
106 lemma to_fract_mult [simp]: "to_fract (x * y) = to_fract x * to_fract y"
107   by (simp add: to_fract_def)
109 lemma to_fract_eq_iff [simp]: "to_fract x = to_fract y \<longleftrightarrow> x = y"
110   by (simp add: to_fract_def eq_fract)
112 lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 \<longleftrightarrow> x = 0"
113   by (simp add: to_fract_def Zero_fract_def eq_fract)
115 lemma snd_quot_of_fract_nonzero [simp]: "snd (quot_of_fract x) \<noteq> 0"
116   by transfer simp
118 lemma Fract_quot_of_fract [simp]: "Fract (fst (quot_of_fract x)) (snd (quot_of_fract x)) = x"
119   by transfer (simp del: fractrel_iff, subst fractrel_normalize_quot_left, simp)
121 lemma to_fract_quot_of_fract:
122   assumes "snd (quot_of_fract x) = 1"
123   shows   "to_fract (fst (quot_of_fract x)) = x"
124 proof -
125   have "x = Fract (fst (quot_of_fract x)) (snd (quot_of_fract x))" by simp
126   also note assms
127   finally show ?thesis by (simp add: to_fract_def)
128 qed
130 lemma snd_quot_of_fract_Fract_whole:
131   assumes "y dvd x"
132   shows   "snd (quot_of_fract (Fract x y)) = 1"
133   using assms by transfer (auto simp: normalize_quot_def Let_def gcd_proj2_if_dvd)
135 lemma Fract_conv_to_fract: "Fract a b = to_fract a / to_fract b"
136   by (simp add: to_fract_def)
138 lemma quot_of_fract_to_fract [simp]: "quot_of_fract (to_fract x) = (x, 1)"
139   unfolding to_fract_def by transfer (simp add: normalize_quot_def)
141 lemma fst_quot_of_fract_eq_0_iff [simp]: "fst (quot_of_fract x) = 0 \<longleftrightarrow> x = 0"
142   by transfer simp
144 lemma snd_quot_of_fract_to_fract [simp]: "snd (quot_of_fract (to_fract x)) = 1"
145   unfolding to_fract_def by (rule snd_quot_of_fract_Fract_whole) simp_all
147 lemma coprime_quot_of_fract:
148   "coprime (fst (quot_of_fract x)) (snd (quot_of_fract x))"
149   by transfer (simp add: coprime_normalize_quot)
151 lemma unit_factor_snd_quot_of_fract: "unit_factor (snd (quot_of_fract x)) = 1"
152   using quot_of_fract_in_normalized_fracts[of x]
153   by (simp add: normalized_fracts_def case_prod_unfold)
155 lemma unit_factor_1_imp_normalized: "unit_factor x = 1 \<Longrightarrow> normalize x = x"
156   by (subst (2) normalize_mult_unit_factor [symmetric, of x])
157      (simp del: normalize_mult_unit_factor)
159 lemma normalize_snd_quot_of_fract: "normalize (snd (quot_of_fract x)) = snd (quot_of_fract x)"
160   by (intro unit_factor_1_imp_normalized unit_factor_snd_quot_of_fract)
163 subsection \<open>Content and primitive part of a polynomial\<close>
165 definition content :: "('a :: semiring_Gcd poly) \<Rightarrow> 'a" where
166   "content p = Gcd (set (coeffs p))"
168 lemma content_0 [simp]: "content 0 = 0"
169   by (simp add: content_def)
171 lemma content_1 [simp]: "content 1 = 1"
172   by (simp add: content_def)
174 lemma content_const [simp]: "content [:c:] = normalize c"
175   by (simp add: content_def cCons_def)
177 lemma const_poly_dvd_iff_dvd_content:
178   fixes c :: "'a :: semiring_Gcd"
179   shows "[:c:] dvd p \<longleftrightarrow> c dvd content p"
180 proof (cases "p = 0")
181   case [simp]: False
182   have "[:c:] dvd p \<longleftrightarrow> (\<forall>n. c dvd coeff p n)" by (rule const_poly_dvd_iff)
183   also have "\<dots> \<longleftrightarrow> (\<forall>a\<in>set (coeffs p). c dvd a)"
184   proof safe
185     fix n :: nat assume "\<forall>a\<in>set (coeffs p). c dvd a"
186     thus "c dvd coeff p n"
187       by (cases "n \<le> degree p") (auto simp: coeff_eq_0 coeffs_def split: if_splits)
188   qed (auto simp: coeffs_def simp del: upt_Suc split: if_splits)
189   also have "\<dots> \<longleftrightarrow> c dvd content p"
190     by (simp add: content_def dvd_Gcd_iff mult.commute [of "unit_factor x" for x]
192   finally show ?thesis .
193 qed simp_all
195 lemma content_dvd [simp]: "[:content p:] dvd p"
196   by (subst const_poly_dvd_iff_dvd_content) simp_all
198 lemma content_dvd_coeff [simp]: "content p dvd coeff p n"
199   by (cases "n \<le> degree p")
200      (auto simp: content_def coeffs_def not_le coeff_eq_0 simp del: upt_Suc intro: Gcd_dvd)
202 lemma content_dvd_coeffs: "c \<in> set (coeffs p) \<Longrightarrow> content p dvd c"
203   by (simp add: content_def Gcd_dvd)
205 lemma normalize_content [simp]: "normalize (content p) = content p"
206   by (simp add: content_def)
208 lemma is_unit_content_iff [simp]: "is_unit (content p) \<longleftrightarrow> content p = 1"
209 proof
210   assume "is_unit (content p)"
211   hence "normalize (content p) = 1" by (simp add: is_unit_normalize del: normalize_content)
212   thus "content p = 1" by simp
213 qed auto
215 lemma content_smult [simp]: "content (smult c p) = normalize c * content p"
216   by (simp add: content_def coeffs_smult Gcd_mult)
218 lemma content_eq_zero_iff [simp]: "content p = 0 \<longleftrightarrow> p = 0"
219   by (auto simp: content_def simp: poly_eq_iff coeffs_def)
221 definition primitive_part :: "'a :: {semiring_Gcd,idom_divide} poly \<Rightarrow> 'a poly" where
222   "primitive_part p = (if p = 0 then 0 else map_poly (\<lambda>x. x div content p) p)"
224 lemma primitive_part_0 [simp]: "primitive_part 0 = 0"
225   by (simp add: primitive_part_def)
227 lemma content_times_primitive_part [simp]:
228   fixes p :: "'a :: {idom_divide, semiring_Gcd} poly"
229   shows "smult (content p) (primitive_part p) = p"
230 proof (cases "p = 0")
231   case False
232   thus ?thesis
233   unfolding primitive_part_def
234   by (auto simp: smult_conv_map_poly map_poly_map_poly o_def content_dvd_coeffs
235            intro: map_poly_idI)
236 qed simp_all
238 lemma primitive_part_eq_0_iff [simp]: "primitive_part p = 0 \<longleftrightarrow> p = 0"
239 proof (cases "p = 0")
240   case False
241   hence "primitive_part p = map_poly (\<lambda>x. x div content p) p"
242     by (simp add:  primitive_part_def)
243   also from False have "\<dots> = 0 \<longleftrightarrow> p = 0"
244     by (intro map_poly_eq_0_iff) (auto simp: dvd_div_eq_0_iff content_dvd_coeffs)
245   finally show ?thesis using False by simp
246 qed simp
248 lemma content_primitive_part [simp]:
249   assumes "p \<noteq> 0"
250   shows   "content (primitive_part p) = 1"
251 proof -
252   have "p = smult (content p) (primitive_part p)" by simp
253   also have "content \<dots> = content p * content (primitive_part p)"
254     by (simp del: content_times_primitive_part)
255   finally show ?thesis using assms by simp
256 qed
258 lemma content_decompose:
259   fixes p :: "'a :: semiring_Gcd poly"
260   obtains p' where "p = smult (content p) p'" "content p' = 1"
261 proof (cases "p = 0")
262   case True
263   thus ?thesis by (intro that[of 1]) simp_all
264 next
265   case False
266   from content_dvd[of p] obtain r where r: "p = [:content p:] * r" by (erule dvdE)
267   have "content p * 1 = content p * content r" by (subst r) simp
268   with False have "content r = 1" by (subst (asm) mult_left_cancel) simp_all
269   with r show ?thesis by (intro that[of r]) simp_all
270 qed
272 lemma smult_content_normalize_primitive_part [simp]:
273   "smult (content p) (normalize (primitive_part p)) = normalize p"
274 proof -
275   have "smult (content p) (normalize (primitive_part p)) =
276           normalize ([:content p:] * primitive_part p)"
277     by (subst normalize_mult) (simp_all add: normalize_const_poly)
278   also have "[:content p:] * primitive_part p = p" by simp
279   finally show ?thesis .
280 qed
282 lemma content_dvd_contentI [intro]:
283   "p dvd q \<Longrightarrow> content p dvd content q"
284   using const_poly_dvd_iff_dvd_content content_dvd dvd_trans by blast
286 lemma primitive_part_const_poly [simp]: "primitive_part [:x:] = [:unit_factor x:]"
287   by (simp add: primitive_part_def map_poly_pCons)
289 lemma primitive_part_prim: "content p = 1 \<Longrightarrow> primitive_part p = p"
290   by (auto simp: primitive_part_def)
292 lemma degree_primitive_part [simp]: "degree (primitive_part p) = degree p"
293 proof (cases "p = 0")
294   case False
295   have "p = smult (content p) (primitive_part p)" by simp
296   also from False have "degree \<dots> = degree (primitive_part p)"
297     by (subst degree_smult_eq) simp_all
298   finally show ?thesis ..
299 qed simp_all
302 subsection \<open>Lifting polynomial coefficients to the field of fractions\<close>
304 abbreviation (input) fract_poly
305   where "fract_poly \<equiv> map_poly to_fract"
307 abbreviation (input) unfract_poly
308   where "unfract_poly \<equiv> map_poly (fst \<circ> quot_of_fract)"
310 lemma fract_poly_smult [simp]: "fract_poly (smult c p) = smult (to_fract c) (fract_poly p)"
311   by (simp add: smult_conv_map_poly map_poly_map_poly o_def)
313 lemma fract_poly_0 [simp]: "fract_poly 0 = 0"
314   by (simp add: poly_eqI coeff_map_poly)
316 lemma fract_poly_1 [simp]: "fract_poly 1 = 1"
317   by (simp add: one_poly_def map_poly_pCons)
319 lemma fract_poly_add [simp]:
320   "fract_poly (p + q) = fract_poly p + fract_poly q"
321   by (intro poly_eqI) (simp_all add: coeff_map_poly)
323 lemma fract_poly_diff [simp]:
324   "fract_poly (p - q) = fract_poly p - fract_poly q"
325   by (intro poly_eqI) (simp_all add: coeff_map_poly)
327 lemma to_fract_sum [simp]: "to_fract (sum f A) = sum (\<lambda>x. to_fract (f x)) A"
328   by (cases "finite A", induction A rule: finite_induct) simp_all
330 lemma fract_poly_mult [simp]:
331   "fract_poly (p * q) = fract_poly p * fract_poly q"
332   by (intro poly_eqI) (simp_all add: coeff_map_poly coeff_mult)
334 lemma fract_poly_eq_iff [simp]: "fract_poly p = fract_poly q \<longleftrightarrow> p = q"
335   by (auto simp: poly_eq_iff coeff_map_poly)
337 lemma fract_poly_eq_0_iff [simp]: "fract_poly p = 0 \<longleftrightarrow> p = 0"
338   using fract_poly_eq_iff[of p 0] by (simp del: fract_poly_eq_iff)
340 lemma fract_poly_dvd: "p dvd q \<Longrightarrow> fract_poly p dvd fract_poly q"
341   by (auto elim!: dvdE)
343 lemma prod_mset_fract_poly:
344   "prod_mset (image_mset (\<lambda>x. fract_poly (f x)) A) = fract_poly (prod_mset (image_mset f A))"
345   by (induction A) (simp_all add: mult_ac)
347 lemma is_unit_fract_poly_iff:
348   "p dvd 1 \<longleftrightarrow> fract_poly p dvd 1 \<and> content p = 1"
349 proof safe
350   assume A: "p dvd 1"
351   with fract_poly_dvd[of p 1] show "is_unit (fract_poly p)" by simp
352   from A show "content p = 1"
353     by (auto simp: is_unit_poly_iff normalize_1_iff)
354 next
355   assume A: "fract_poly p dvd 1" and B: "content p = 1"
356   from A obtain c where c: "fract_poly p = [:c:]" by (auto simp: is_unit_poly_iff)
357   {
358     fix n :: nat assume "n > 0"
359     have "to_fract (coeff p n) = coeff (fract_poly p) n" by (simp add: coeff_map_poly)
360     also note c
361     also from \<open>n > 0\<close> have "coeff [:c:] n = 0" by (simp add: coeff_pCons split: nat.splits)
362     finally have "coeff p n = 0" by simp
363   }
364   hence "degree p \<le> 0" by (intro degree_le) simp_all
365   with B show "p dvd 1" by (auto simp: is_unit_poly_iff normalize_1_iff elim!: degree_eq_zeroE)
366 qed
368 lemma fract_poly_is_unit: "p dvd 1 \<Longrightarrow> fract_poly p dvd 1"
369   using fract_poly_dvd[of p 1] by simp
371 lemma fract_poly_smult_eqE:
372   fixes c :: "'a :: {idom_divide,ring_gcd} fract"
373   assumes "fract_poly p = smult c (fract_poly q)"
374   obtains a b
375     where "c = to_fract b / to_fract a" "smult a p = smult b q" "coprime a b" "normalize a = a"
376 proof -
377   define a b where "a = fst (quot_of_fract c)" and "b = snd (quot_of_fract c)"
378   have "smult (to_fract a) (fract_poly q) = smult (to_fract b) (fract_poly p)"
379     by (subst smult_eq_iff) (simp_all add: a_def b_def Fract_conv_to_fract [symmetric] assms)
380   hence "fract_poly (smult a q) = fract_poly (smult b p)" by (simp del: fract_poly_eq_iff)
381   hence "smult b p = smult a q" by (simp only: fract_poly_eq_iff)
382   moreover have "c = to_fract a / to_fract b" "coprime b a" "normalize b = b"
383     by (simp_all add: a_def b_def coprime_quot_of_fract gcd.commute
384           normalize_snd_quot_of_fract Fract_conv_to_fract [symmetric])
385   ultimately show ?thesis by (intro that[of a b])
386 qed
389 subsection \<open>Fractional content\<close>
391 abbreviation (input) Lcm_coeff_denoms
392     :: "'a :: {semiring_Gcd,idom_divide,ring_gcd} fract poly \<Rightarrow> 'a"
393   where "Lcm_coeff_denoms p \<equiv> Lcm (snd ` quot_of_fract ` set (coeffs p))"
395 definition fract_content ::
396       "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a fract" where
397   "fract_content p =
398      (let d = Lcm_coeff_denoms p in Fract (content (unfract_poly (smult (to_fract d) p))) d)"
400 definition primitive_part_fract ::
401       "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a poly" where
402   "primitive_part_fract p =
403      primitive_part (unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p))"
405 lemma primitive_part_fract_0 [simp]: "primitive_part_fract 0 = 0"
406   by (simp add: primitive_part_fract_def)
408 lemma fract_content_eq_0_iff [simp]:
409   "fract_content p = 0 \<longleftrightarrow> p = 0"
410   unfolding fract_content_def Let_def Zero_fract_def
411   by (subst eq_fract) (auto simp: Lcm_0_iff map_poly_eq_0_iff)
413 lemma content_primitive_part_fract [simp]: "p \<noteq> 0 \<Longrightarrow> content (primitive_part_fract p) = 1"
414   unfolding primitive_part_fract_def
415   by (rule content_primitive_part)
416      (auto simp: primitive_part_fract_def map_poly_eq_0_iff Lcm_0_iff)
418 lemma content_times_primitive_part_fract:
419   "smult (fract_content p) (fract_poly (primitive_part_fract p)) = p"
420 proof -
421   define p' where "p' = unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p)"
422   have "fract_poly p' =
423           map_poly (to_fract \<circ> fst \<circ> quot_of_fract) (smult (to_fract (Lcm_coeff_denoms p)) p)"
424     unfolding primitive_part_fract_def p'_def
425     by (subst map_poly_map_poly) (simp_all add: o_assoc)
426   also have "\<dots> = smult (to_fract (Lcm_coeff_denoms p)) p"
427   proof (intro map_poly_idI, unfold o_apply)
428     fix c assume "c \<in> set (coeffs (smult (to_fract (Lcm_coeff_denoms p)) p))"
429     then obtain c' where c: "c' \<in> set (coeffs p)" "c = to_fract (Lcm_coeff_denoms p) * c'"
430       by (auto simp add: Lcm_0_iff coeffs_smult split: if_splits)
431     note c(2)
432     also have "c' = Fract (fst (quot_of_fract c')) (snd (quot_of_fract c'))"
433       by simp
434     also have "to_fract (Lcm_coeff_denoms p) * \<dots> =
435                  Fract (Lcm_coeff_denoms p * fst (quot_of_fract c')) (snd (quot_of_fract c'))"
436       unfolding to_fract_def by (subst mult_fract) simp_all
437     also have "snd (quot_of_fract \<dots>) = 1"
438       by (intro snd_quot_of_fract_Fract_whole dvd_mult2 dvd_Lcm) (insert c(1), auto)
439     finally show "to_fract (fst (quot_of_fract c)) = c"
440       by (rule to_fract_quot_of_fract)
441   qed
442   also have "p' = smult (content p') (primitive_part p')"
443     by (rule content_times_primitive_part [symmetric])
444   also have "primitive_part p' = primitive_part_fract p"
445     by (simp add: primitive_part_fract_def p'_def)
446   also have "fract_poly (smult (content p') (primitive_part_fract p)) =
447                smult (to_fract (content p')) (fract_poly (primitive_part_fract p))" by simp
448   finally have "smult (to_fract (content p')) (fract_poly (primitive_part_fract p)) =
449                       smult (to_fract (Lcm_coeff_denoms p)) p" .
450   thus ?thesis
451     by (subst (asm) smult_eq_iff)
452        (auto simp add: Let_def p'_def Fract_conv_to_fract field_simps Lcm_0_iff fract_content_def)
453 qed
455 lemma fract_content_fract_poly [simp]: "fract_content (fract_poly p) = to_fract (content p)"
456 proof -
457   have "Lcm_coeff_denoms (fract_poly p) = 1"
458     by (auto simp: set_coeffs_map_poly)
459   hence "fract_content (fract_poly p) =
460            to_fract (content (map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p))"
461     by (simp add: fract_content_def to_fract_def fract_collapse map_poly_map_poly del: Lcm_1_iff)
462   also have "map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p = p"
463     by (intro map_poly_idI) simp_all
464   finally show ?thesis .
465 qed
467 lemma content_decompose_fract:
468   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly"
469   obtains c p' where "p = smult c (map_poly to_fract p')" "content p' = 1"
470 proof (cases "p = 0")
471   case True
472   hence "p = smult 0 (map_poly to_fract 1)" "content 1 = 1" by simp_all
473   thus ?thesis ..
474 next
475   case False
476   thus ?thesis
477     by (rule that[OF content_times_primitive_part_fract [symmetric] content_primitive_part_fract])
478 qed
481 subsection \<open>More properties of content and primitive part\<close>
483 lemma lift_prime_elem_poly:
484   assumes "prime_elem (c :: 'a :: semidom)"
485   shows   "prime_elem [:c:]"
486 proof (rule prime_elemI)
487   fix a b assume *: "[:c:] dvd a * b"
488   from * have dvd: "c dvd coeff (a * b) n" for n
489     by (subst (asm) const_poly_dvd_iff) blast
490   {
491     define m where "m = (GREATEST m. \<not>c dvd coeff b m)"
492     assume "\<not>[:c:] dvd b"
493     hence A: "\<exists>i. \<not>c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
494     have B: "\<forall>i. \<not>c dvd coeff b i \<longrightarrow> i < Suc (degree b)"
495       by (auto intro: le_degree simp: less_Suc_eq_le)
496     have coeff_m: "\<not>c dvd coeff b m" unfolding m_def by (rule GreatestI_ex[OF A B])
497     have "i \<le> m" if "\<not>c dvd coeff b i" for i
498       unfolding m_def by (rule Greatest_le[OF that B])
499     hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force
501     have "c dvd coeff a i" for i
502     proof (induction i rule: nat_descend_induct[of "degree a"])
503       case (base i)
504       thus ?case by (simp add: coeff_eq_0)
505     next
506       case (descend i)
507       let ?A = "{..i+m} - {i}"
508       have "c dvd coeff (a * b) (i + m)" by (rule dvd)
509       also have "coeff (a * b) (i + m) = (\<Sum>k\<le>i + m. coeff a k * coeff b (i + m - k))"
510         by (simp add: coeff_mult)
511       also have "{..i+m} = insert i ?A" by auto
512       also have "(\<Sum>k\<in>\<dots>. coeff a k * coeff b (i + m - k)) =
513                    coeff a i * coeff b m + (\<Sum>k\<in>?A. coeff a k * coeff b (i + m - k))"
514         (is "_ = _ + ?S")
515         by (subst sum.insert) simp_all
516       finally have eq: "c dvd coeff a i * coeff b m + ?S" .
517       moreover have "c dvd ?S"
518       proof (rule dvd_sum)
519         fix k assume k: "k \<in> {..i+m} - {i}"
520         show "c dvd coeff a k * coeff b (i + m - k)"
521         proof (cases "k < i")
522           case False
523           with k have "c dvd coeff a k" by (intro descend.IH) simp
524           thus ?thesis by simp
525         next
526           case True
527           hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp
528           thus ?thesis by simp
529         qed
530       qed
531       ultimately have "c dvd coeff a i * coeff b m"
533       with assms coeff_m show "c dvd coeff a i"
534         by (simp add: prime_elem_dvd_mult_iff)
535     qed
536     hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast
537   }
538   thus "[:c:] dvd a \<or> [:c:] dvd b" by blast
539 qed (insert assms, simp_all add: prime_elem_def one_poly_def)
541 lemma prime_elem_const_poly_iff:
542   fixes c :: "'a :: semidom"
543   shows   "prime_elem [:c:] \<longleftrightarrow> prime_elem c"
544 proof
545   assume A: "prime_elem [:c:]"
546   show "prime_elem c"
547   proof (rule prime_elemI)
548     fix a b assume "c dvd a * b"
549     hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac)
550     from A and this have "[:c:] dvd [:a:] \<or> [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
551     thus "c dvd a \<or> c dvd b" by simp
552   qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
553 qed (auto intro: lift_prime_elem_poly)
555 context
556 begin
558 private lemma content_1_mult:
559   fixes f g :: "'a :: {semiring_Gcd,factorial_semiring} poly"
560   assumes "content f = 1" "content g = 1"
561   shows   "content (f * g) = 1"
562 proof (cases "f * g = 0")
563   case False
564   from assms have "f \<noteq> 0" "g \<noteq> 0" by auto
566   hence "f * g \<noteq> 0" by auto
567   {
568     assume "\<not>is_unit (content (f * g))"
569     with False have "\<exists>p. p dvd content (f * g) \<and> prime p"
570       by (intro prime_divisor_exists) simp_all
571     then obtain p where "p dvd content (f * g)" "prime p" by blast
572     from \<open>p dvd content (f * g)\<close> have "[:p:] dvd f * g"
573       by (simp add: const_poly_dvd_iff_dvd_content)
574     moreover from \<open>prime p\<close> have "prime_elem [:p:]" by (simp add: lift_prime_elem_poly)
575     ultimately have "[:p:] dvd f \<or> [:p:] dvd g"
576       by (simp add: prime_elem_dvd_mult_iff)
577     with assms have "is_unit p" by (simp add: const_poly_dvd_iff_dvd_content)
578     with \<open>prime p\<close> have False by simp
579   }
580   hence "is_unit (content (f * g))" by blast
581   hence "normalize (content (f * g)) = 1" by (simp add: is_unit_normalize del: normalize_content)
582   thus ?thesis by simp
583 qed (insert assms, auto)
585 lemma content_mult:
586   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
587   shows "content (p * q) = content p * content q"
588 proof -
589   from content_decompose[of p] guess p' . note p = this
590   from content_decompose[of q] guess q' . note q = this
591   have "content (p * q) = content p * content q * content (p' * q')"
592     by (subst p, subst q) (simp add: mult_ac normalize_mult)
593   also from p q have "content (p' * q') = 1" by (intro content_1_mult)
594   finally show ?thesis by simp
595 qed
597 lemma primitive_part_mult:
598   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
599   shows "primitive_part (p * q) = primitive_part p * primitive_part q"
600 proof -
601   have "primitive_part (p * q) = p * q div [:content (p * q):]"
602     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
603   also have "\<dots> = (p div [:content p:]) * (q div [:content q:])"
604     by (subst div_mult_div_if_dvd) (simp_all add: content_mult mult_ac)
605   also have "\<dots> = primitive_part p * primitive_part q"
606     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
607   finally show ?thesis .
608 qed
610 lemma primitive_part_smult:
611   fixes p :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
612   shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
613 proof -
614   have "smult a p = [:a:] * p" by simp
615   also have "primitive_part \<dots> = smult (unit_factor a) (primitive_part p)"
616     by (subst primitive_part_mult) simp_all
617   finally show ?thesis .
618 qed
620 lemma primitive_part_dvd_primitive_partI [intro]:
621   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
622   shows "p dvd q \<Longrightarrow> primitive_part p dvd primitive_part q"
623   by (auto elim!: dvdE simp: primitive_part_mult)
625 lemma content_prod_mset:
626   fixes A :: "'a :: {factorial_semiring, semiring_Gcd} poly multiset"
627   shows "content (prod_mset A) = prod_mset (image_mset content A)"
628   by (induction A) (simp_all add: content_mult mult_ac)
630 lemma fract_poly_dvdD:
631   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
632   assumes "fract_poly p dvd fract_poly q" "content p = 1"
633   shows   "p dvd q"
634 proof -
635   from assms(1) obtain r where r: "fract_poly q = fract_poly p * r" by (erule dvdE)
636   from content_decompose_fract[of r] guess c r' . note r' = this
637   from r r' have eq: "fract_poly q = smult c (fract_poly (p * r'))" by simp
638   from fract_poly_smult_eqE[OF this] guess a b . note ab = this
639   have "content (smult a q) = content (smult b (p * r'))" by (simp only: ab(2))
640   hence eq': "normalize b = a * content q" by (simp add: assms content_mult r' ab(4))
641   have "1 = gcd a (normalize b)" by (simp add: ab)
642   also note eq'
643   also have "gcd a (a * content q) = a" by (simp add: gcd_proj1_if_dvd ab(4))
644   finally have [simp]: "a = 1" by simp
645   from eq ab have "q = p * ([:b:] * r')" by simp
646   thus ?thesis by (rule dvdI)
647 qed
649 lemma content_prod_eq_1_iff:
650   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
651   shows "content (p * q) = 1 \<longleftrightarrow> content p = 1 \<and> content q = 1"
652 proof safe
653   assume A: "content (p * q) = 1"
654   {
655     fix p q :: "'a poly" assume "content p * content q = 1"
656     hence "1 = content p * content q" by simp
657     hence "content p dvd 1" by (rule dvdI)
658     hence "content p = 1" by simp
659   } note B = this
660   from A B[of p q] B [of q p] show "content p = 1" "content q = 1"
661     by (simp_all add: content_mult mult_ac)
662 qed (auto simp: content_mult)
664 end
667 subsection \<open>Polynomials over a field are a Euclidean ring\<close>
669 definition unit_factor_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
670   "unit_factor_field_poly p = [:lead_coeff p:]"
672 definition normalize_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
673   "normalize_field_poly p = smult (inverse (lead_coeff p)) p"
675 definition euclidean_size_field_poly :: "'a :: field poly \<Rightarrow> nat" where
676   "euclidean_size_field_poly p = (if p = 0 then 0 else 2 ^ degree p)"
678 lemma dvd_field_poly: "dvd.dvd (op * :: 'a :: field poly \<Rightarrow> _) = op dvd"
679   by (intro ext) (simp_all add: dvd.dvd_def dvd_def)
681 interpretation field_poly:
682   unique_euclidean_ring where zero = "0 :: 'a :: field poly"
683     and one = 1 and plus = plus and uminus = uminus and minus = minus
684     and times = times
685     and normalize = normalize_field_poly and unit_factor = unit_factor_field_poly
686     and euclidean_size = euclidean_size_field_poly
687     and uniqueness_constraint = top
688     and divide = divide and modulo = modulo
689 proof (standard, unfold dvd_field_poly)
690   fix p :: "'a poly"
691   show "unit_factor_field_poly p * normalize_field_poly p = p"
692     by (cases "p = 0")
694 next
695   fix p :: "'a poly" assume "is_unit p"
696   thus "normalize_field_poly p = 1"
697     by (elim is_unit_polyE) (auto simp: normalize_field_poly_def monom_0 one_poly_def field_simps)
698 next
699   fix p :: "'a poly" assume "p \<noteq> 0"
700   thus "is_unit (unit_factor_field_poly p)"
701     by (simp add: unit_factor_field_poly_def lead_coeff_nonzero is_unit_pCons_iff)
702 next
703   fix p q s :: "'a poly" assume "s \<noteq> 0"
704   moreover assume "euclidean_size_field_poly p < euclidean_size_field_poly q"
705   ultimately show "euclidean_size_field_poly (p * s) < euclidean_size_field_poly (q * s)"
706     by (auto simp add: euclidean_size_field_poly_def degree_mult_eq)
707 next
708   fix p q r :: "'a poly" assume "p \<noteq> 0"
709   moreover assume "euclidean_size_field_poly r < euclidean_size_field_poly p"
710   ultimately show "(q * p + r) div p = q"
711     by (cases "r = 0")
712       (auto simp add: unit_factor_field_poly_def euclidean_size_field_poly_def div_poly_less)
713 qed (auto simp: unit_factor_field_poly_def normalize_field_poly_def lead_coeff_mult
714        euclidean_size_field_poly_def Rings.div_mult_mod_eq intro!: degree_mod_less' degree_mult_right_le)
716 lemma field_poly_irreducible_imp_prime:
717   assumes "irreducible (p :: 'a :: field poly)"
718   shows   "prime_elem p"
719 proof -
720   have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
721   from field_poly.irreducible_imp_prime_elem[of p] assms
722     show ?thesis unfolding irreducible_def prime_elem_def dvd_field_poly
723       comm_semiring_1.irreducible_def[OF A] comm_semiring_1.prime_elem_def[OF A] by blast
724 qed
726 lemma field_poly_prod_mset_prime_factorization:
727   assumes "(x :: 'a :: field poly) \<noteq> 0"
728   shows   "prod_mset (field_poly.prime_factorization x) = normalize_field_poly x"
729 proof -
730   have A: "class.comm_monoid_mult op * (1 :: 'a poly)" ..
731   have "comm_monoid_mult.prod_mset op * (1 :: 'a poly) = prod_mset"
732     by (intro ext) (simp add: comm_monoid_mult.prod_mset_def[OF A] prod_mset_def)
733   with field_poly.prod_mset_prime_factorization[OF assms] show ?thesis by simp
734 qed
736 lemma field_poly_in_prime_factorization_imp_prime:
737   assumes "(p :: 'a :: field poly) \<in># field_poly.prime_factorization x"
738   shows   "prime_elem p"
739 proof -
740   have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
741   have B: "class.normalization_semidom op div op + op - (0 :: 'a poly) op * 1
742              normalize_field_poly unit_factor_field_poly" ..
743   from field_poly.in_prime_factors_imp_prime [of p x] assms
744     show ?thesis unfolding prime_elem_def dvd_field_poly
745       comm_semiring_1.prime_elem_def[OF A] normalization_semidom.prime_def[OF B] by blast
746 qed
749 subsection \<open>Primality and irreducibility in polynomial rings\<close>
751 lemma nonconst_poly_irreducible_iff:
752   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
753   assumes "degree p \<noteq> 0"
754   shows   "irreducible p \<longleftrightarrow> irreducible (fract_poly p) \<and> content p = 1"
755 proof safe
756   assume p: "irreducible p"
758   from content_decompose[of p] guess p' . note p' = this
759   hence "p = [:content p:] * p'" by simp
760   from p this have "[:content p:] dvd 1 \<or> p' dvd 1" by (rule irreducibleD)
761   moreover have "\<not>p' dvd 1"
762   proof
763     assume "p' dvd 1"
764     hence "degree p = 0" by (subst p') (auto simp: is_unit_poly_iff)
765     with assms show False by contradiction
766   qed
767   ultimately show [simp]: "content p = 1" by (simp add: is_unit_const_poly_iff)
769   show "irreducible (map_poly to_fract p)"
770   proof (rule irreducibleI)
771     have "fract_poly p = 0 \<longleftrightarrow> p = 0" by (intro map_poly_eq_0_iff) auto
772     with assms show "map_poly to_fract p \<noteq> 0" by auto
773   next
774     show "\<not>is_unit (fract_poly p)"
775     proof
776       assume "is_unit (map_poly to_fract p)"
777       hence "degree (map_poly to_fract p) = 0"
778         by (auto simp: is_unit_poly_iff)
779       hence "degree p = 0" by (simp add: degree_map_poly)
780       with assms show False by contradiction
781    qed
782  next
783    fix q r assume qr: "fract_poly p = q * r"
784    from content_decompose_fract[of q] guess cg q' . note q = this
785    from content_decompose_fract[of r] guess cr r' . note r = this
786    from qr q r p have nz: "cg \<noteq> 0" "cr \<noteq> 0" by auto
787    from qr have eq: "fract_poly p = smult (cr * cg) (fract_poly (q' * r'))"
788      by (simp add: q r)
789    from fract_poly_smult_eqE[OF this] guess a b . note ab = this
790    hence "content (smult a p) = content (smult b (q' * r'))" by (simp only:)
791    with ab(4) have a: "a = normalize b" by (simp add: content_mult q r)
792    hence "normalize b = gcd a b" by simp
793    also from ab(3) have "\<dots> = 1" .
794    finally have "a = 1" "is_unit b" by (simp_all add: a normalize_1_iff)
796    note eq
797    also from ab(1) \<open>a = 1\<close> have "cr * cg = to_fract b" by simp
798    also have "smult \<dots> (fract_poly (q' * r')) = fract_poly (smult b (q' * r'))" by simp
799    finally have "p = ([:b:] * q') * r'" by (simp del: fract_poly_smult)
800    from p and this have "([:b:] * q') dvd 1 \<or> r' dvd 1" by (rule irreducibleD)
801    hence "q' dvd 1 \<or> r' dvd 1" by (auto dest: dvd_mult_right simp del: mult_pCons_left)
802    hence "fract_poly q' dvd 1 \<or> fract_poly r' dvd 1" by (auto simp: fract_poly_is_unit)
803    with q r show "is_unit q \<or> is_unit r"
804      by (auto simp add: is_unit_smult_iff dvd_field_iff nz)
805  qed
807 next
809   assume irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
810   show "irreducible p"
811   proof (rule irreducibleI)
812     from irred show "p \<noteq> 0" by auto
813   next
814     from irred show "\<not>p dvd 1"
815       by (auto simp: irreducible_def dest: fract_poly_is_unit)
816   next
817     fix q r assume qr: "p = q * r"
818     hence "fract_poly p = fract_poly q * fract_poly r" by simp
819     from irred and this have "fract_poly q dvd 1 \<or> fract_poly r dvd 1"
820       by (rule irreducibleD)
821     with primitive qr show "q dvd 1 \<or> r dvd 1"
822       by (auto simp:  content_prod_eq_1_iff is_unit_fract_poly_iff)
823   qed
824 qed
826 context
827 begin
829 private lemma irreducible_imp_prime_poly:
830   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
831   assumes "irreducible p"
832   shows   "prime_elem p"
833 proof (cases "degree p = 0")
834   case True
835   with assms show ?thesis
836     by (auto simp: prime_elem_const_poly_iff irreducible_const_poly_iff
837              intro!: irreducible_imp_prime_elem elim!: degree_eq_zeroE)
838 next
839   case False
840   from assms False have irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
841     by (simp_all add: nonconst_poly_irreducible_iff)
842   from irred have prime: "prime_elem (fract_poly p)" by (rule field_poly_irreducible_imp_prime)
843   show ?thesis
844   proof (rule prime_elemI)
845     fix q r assume "p dvd q * r"
846     hence "fract_poly p dvd fract_poly (q * r)" by (rule fract_poly_dvd)
847     hence "fract_poly p dvd fract_poly q * fract_poly r" by simp
848     from prime and this have "fract_poly p dvd fract_poly q \<or> fract_poly p dvd fract_poly r"
849       by (rule prime_elem_dvd_multD)
850     with primitive show "p dvd q \<or> p dvd r" by (auto dest: fract_poly_dvdD)
851   qed (insert assms, auto simp: irreducible_def)
852 qed
855 lemma degree_primitive_part_fract [simp]:
856   "degree (primitive_part_fract p) = degree p"
857 proof -
858   have "p = smult (fract_content p) (fract_poly (primitive_part_fract p))"
859     by (simp add: content_times_primitive_part_fract)
860   also have "degree \<dots> = degree (primitive_part_fract p)"
861     by (auto simp: degree_map_poly)
862   finally show ?thesis ..
863 qed
865 lemma irreducible_primitive_part_fract:
866   fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
867   assumes "irreducible p"
868   shows   "irreducible (primitive_part_fract p)"
869 proof -
870   from assms have deg: "degree (primitive_part_fract p) \<noteq> 0"
871     by (intro notI)
872        (auto elim!: degree_eq_zeroE simp: irreducible_def is_unit_poly_iff dvd_field_iff)
873   hence [simp]: "p \<noteq> 0" by auto
875   note \<open>irreducible p\<close>
876   also have "p = [:fract_content p:] * fract_poly (primitive_part_fract p)"
877     by (simp add: content_times_primitive_part_fract)
878   also have "irreducible \<dots> \<longleftrightarrow> irreducible (fract_poly (primitive_part_fract p))"
879     by (intro irreducible_mult_unit_left) (simp_all add: is_unit_poly_iff dvd_field_iff)
880   finally show ?thesis using deg
881     by (simp add: nonconst_poly_irreducible_iff)
882 qed
884 lemma prime_elem_primitive_part_fract:
885   fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
886   shows "irreducible p \<Longrightarrow> prime_elem (primitive_part_fract p)"
887   by (intro irreducible_imp_prime_poly irreducible_primitive_part_fract)
889 lemma irreducible_linear_field_poly:
890   fixes a b :: "'a::field"
891   assumes "b \<noteq> 0"
892   shows "irreducible [:a,b:]"
893 proof (rule irreducibleI)
894   fix p q assume pq: "[:a,b:] = p * q"
895   also from pq assms have "degree \<dots> = degree p + degree q"
896     by (intro degree_mult_eq) auto
897   finally have "degree p = 0 \<or> degree q = 0" using assms by auto
898   with assms pq show "is_unit p \<or> is_unit q"
899     by (auto simp: is_unit_const_poly_iff dvd_field_iff elim!: degree_eq_zeroE)
900 qed (insert assms, auto simp: is_unit_poly_iff)
902 lemma prime_elem_linear_field_poly:
903   "(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"
904   by (rule field_poly_irreducible_imp_prime, rule irreducible_linear_field_poly)
906 lemma irreducible_linear_poly:
907   fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
908   shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> irreducible [:a,b:]"
909   by (auto intro!: irreducible_linear_field_poly
910            simp:   nonconst_poly_irreducible_iff content_def map_poly_pCons)
912 lemma prime_elem_linear_poly:
913   fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
914   shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> prime_elem [:a,b:]"
915   by (rule irreducible_imp_prime_poly, rule irreducible_linear_poly)
917 end
920 subsection \<open>Prime factorisation of polynomials\<close>
922 context
923 begin
925 private lemma poly_prime_factorization_exists_content_1:
926   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
927   assumes "p \<noteq> 0" "content p = 1"
928   shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
929 proof -
930   let ?P = "field_poly.prime_factorization (fract_poly p)"
931   define c where "c = prod_mset (image_mset fract_content ?P)"
932   define c' where "c' = c * to_fract (lead_coeff p)"
933   define e where "e = prod_mset (image_mset primitive_part_fract ?P)"
934   define A where "A = image_mset (normalize \<circ> primitive_part_fract) ?P"
935   have "content e = (\<Prod>x\<in>#field_poly.prime_factorization (map_poly to_fract p).
936                       content (primitive_part_fract x))"
937     by (simp add: e_def content_prod_mset multiset.map_comp o_def)
938   also have "image_mset (\<lambda>x. content (primitive_part_fract x)) ?P = image_mset (\<lambda>_. 1) ?P"
939     by (intro image_mset_cong content_primitive_part_fract) auto
940   finally have content_e: "content e = 1"
941     by simp
943   have "fract_poly p = unit_factor_field_poly (fract_poly p) *
944           normalize_field_poly (fract_poly p)" by simp
945   also have "unit_factor_field_poly (fract_poly p) = [:to_fract (lead_coeff p):]"
946     by (simp add: unit_factor_field_poly_def lead_coeff_def monom_0 degree_map_poly coeff_map_poly)
947   also from assms have "normalize_field_poly (fract_poly p) = prod_mset ?P"
948     by (subst field_poly_prod_mset_prime_factorization) simp_all
949   also have "\<dots> = prod_mset (image_mset id ?P)" by simp
950   also have "image_mset id ?P =
951                image_mset (\<lambda>x. [:fract_content x:] * fract_poly (primitive_part_fract x)) ?P"
952     by (intro image_mset_cong) (auto simp: content_times_primitive_part_fract)
953   also have "prod_mset \<dots> = smult c (fract_poly e)"
954     by (subst prod_mset.distrib) (simp_all add: prod_mset_fract_poly prod_mset_const_poly c_def e_def)
955   also have "[:to_fract (lead_coeff p):] * \<dots> = smult c' (fract_poly e)"
956     by (simp add: c'_def)
957   finally have eq: "fract_poly p = smult c' (fract_poly e)" .
958   also obtain b where b: "c' = to_fract b" "is_unit b"
959   proof -
960     from fract_poly_smult_eqE[OF eq] guess a b . note ab = this
961     from ab(2) have "content (smult a p) = content (smult b e)" by (simp only: )
962     with assms content_e have "a = normalize b" by (simp add: ab(4))
963     with ab have ab': "a = 1" "is_unit b" by (simp_all add: normalize_1_iff)
964     with ab ab' have "c' = to_fract b" by auto
965     from this and \<open>is_unit b\<close> show ?thesis by (rule that)
966   qed
967   hence "smult c' (fract_poly e) = fract_poly (smult b e)" by simp
968   finally have "p = smult b e" by (simp only: fract_poly_eq_iff)
969   hence "p = [:b:] * e" by simp
970   with b have "normalize p = normalize e"
971     by (simp only: normalize_mult) (simp add: is_unit_normalize is_unit_poly_iff)
972   also have "normalize e = prod_mset A"
973     by (simp add: multiset.map_comp e_def A_def normalize_prod_mset)
974   finally have "prod_mset A = normalize p" ..
976   have "prime_elem p" if "p \<in># A" for p
977     using that by (auto simp: A_def prime_elem_primitive_part_fract prime_elem_imp_irreducible
978                         dest!: field_poly_in_prime_factorization_imp_prime )
979   from this and \<open>prod_mset A = normalize p\<close> show ?thesis
980     by (intro exI[of _ A]) blast
981 qed
983 lemma poly_prime_factorization_exists:
984   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
985   assumes "p \<noteq> 0"
986   shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
987 proof -
988   define B where "B = image_mset (\<lambda>x. [:x:]) (prime_factorization (content p))"
989   have "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize (primitive_part p)"
990     by (rule poly_prime_factorization_exists_content_1) (insert assms, simp_all)
991   then guess A by (elim exE conjE) note A = this
992   moreover from assms have "prod_mset B = [:content p:]"
993     by (simp add: B_def prod_mset_const_poly prod_mset_prime_factorization)
994   moreover have "\<forall>p. p \<in># B \<longrightarrow> prime_elem p"
995     by (auto simp: B_def intro!: lift_prime_elem_poly dest: in_prime_factors_imp_prime)
996   ultimately show ?thesis by (intro exI[of _ "B + A"]) auto
997 qed
999 end
1002 subsection \<open>Typeclass instances\<close>
1004 instance poly :: (factorial_ring_gcd) factorial_semiring
1005   by standard (rule poly_prime_factorization_exists)
1007 instantiation poly :: (factorial_ring_gcd) factorial_ring_gcd
1008 begin
1010 definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
1011   [code del]: "gcd_poly = gcd_factorial"
1013 definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
1014   [code del]: "lcm_poly = lcm_factorial"
1016 definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
1017  [code del]: "Gcd_poly = Gcd_factorial"
1019 definition Lcm_poly :: "'a poly set \<Rightarrow> 'a poly" where
1020  [code del]: "Lcm_poly = Lcm_factorial"
1022 instance by standard (simp_all add: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)
1024 end
1026 instantiation poly :: ("{field,factorial_ring_gcd}") unique_euclidean_ring
1027 begin
1029 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
1030   where "euclidean_size_poly p = (if p = 0 then 0 else 2 ^ degree p)"
1032 definition uniqueness_constraint_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
1033   where [simp]: "uniqueness_constraint_poly = top"
1035 instance
1036   by standard
1037    (auto simp: euclidean_size_poly_def Rings.div_mult_mod_eq div_poly_less degree_mult_eq intro!: degree_mod_less' degree_mult_right_le
1038     split: if_splits)
1040 end
1042 instance poly :: ("{field,factorial_ring_gcd}") euclidean_ring_gcd
1043   by (rule euclidean_ring_gcd_class.intro, rule factorial_euclidean_semiring_gcdI)
1044     standard
1047 subsection \<open>Polynomial GCD\<close>
1049 lemma gcd_poly_decompose:
1050   fixes p q :: "'a :: factorial_ring_gcd poly"
1051   shows "gcd p q =
1052            smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
1053 proof (rule sym, rule gcdI)
1054   have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
1055           [:content p:] * primitive_part p" by (intro mult_dvd_mono) simp_all
1056   thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd p"
1057     by simp
1058 next
1059   have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
1060           [:content q:] * primitive_part q" by (intro mult_dvd_mono) simp_all
1061   thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd q"
1062     by simp
1063 next
1064   fix d assume "d dvd p" "d dvd q"
1065   hence "[:content d:] * primitive_part d dvd
1066            [:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q)"
1067     by (intro mult_dvd_mono) auto
1068   thus "d dvd smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
1069     by simp
1070 qed (auto simp: normalize_smult)
1073 lemma gcd_poly_pseudo_mod:
1074   fixes p q :: "'a :: factorial_ring_gcd poly"
1075   assumes nz: "q \<noteq> 0" and prim: "content p = 1" "content q = 1"
1076   shows   "gcd p q = gcd q (primitive_part (pseudo_mod p q))"
1077 proof -
1078   define r s where "r = fst (pseudo_divmod p q)" and "s = snd (pseudo_divmod p q)"
1079   define a where "a = [:coeff q (degree q) ^ (Suc (degree p) - degree q):]"
1080   have [simp]: "primitive_part a = unit_factor a"
1081     by (simp add: a_def unit_factor_poly_def unit_factor_power monom_0)
1082   from nz have [simp]: "a \<noteq> 0" by (auto simp: a_def)
1084   have rs: "pseudo_divmod p q = (r, s)" by (simp add: r_def s_def)
1085   have "gcd (q * r + s) q = gcd q s"
1086     using gcd_add_mult[of q r s] by (simp add: gcd.commute add_ac mult_ac)
1087   with pseudo_divmod(1)[OF nz rs]
1088     have "gcd (p * a) q = gcd q s" by (simp add: a_def)
1089   also from prim have "gcd (p * a) q = gcd p q"
1090     by (subst gcd_poly_decompose)
1091        (auto simp: primitive_part_mult gcd_mult_unit1 primitive_part_prim
1092              simp del: mult_pCons_right )
1093   also from prim have "gcd q s = gcd q (primitive_part s)"
1094     by (subst gcd_poly_decompose) (simp_all add: primitive_part_prim)
1095   also have "s = pseudo_mod p q" by (simp add: s_def pseudo_mod_def)
1096   finally show ?thesis .
1097 qed
1099 lemma degree_pseudo_mod_less:
1100   assumes "q \<noteq> 0" "pseudo_mod p q \<noteq> 0"
1101   shows   "degree (pseudo_mod p q) < degree q"
1102   using pseudo_mod(2)[of q p] assms by auto
1104 function gcd_poly_code_aux :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
1105   "gcd_poly_code_aux p q =
1106      (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
1107 by auto
1108 termination
1109   by (relation "measure ((\<lambda>p. if p = 0 then 0 else Suc (degree p)) \<circ> snd)")
1110      (auto simp: degree_pseudo_mod_less)
1112 declare gcd_poly_code_aux.simps [simp del]
1114 lemma gcd_poly_code_aux_correct:
1115   assumes "content p = 1" "q = 0 \<or> content q = 1"
1116   shows   "gcd_poly_code_aux p q = gcd p q"
1117   using assms
1118 proof (induction p q rule: gcd_poly_code_aux.induct)
1119   case (1 p q)
1120   show ?case
1121   proof (cases "q = 0")
1122     case True
1123     thus ?thesis by (subst gcd_poly_code_aux.simps) auto
1124   next
1125     case False
1126     hence "gcd_poly_code_aux p q = gcd_poly_code_aux q (primitive_part (pseudo_mod p q))"
1127       by (subst gcd_poly_code_aux.simps) simp_all
1128     also from "1.prems" False
1129       have "primitive_part (pseudo_mod p q) = 0 \<or>
1130               content (primitive_part (pseudo_mod p q)) = 1"
1131       by (cases "pseudo_mod p q = 0") auto
1132     with "1.prems" False
1133       have "gcd_poly_code_aux q (primitive_part (pseudo_mod p q)) =
1134               gcd q (primitive_part (pseudo_mod p q))"
1135       by (intro 1) simp_all
1136     also from "1.prems" False
1137       have "\<dots> = gcd p q" by (intro gcd_poly_pseudo_mod [symmetric]) auto
1138     finally show ?thesis .
1139   qed
1140 qed
1142 definition gcd_poly_code
1143     :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
1144   where "gcd_poly_code p q =
1145            (if p = 0 then normalize q else if q = 0 then normalize p else
1146               smult (gcd (content p) (content q))
1147                 (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
1149 lemma gcd_poly_code [code]: "gcd p q = gcd_poly_code p q"
1150   by (simp add: gcd_poly_code_def gcd_poly_code_aux_correct gcd_poly_decompose [symmetric])
1152 lemma lcm_poly_code [code]:
1153   fixes p q :: "'a :: factorial_ring_gcd poly"
1154   shows "lcm p q = normalize (p * q) div gcd p q"
1155   by (fact lcm_gcd)
1157 declare Gcd_set
1158   [where ?'a = "'a :: factorial_ring_gcd poly", code]
1160 declare Lcm_set
1161   [where ?'a = "'a :: factorial_ring_gcd poly", code]
1163 text \<open>Example:
1164   @{lemma "Lcm {[:1, 2, 3:], [:2, 3, 4:]} = [:[:2:], [:7:], [:16:], [:17:], [:12 :: int:]:]" by eval}
1165 \<close>
1167 end