src/HOL/Library/Polynomial_Factorial.thy
 author wenzelm Wed Mar 08 10:50:59 2017 +0100 (2017-03-08) changeset 65151 a7394aa4d21c parent 64911 f0e07600de47 child 65366 10ca63a18e56 permissions -rw-r--r--
tuned proofs;
```     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 *)
```
```     8
```
```     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
```
```    16
```
```    17 subsection \<open>Various facts about polynomials\<close>
```
```    18
```
```    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)
```
```    21
```
```    22 lemma irreducible_const_poly_iff:
```
```    23   fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}"
```
```    24   shows "irreducible [:c:] \<longleftrightarrow> irreducible c"
```
```    25 proof
```
```    26   assume A: "irreducible c"
```
```    27   show "irreducible [:c:]"
```
```    28   proof (rule irreducibleI)
```
```    29     fix a b assume ab: "[:c:] = a * b"
```
```    30     hence "degree [:c:] = degree (a * b)" by (simp only: )
```
```    31     also from A ab have "a \<noteq> 0" "b \<noteq> 0" by auto
```
```    32     hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq)
```
```    33     finally have "degree a = 0" "degree b = 0" by auto
```
```    34     then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE)
```
```    35     from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: )
```
```    36     hence "c = a' * b'" by (simp add: ab' mult_ac)
```
```    37     from A and this have "a' dvd 1 \<or> b' dvd 1" by (rule irreducibleD)
```
```    38     with ab' show "a dvd 1 \<or> b dvd 1" by (auto simp: one_poly_def)
```
```    39   qed (insert A, auto simp: irreducible_def is_unit_poly_iff)
```
```    40 next
```
```    41   assume A: "irreducible [:c:]"
```
```    42   show "irreducible c"
```
```    43   proof (rule irreducibleI)
```
```    44     fix a b assume ab: "c = a * b"
```
```    45     hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac)
```
```    46     from A and this have "[:a:] dvd 1 \<or> [:b:] dvd 1" by (rule irreducibleD)
```
```    47     thus "a dvd 1 \<or> b dvd 1" by (simp add: one_poly_def)
```
```    48   qed (insert A, auto simp: irreducible_def one_poly_def)
```
```    49 qed
```
```    50
```
```    51
```
```    52 subsection \<open>Lifting elements into the field of fractions\<close>
```
```    53
```
```    54 definition to_fract :: "'a :: idom \<Rightarrow> 'a fract" where "to_fract x = Fract x 1"
```
```    55   \<comment> \<open>FIXME: name \<open>of_idom\<close>, abbreviation\<close>
```
```    56
```
```    57 lemma to_fract_0 [simp]: "to_fract 0 = 0"
```
```    58   by (simp add: to_fract_def eq_fract Zero_fract_def)
```
```    59
```
```    60 lemma to_fract_1 [simp]: "to_fract 1 = 1"
```
```    61   by (simp add: to_fract_def eq_fract One_fract_def)
```
```    62
```
```    63 lemma to_fract_add [simp]: "to_fract (x + y) = to_fract x + to_fract y"
```
```    64   by (simp add: to_fract_def)
```
```    65
```
```    66 lemma to_fract_diff [simp]: "to_fract (x - y) = to_fract x - to_fract y"
```
```    67   by (simp add: to_fract_def)
```
```    68
```
```    69 lemma to_fract_uminus [simp]: "to_fract (-x) = -to_fract x"
```
```    70   by (simp add: to_fract_def)
```
```    71
```
```    72 lemma to_fract_mult [simp]: "to_fract (x * y) = to_fract x * to_fract y"
```
```    73   by (simp add: to_fract_def)
```
```    74
```
```    75 lemma to_fract_eq_iff [simp]: "to_fract x = to_fract y \<longleftrightarrow> x = y"
```
```    76   by (simp add: to_fract_def eq_fract)
```
```    77
```
```    78 lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 \<longleftrightarrow> x = 0"
```
```    79   by (simp add: to_fract_def Zero_fract_def eq_fract)
```
```    80
```
```    81 lemma snd_quot_of_fract_nonzero [simp]: "snd (quot_of_fract x) \<noteq> 0"
```
```    82   by transfer simp
```
```    83
```
```    84 lemma Fract_quot_of_fract [simp]: "Fract (fst (quot_of_fract x)) (snd (quot_of_fract x)) = x"
```
```    85   by transfer (simp del: fractrel_iff, subst fractrel_normalize_quot_left, simp)
```
```    86
```
```    87 lemma to_fract_quot_of_fract:
```
```    88   assumes "snd (quot_of_fract x) = 1"
```
```    89   shows   "to_fract (fst (quot_of_fract x)) = x"
```
```    90 proof -
```
```    91   have "x = Fract (fst (quot_of_fract x)) (snd (quot_of_fract x))" by simp
```
```    92   also note assms
```
```    93   finally show ?thesis by (simp add: to_fract_def)
```
```    94 qed
```
```    95
```
```    96 lemma snd_quot_of_fract_Fract_whole:
```
```    97   assumes "y dvd x"
```
```    98   shows   "snd (quot_of_fract (Fract x y)) = 1"
```
```    99   using assms by transfer (auto simp: normalize_quot_def Let_def gcd_proj2_if_dvd)
```
```   100
```
```   101 lemma Fract_conv_to_fract: "Fract a b = to_fract a / to_fract b"
```
```   102   by (simp add: to_fract_def)
```
```   103
```
```   104 lemma quot_of_fract_to_fract [simp]: "quot_of_fract (to_fract x) = (x, 1)"
```
```   105   unfolding to_fract_def by transfer (simp add: normalize_quot_def)
```
```   106
```
```   107 lemma fst_quot_of_fract_eq_0_iff [simp]: "fst (quot_of_fract x) = 0 \<longleftrightarrow> x = 0"
```
```   108   by transfer simp
```
```   109
```
```   110 lemma snd_quot_of_fract_to_fract [simp]: "snd (quot_of_fract (to_fract x)) = 1"
```
```   111   unfolding to_fract_def by (rule snd_quot_of_fract_Fract_whole) simp_all
```
```   112
```
```   113 lemma coprime_quot_of_fract:
```
```   114   "coprime (fst (quot_of_fract x)) (snd (quot_of_fract x))"
```
```   115   by transfer (simp add: coprime_normalize_quot)
```
```   116
```
```   117 lemma unit_factor_snd_quot_of_fract: "unit_factor (snd (quot_of_fract x)) = 1"
```
```   118   using quot_of_fract_in_normalized_fracts[of x]
```
```   119   by (simp add: normalized_fracts_def case_prod_unfold)
```
```   120
```
```   121 lemma unit_factor_1_imp_normalized: "unit_factor x = 1 \<Longrightarrow> normalize x = x"
```
```   122   by (subst (2) normalize_mult_unit_factor [symmetric, of x])
```
```   123      (simp del: normalize_mult_unit_factor)
```
```   124
```
```   125 lemma normalize_snd_quot_of_fract: "normalize (snd (quot_of_fract x)) = snd (quot_of_fract x)"
```
```   126   by (intro unit_factor_1_imp_normalized unit_factor_snd_quot_of_fract)
```
```   127
```
```   128
```
```   129 subsection \<open>Lifting polynomial coefficients to the field of fractions\<close>
```
```   130
```
```   131 abbreviation (input) fract_poly
```
```   132   where "fract_poly \<equiv> map_poly to_fract"
```
```   133
```
```   134 abbreviation (input) unfract_poly
```
```   135   where "unfract_poly \<equiv> map_poly (fst \<circ> quot_of_fract)"
```
```   136
```
```   137 lemma fract_poly_smult [simp]: "fract_poly (smult c p) = smult (to_fract c) (fract_poly p)"
```
```   138   by (simp add: smult_conv_map_poly map_poly_map_poly o_def)
```
```   139
```
```   140 lemma fract_poly_0 [simp]: "fract_poly 0 = 0"
```
```   141   by (simp add: poly_eqI coeff_map_poly)
```
```   142
```
```   143 lemma fract_poly_1 [simp]: "fract_poly 1 = 1"
```
```   144   by (simp add: one_poly_def map_poly_pCons)
```
```   145
```
```   146 lemma fract_poly_add [simp]:
```
```   147   "fract_poly (p + q) = fract_poly p + fract_poly q"
```
```   148   by (intro poly_eqI) (simp_all add: coeff_map_poly)
```
```   149
```
```   150 lemma fract_poly_diff [simp]:
```
```   151   "fract_poly (p - q) = fract_poly p - fract_poly q"
```
```   152   by (intro poly_eqI) (simp_all add: coeff_map_poly)
```
```   153
```
```   154 lemma to_fract_sum [simp]: "to_fract (sum f A) = sum (\<lambda>x. to_fract (f x)) A"
```
```   155   by (cases "finite A", induction A rule: finite_induct) simp_all
```
```   156
```
```   157 lemma fract_poly_mult [simp]:
```
```   158   "fract_poly (p * q) = fract_poly p * fract_poly q"
```
```   159   by (intro poly_eqI) (simp_all add: coeff_map_poly coeff_mult)
```
```   160
```
```   161 lemma fract_poly_eq_iff [simp]: "fract_poly p = fract_poly q \<longleftrightarrow> p = q"
```
```   162   by (auto simp: poly_eq_iff coeff_map_poly)
```
```   163
```
```   164 lemma fract_poly_eq_0_iff [simp]: "fract_poly p = 0 \<longleftrightarrow> p = 0"
```
```   165   using fract_poly_eq_iff[of p 0] by (simp del: fract_poly_eq_iff)
```
```   166
```
```   167 lemma fract_poly_dvd: "p dvd q \<Longrightarrow> fract_poly p dvd fract_poly q"
```
```   168   by (auto elim!: dvdE)
```
```   169
```
```   170 lemma prod_mset_fract_poly:
```
```   171   "prod_mset (image_mset (\<lambda>x. fract_poly (f x)) A) = fract_poly (prod_mset (image_mset f A))"
```
```   172   by (induction A) (simp_all add: mult_ac)
```
```   173
```
```   174 lemma is_unit_fract_poly_iff:
```
```   175   "p dvd 1 \<longleftrightarrow> fract_poly p dvd 1 \<and> content p = 1"
```
```   176 proof safe
```
```   177   assume A: "p dvd 1"
```
```   178   with fract_poly_dvd[of p 1] show "is_unit (fract_poly p)" by simp
```
```   179   from A show "content p = 1"
```
```   180     by (auto simp: is_unit_poly_iff normalize_1_iff)
```
```   181 next
```
```   182   assume A: "fract_poly p dvd 1" and B: "content p = 1"
```
```   183   from A obtain c where c: "fract_poly p = [:c:]" by (auto simp: is_unit_poly_iff)
```
```   184   {
```
```   185     fix n :: nat assume "n > 0"
```
```   186     have "to_fract (coeff p n) = coeff (fract_poly p) n" by (simp add: coeff_map_poly)
```
```   187     also note c
```
```   188     also from \<open>n > 0\<close> have "coeff [:c:] n = 0" by (simp add: coeff_pCons split: nat.splits)
```
```   189     finally have "coeff p n = 0" by simp
```
```   190   }
```
```   191   hence "degree p \<le> 0" by (intro degree_le) simp_all
```
```   192   with B show "p dvd 1" by (auto simp: is_unit_poly_iff normalize_1_iff elim!: degree_eq_zeroE)
```
```   193 qed
```
```   194
```
```   195 lemma fract_poly_is_unit: "p dvd 1 \<Longrightarrow> fract_poly p dvd 1"
```
```   196   using fract_poly_dvd[of p 1] by simp
```
```   197
```
```   198 lemma fract_poly_smult_eqE:
```
```   199   fixes c :: "'a :: {idom_divide,ring_gcd} fract"
```
```   200   assumes "fract_poly p = smult c (fract_poly q)"
```
```   201   obtains a b
```
```   202     where "c = to_fract b / to_fract a" "smult a p = smult b q" "coprime a b" "normalize a = a"
```
```   203 proof -
```
```   204   define a b where "a = fst (quot_of_fract c)" and "b = snd (quot_of_fract c)"
```
```   205   have "smult (to_fract a) (fract_poly q) = smult (to_fract b) (fract_poly p)"
```
```   206     by (subst smult_eq_iff) (simp_all add: a_def b_def Fract_conv_to_fract [symmetric] assms)
```
```   207   hence "fract_poly (smult a q) = fract_poly (smult b p)" by (simp del: fract_poly_eq_iff)
```
```   208   hence "smult b p = smult a q" by (simp only: fract_poly_eq_iff)
```
```   209   moreover have "c = to_fract a / to_fract b" "coprime b a" "normalize b = b"
```
```   210     by (simp_all add: a_def b_def coprime_quot_of_fract gcd.commute
```
```   211           normalize_snd_quot_of_fract Fract_conv_to_fract [symmetric])
```
```   212   ultimately show ?thesis by (intro that[of a b])
```
```   213 qed
```
```   214
```
```   215
```
```   216 subsection \<open>Fractional content\<close>
```
```   217
```
```   218 abbreviation (input) Lcm_coeff_denoms
```
```   219     :: "'a :: {semiring_Gcd,idom_divide,ring_gcd} fract poly \<Rightarrow> 'a"
```
```   220   where "Lcm_coeff_denoms p \<equiv> Lcm (snd ` quot_of_fract ` set (coeffs p))"
```
```   221
```
```   222 definition fract_content ::
```
```   223       "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a fract" where
```
```   224   "fract_content p =
```
```   225      (let d = Lcm_coeff_denoms p in Fract (content (unfract_poly (smult (to_fract d) p))) d)"
```
```   226
```
```   227 definition primitive_part_fract ::
```
```   228       "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a poly" where
```
```   229   "primitive_part_fract p =
```
```   230      primitive_part (unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p))"
```
```   231
```
```   232 lemma primitive_part_fract_0 [simp]: "primitive_part_fract 0 = 0"
```
```   233   by (simp add: primitive_part_fract_def)
```
```   234
```
```   235 lemma fract_content_eq_0_iff [simp]:
```
```   236   "fract_content p = 0 \<longleftrightarrow> p = 0"
```
```   237   unfolding fract_content_def Let_def Zero_fract_def
```
```   238   by (subst eq_fract) (auto simp: Lcm_0_iff map_poly_eq_0_iff)
```
```   239
```
```   240 lemma content_primitive_part_fract [simp]: "p \<noteq> 0 \<Longrightarrow> content (primitive_part_fract p) = 1"
```
```   241   unfolding primitive_part_fract_def
```
```   242   by (rule content_primitive_part)
```
```   243      (auto simp: primitive_part_fract_def map_poly_eq_0_iff Lcm_0_iff)
```
```   244
```
```   245 lemma content_times_primitive_part_fract:
```
```   246   "smult (fract_content p) (fract_poly (primitive_part_fract p)) = p"
```
```   247 proof -
```
```   248   define p' where "p' = unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p)"
```
```   249   have "fract_poly p' =
```
```   250           map_poly (to_fract \<circ> fst \<circ> quot_of_fract) (smult (to_fract (Lcm_coeff_denoms p)) p)"
```
```   251     unfolding primitive_part_fract_def p'_def
```
```   252     by (subst map_poly_map_poly) (simp_all add: o_assoc)
```
```   253   also have "\<dots> = smult (to_fract (Lcm_coeff_denoms p)) p"
```
```   254   proof (intro map_poly_idI, unfold o_apply)
```
```   255     fix c assume "c \<in> set (coeffs (smult (to_fract (Lcm_coeff_denoms p)) p))"
```
```   256     then obtain c' where c: "c' \<in> set (coeffs p)" "c = to_fract (Lcm_coeff_denoms p) * c'"
```
```   257       by (auto simp add: Lcm_0_iff coeffs_smult split: if_splits)
```
```   258     note c(2)
```
```   259     also have "c' = Fract (fst (quot_of_fract c')) (snd (quot_of_fract c'))"
```
```   260       by simp
```
```   261     also have "to_fract (Lcm_coeff_denoms p) * \<dots> =
```
```   262                  Fract (Lcm_coeff_denoms p * fst (quot_of_fract c')) (snd (quot_of_fract c'))"
```
```   263       unfolding to_fract_def by (subst mult_fract) simp_all
```
```   264     also have "snd (quot_of_fract \<dots>) = 1"
```
```   265       by (intro snd_quot_of_fract_Fract_whole dvd_mult2 dvd_Lcm) (insert c(1), auto)
```
```   266     finally show "to_fract (fst (quot_of_fract c)) = c"
```
```   267       by (rule to_fract_quot_of_fract)
```
```   268   qed
```
```   269   also have "p' = smult (content p') (primitive_part p')"
```
```   270     by (rule content_times_primitive_part [symmetric])
```
```   271   also have "primitive_part p' = primitive_part_fract p"
```
```   272     by (simp add: primitive_part_fract_def p'_def)
```
```   273   also have "fract_poly (smult (content p') (primitive_part_fract p)) =
```
```   274                smult (to_fract (content p')) (fract_poly (primitive_part_fract p))" by simp
```
```   275   finally have "smult (to_fract (content p')) (fract_poly (primitive_part_fract p)) =
```
```   276                       smult (to_fract (Lcm_coeff_denoms p)) p" .
```
```   277   thus ?thesis
```
```   278     by (subst (asm) smult_eq_iff)
```
```   279        (auto simp add: Let_def p'_def Fract_conv_to_fract field_simps Lcm_0_iff fract_content_def)
```
```   280 qed
```
```   281
```
```   282 lemma fract_content_fract_poly [simp]: "fract_content (fract_poly p) = to_fract (content p)"
```
```   283 proof -
```
```   284   have "Lcm_coeff_denoms (fract_poly p) = 1"
```
```   285     by (auto simp: set_coeffs_map_poly)
```
```   286   hence "fract_content (fract_poly p) =
```
```   287            to_fract (content (map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p))"
```
```   288     by (simp add: fract_content_def to_fract_def fract_collapse map_poly_map_poly del: Lcm_1_iff)
```
```   289   also have "map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p = p"
```
```   290     by (intro map_poly_idI) simp_all
```
```   291   finally show ?thesis .
```
```   292 qed
```
```   293
```
```   294 lemma content_decompose_fract:
```
```   295   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly"
```
```   296   obtains c p' where "p = smult c (map_poly to_fract p')" "content p' = 1"
```
```   297 proof (cases "p = 0")
```
```   298   case True
```
```   299   hence "p = smult 0 (map_poly to_fract 1)" "content 1 = 1" by simp_all
```
```   300   thus ?thesis ..
```
```   301 next
```
```   302   case False
```
```   303   thus ?thesis
```
```   304     by (rule that[OF content_times_primitive_part_fract [symmetric] content_primitive_part_fract])
```
```   305 qed
```
```   306
```
```   307
```
```   308 subsection \<open>More properties of content and primitive part\<close>
```
```   309
```
```   310 lemma lift_prime_elem_poly:
```
```   311   assumes "prime_elem (c :: 'a :: semidom)"
```
```   312   shows   "prime_elem [:c:]"
```
```   313 proof (rule prime_elemI)
```
```   314   fix a b assume *: "[:c:] dvd a * b"
```
```   315   from * have dvd: "c dvd coeff (a * b) n" for n
```
```   316     by (subst (asm) const_poly_dvd_iff) blast
```
```   317   {
```
```   318     define m where "m = (GREATEST m. \<not>c dvd coeff b m)"
```
```   319     assume "\<not>[:c:] dvd b"
```
```   320     hence A: "\<exists>i. \<not>c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
```
```   321     have B: "\<forall>i. \<not>c dvd coeff b i \<longrightarrow> i < Suc (degree b)"
```
```   322       by (auto intro: le_degree simp: less_Suc_eq_le)
```
```   323     have coeff_m: "\<not>c dvd coeff b m" unfolding m_def by (rule GreatestI_ex[OF A B])
```
```   324     have "i \<le> m" if "\<not>c dvd coeff b i" for i
```
```   325       unfolding m_def by (rule Greatest_le[OF that B])
```
```   326     hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force
```
```   327
```
```   328     have "c dvd coeff a i" for i
```
```   329     proof (induction i rule: nat_descend_induct[of "degree a"])
```
```   330       case (base i)
```
```   331       thus ?case by (simp add: coeff_eq_0)
```
```   332     next
```
```   333       case (descend i)
```
```   334       let ?A = "{..i+m} - {i}"
```
```   335       have "c dvd coeff (a * b) (i + m)" by (rule dvd)
```
```   336       also have "coeff (a * b) (i + m) = (\<Sum>k\<le>i + m. coeff a k * coeff b (i + m - k))"
```
```   337         by (simp add: coeff_mult)
```
```   338       also have "{..i+m} = insert i ?A" by auto
```
```   339       also have "(\<Sum>k\<in>\<dots>. coeff a k * coeff b (i + m - k)) =
```
```   340                    coeff a i * coeff b m + (\<Sum>k\<in>?A. coeff a k * coeff b (i + m - k))"
```
```   341         (is "_ = _ + ?S")
```
```   342         by (subst sum.insert) simp_all
```
```   343       finally have eq: "c dvd coeff a i * coeff b m + ?S" .
```
```   344       moreover have "c dvd ?S"
```
```   345       proof (rule dvd_sum)
```
```   346         fix k assume k: "k \<in> {..i+m} - {i}"
```
```   347         show "c dvd coeff a k * coeff b (i + m - k)"
```
```   348         proof (cases "k < i")
```
```   349           case False
```
```   350           with k have "c dvd coeff a k" by (intro descend.IH) simp
```
```   351           thus ?thesis by simp
```
```   352         next
```
```   353           case True
```
```   354           hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp
```
```   355           thus ?thesis by simp
```
```   356         qed
```
```   357       qed
```
```   358       ultimately have "c dvd coeff a i * coeff b m"
```
```   359         by (simp add: dvd_add_left_iff)
```
```   360       with assms coeff_m show "c dvd coeff a i"
```
```   361         by (simp add: prime_elem_dvd_mult_iff)
```
```   362     qed
```
```   363     hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast
```
```   364   }
```
```   365   thus "[:c:] dvd a \<or> [:c:] dvd b" by blast
```
```   366 qed (insert assms, simp_all add: prime_elem_def one_poly_def)
```
```   367
```
```   368 lemma prime_elem_const_poly_iff:
```
```   369   fixes c :: "'a :: semidom"
```
```   370   shows   "prime_elem [:c:] \<longleftrightarrow> prime_elem c"
```
```   371 proof
```
```   372   assume A: "prime_elem [:c:]"
```
```   373   show "prime_elem c"
```
```   374   proof (rule prime_elemI)
```
```   375     fix a b assume "c dvd a * b"
```
```   376     hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac)
```
```   377     from A and this have "[:c:] dvd [:a:] \<or> [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
```
```   378     thus "c dvd a \<or> c dvd b" by simp
```
```   379   qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
```
```   380 qed (auto intro: lift_prime_elem_poly)
```
```   381
```
```   382 context
```
```   383 begin
```
```   384
```
```   385 private lemma content_1_mult:
```
```   386   fixes f g :: "'a :: {semiring_Gcd,factorial_semiring} poly"
```
```   387   assumes "content f = 1" "content g = 1"
```
```   388   shows   "content (f * g) = 1"
```
```   389 proof (cases "f * g = 0")
```
```   390   case False
```
```   391   from assms have "f \<noteq> 0" "g \<noteq> 0" by auto
```
```   392
```
```   393   hence "f * g \<noteq> 0" by auto
```
```   394   {
```
```   395     assume "\<not>is_unit (content (f * g))"
```
```   396     with False have "\<exists>p. p dvd content (f * g) \<and> prime p"
```
```   397       by (intro prime_divisor_exists) simp_all
```
```   398     then obtain p where "p dvd content (f * g)" "prime p" by blast
```
```   399     from \<open>p dvd content (f * g)\<close> have "[:p:] dvd f * g"
```
```   400       by (simp add: const_poly_dvd_iff_dvd_content)
```
```   401     moreover from \<open>prime p\<close> have "prime_elem [:p:]" by (simp add: lift_prime_elem_poly)
```
```   402     ultimately have "[:p:] dvd f \<or> [:p:] dvd g"
```
```   403       by (simp add: prime_elem_dvd_mult_iff)
```
```   404     with assms have "is_unit p" by (simp add: const_poly_dvd_iff_dvd_content)
```
```   405     with \<open>prime p\<close> have False by simp
```
```   406   }
```
```   407   hence "is_unit (content (f * g))" by blast
```
```   408   hence "normalize (content (f * g)) = 1" by (simp add: is_unit_normalize del: normalize_content)
```
```   409   thus ?thesis by simp
```
```   410 qed (insert assms, auto)
```
```   411
```
```   412 lemma content_mult:
```
```   413   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
```
```   414   shows "content (p * q) = content p * content q"
```
```   415 proof -
```
```   416   from content_decompose[of p] guess p' . note p = this
```
```   417   from content_decompose[of q] guess q' . note q = this
```
```   418   have "content (p * q) = content p * content q * content (p' * q')"
```
```   419     by (subst p, subst q) (simp add: mult_ac normalize_mult)
```
```   420   also from p q have "content (p' * q') = 1" by (intro content_1_mult)
```
```   421   finally show ?thesis by simp
```
```   422 qed
```
```   423
```
```   424 lemma primitive_part_mult:
```
```   425   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```   426   shows "primitive_part (p * q) = primitive_part p * primitive_part q"
```
```   427 proof -
```
```   428   have "primitive_part (p * q) = p * q div [:content (p * q):]"
```
```   429     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
```
```   430   also have "\<dots> = (p div [:content p:]) * (q div [:content q:])"
```
```   431     by (subst div_mult_div_if_dvd) (simp_all add: content_mult mult_ac)
```
```   432   also have "\<dots> = primitive_part p * primitive_part q"
```
```   433     by (simp add: primitive_part_def div_const_poly_conv_map_poly)
```
```   434   finally show ?thesis .
```
```   435 qed
```
```   436
```
```   437 lemma primitive_part_smult:
```
```   438   fixes p :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```   439   shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
```
```   440 proof -
```
```   441   have "smult a p = [:a:] * p" by simp
```
```   442   also have "primitive_part \<dots> = smult (unit_factor a) (primitive_part p)"
```
```   443     by (subst primitive_part_mult) simp_all
```
```   444   finally show ?thesis .
```
```   445 qed
```
```   446
```
```   447 lemma primitive_part_dvd_primitive_partI [intro]:
```
```   448   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
```
```   449   shows "p dvd q \<Longrightarrow> primitive_part p dvd primitive_part q"
```
```   450   by (auto elim!: dvdE simp: primitive_part_mult)
```
```   451
```
```   452 lemma content_prod_mset:
```
```   453   fixes A :: "'a :: {factorial_semiring, semiring_Gcd} poly multiset"
```
```   454   shows "content (prod_mset A) = prod_mset (image_mset content A)"
```
```   455   by (induction A) (simp_all add: content_mult mult_ac)
```
```   456
```
```   457 lemma fract_poly_dvdD:
```
```   458   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
```
```   459   assumes "fract_poly p dvd fract_poly q" "content p = 1"
```
```   460   shows   "p dvd q"
```
```   461 proof -
```
```   462   from assms(1) obtain r where r: "fract_poly q = fract_poly p * r" by (erule dvdE)
```
```   463   from content_decompose_fract[of r] guess c r' . note r' = this
```
```   464   from r r' have eq: "fract_poly q = smult c (fract_poly (p * r'))" by simp
```
```   465   from fract_poly_smult_eqE[OF this] guess a b . note ab = this
```
```   466   have "content (smult a q) = content (smult b (p * r'))" by (simp only: ab(2))
```
```   467   hence eq': "normalize b = a * content q" by (simp add: assms content_mult r' ab(4))
```
```   468   have "1 = gcd a (normalize b)" by (simp add: ab)
```
```   469   also note eq'
```
```   470   also have "gcd a (a * content q) = a" by (simp add: gcd_proj1_if_dvd ab(4))
```
```   471   finally have [simp]: "a = 1" by simp
```
```   472   from eq ab have "q = p * ([:b:] * r')" by simp
```
```   473   thus ?thesis by (rule dvdI)
```
```   474 qed
```
```   475
```
```   476 lemma content_prod_eq_1_iff:
```
```   477   fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
```
```   478   shows "content (p * q) = 1 \<longleftrightarrow> content p = 1 \<and> content q = 1"
```
```   479 proof safe
```
```   480   assume A: "content (p * q) = 1"
```
```   481   {
```
```   482     fix p q :: "'a poly" assume "content p * content q = 1"
```
```   483     hence "1 = content p * content q" by simp
```
```   484     hence "content p dvd 1" by (rule dvdI)
```
```   485     hence "content p = 1" by simp
```
```   486   } note B = this
```
```   487   from A B[of p q] B [of q p] show "content p = 1" "content q = 1"
```
```   488     by (simp_all add: content_mult mult_ac)
```
```   489 qed (auto simp: content_mult)
```
```   490
```
```   491 end
```
```   492
```
```   493
```
```   494 subsection \<open>Polynomials over a field are a Euclidean ring\<close>
```
```   495
```
```   496 definition unit_factor_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
```
```   497   "unit_factor_field_poly p = [:lead_coeff p:]"
```
```   498
```
```   499 definition normalize_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
```
```   500   "normalize_field_poly p = smult (inverse (lead_coeff p)) p"
```
```   501
```
```   502 definition euclidean_size_field_poly :: "'a :: field poly \<Rightarrow> nat" where
```
```   503   "euclidean_size_field_poly p = (if p = 0 then 0 else 2 ^ degree p)"
```
```   504
```
```   505 lemma dvd_field_poly: "dvd.dvd (op * :: 'a :: field poly \<Rightarrow> _) = op dvd"
```
```   506   by (intro ext) (simp_all add: dvd.dvd_def dvd_def)
```
```   507
```
```   508 interpretation field_poly:
```
```   509   unique_euclidean_ring where zero = "0 :: 'a :: field poly"
```
```   510     and one = 1 and plus = plus and uminus = uminus and minus = minus
```
```   511     and times = times
```
```   512     and normalize = normalize_field_poly and unit_factor = unit_factor_field_poly
```
```   513     and euclidean_size = euclidean_size_field_poly
```
```   514     and uniqueness_constraint = top
```
```   515     and divide = divide and modulo = modulo
```
```   516 proof (standard, unfold dvd_field_poly)
```
```   517   fix p :: "'a poly"
```
```   518   show "unit_factor_field_poly p * normalize_field_poly p = p"
```
```   519     by (cases "p = 0")
```
```   520        (simp_all add: unit_factor_field_poly_def normalize_field_poly_def)
```
```   521 next
```
```   522   fix p :: "'a poly" assume "is_unit p"
```
```   523   then show "unit_factor_field_poly p = p"
```
```   524     by (elim is_unit_polyE) (auto simp: unit_factor_field_poly_def monom_0 one_poly_def field_simps)
```
```   525 next
```
```   526   fix p :: "'a poly" assume "p \<noteq> 0"
```
```   527   thus "is_unit (unit_factor_field_poly p)"
```
```   528     by (simp add: unit_factor_field_poly_def is_unit_pCons_iff)
```
```   529 next
```
```   530   fix p q s :: "'a poly" assume "s \<noteq> 0"
```
```   531   moreover assume "euclidean_size_field_poly p < euclidean_size_field_poly q"
```
```   532   ultimately show "euclidean_size_field_poly (p * s) < euclidean_size_field_poly (q * s)"
```
```   533     by (auto simp add: euclidean_size_field_poly_def degree_mult_eq)
```
```   534 next
```
```   535   fix p q r :: "'a poly" assume "p \<noteq> 0"
```
```   536   moreover assume "euclidean_size_field_poly r < euclidean_size_field_poly p"
```
```   537   ultimately show "(q * p + r) div p = q"
```
```   538     by (cases "r = 0")
```
```   539       (auto simp add: unit_factor_field_poly_def euclidean_size_field_poly_def div_poly_less)
```
```   540 qed (auto simp: unit_factor_field_poly_def normalize_field_poly_def lead_coeff_mult
```
```   541        euclidean_size_field_poly_def Rings.div_mult_mod_eq intro!: degree_mod_less' degree_mult_right_le)
```
```   542
```
```   543 lemma field_poly_irreducible_imp_prime:
```
```   544   assumes "irreducible (p :: 'a :: field poly)"
```
```   545   shows   "prime_elem p"
```
```   546 proof -
```
```   547   have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
```
```   548   from field_poly.irreducible_imp_prime_elem[of p] assms
```
```   549     show ?thesis unfolding irreducible_def prime_elem_def dvd_field_poly
```
```   550       comm_semiring_1.irreducible_def[OF A] comm_semiring_1.prime_elem_def[OF A] by blast
```
```   551 qed
```
```   552
```
```   553 lemma field_poly_prod_mset_prime_factorization:
```
```   554   assumes "(x :: 'a :: field poly) \<noteq> 0"
```
```   555   shows   "prod_mset (field_poly.prime_factorization x) = normalize_field_poly x"
```
```   556 proof -
```
```   557   have A: "class.comm_monoid_mult op * (1 :: 'a poly)" ..
```
```   558   have "comm_monoid_mult.prod_mset op * (1 :: 'a poly) = prod_mset"
```
```   559     by (intro ext) (simp add: comm_monoid_mult.prod_mset_def[OF A] prod_mset_def)
```
```   560   with field_poly.prod_mset_prime_factorization[OF assms] show ?thesis by simp
```
```   561 qed
```
```   562
```
```   563 lemma field_poly_in_prime_factorization_imp_prime:
```
```   564   assumes "(p :: 'a :: field poly) \<in># field_poly.prime_factorization x"
```
```   565   shows   "prime_elem p"
```
```   566 proof -
```
```   567   have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
```
```   568   have B: "class.normalization_semidom op div op + op - (0 :: 'a poly) op * 1
```
```   569              unit_factor_field_poly normalize_field_poly" ..
```
```   570   from field_poly.in_prime_factors_imp_prime [of p x] assms
```
```   571     show ?thesis unfolding prime_elem_def dvd_field_poly
```
```   572       comm_semiring_1.prime_elem_def[OF A] normalization_semidom.prime_def[OF B] by blast
```
```   573 qed
```
```   574
```
```   575
```
```   576 subsection \<open>Primality and irreducibility in polynomial rings\<close>
```
```   577
```
```   578 lemma nonconst_poly_irreducible_iff:
```
```   579   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
```
```   580   assumes "degree p \<noteq> 0"
```
```   581   shows   "irreducible p \<longleftrightarrow> irreducible (fract_poly p) \<and> content p = 1"
```
```   582 proof safe
```
```   583   assume p: "irreducible p"
```
```   584
```
```   585   from content_decompose[of p] guess p' . note p' = this
```
```   586   hence "p = [:content p:] * p'" by simp
```
```   587   from p this have "[:content p:] dvd 1 \<or> p' dvd 1" by (rule irreducibleD)
```
```   588   moreover have "\<not>p' dvd 1"
```
```   589   proof
```
```   590     assume "p' dvd 1"
```
```   591     hence "degree p = 0" by (subst p') (auto simp: is_unit_poly_iff)
```
```   592     with assms show False by contradiction
```
```   593   qed
```
```   594   ultimately show [simp]: "content p = 1" by (simp add: is_unit_const_poly_iff)
```
```   595
```
```   596   show "irreducible (map_poly to_fract p)"
```
```   597   proof (rule irreducibleI)
```
```   598     have "fract_poly p = 0 \<longleftrightarrow> p = 0" by (intro map_poly_eq_0_iff) auto
```
```   599     with assms show "map_poly to_fract p \<noteq> 0" by auto
```
```   600   next
```
```   601     show "\<not>is_unit (fract_poly p)"
```
```   602     proof
```
```   603       assume "is_unit (map_poly to_fract p)"
```
```   604       hence "degree (map_poly to_fract p) = 0"
```
```   605         by (auto simp: is_unit_poly_iff)
```
```   606       hence "degree p = 0" by (simp add: degree_map_poly)
```
```   607       with assms show False by contradiction
```
```   608    qed
```
```   609  next
```
```   610    fix q r assume qr: "fract_poly p = q * r"
```
```   611    from content_decompose_fract[of q] guess cg q' . note q = this
```
```   612    from content_decompose_fract[of r] guess cr r' . note r = this
```
```   613    from qr q r p have nz: "cg \<noteq> 0" "cr \<noteq> 0" by auto
```
```   614    from qr have eq: "fract_poly p = smult (cr * cg) (fract_poly (q' * r'))"
```
```   615      by (simp add: q r)
```
```   616    from fract_poly_smult_eqE[OF this] guess a b . note ab = this
```
```   617    hence "content (smult a p) = content (smult b (q' * r'))" by (simp only:)
```
```   618    with ab(4) have a: "a = normalize b" by (simp add: content_mult q r)
```
```   619    hence "normalize b = gcd a b" by simp
```
```   620    also from ab(3) have "\<dots> = 1" .
```
```   621    finally have "a = 1" "is_unit b" by (simp_all add: a normalize_1_iff)
```
```   622
```
```   623    note eq
```
```   624    also from ab(1) \<open>a = 1\<close> have "cr * cg = to_fract b" by simp
```
```   625    also have "smult \<dots> (fract_poly (q' * r')) = fract_poly (smult b (q' * r'))" by simp
```
```   626    finally have "p = ([:b:] * q') * r'" by (simp del: fract_poly_smult)
```
```   627    from p and this have "([:b:] * q') dvd 1 \<or> r' dvd 1" by (rule irreducibleD)
```
```   628    hence "q' dvd 1 \<or> r' dvd 1" by (auto dest: dvd_mult_right simp del: mult_pCons_left)
```
```   629    hence "fract_poly q' dvd 1 \<or> fract_poly r' dvd 1" by (auto simp: fract_poly_is_unit)
```
```   630    with q r show "is_unit q \<or> is_unit r"
```
```   631      by (auto simp add: is_unit_smult_iff dvd_field_iff nz)
```
```   632  qed
```
```   633
```
```   634 next
```
```   635
```
```   636   assume irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
```
```   637   show "irreducible p"
```
```   638   proof (rule irreducibleI)
```
```   639     from irred show "p \<noteq> 0" by auto
```
```   640   next
```
```   641     from irred show "\<not>p dvd 1"
```
```   642       by (auto simp: irreducible_def dest: fract_poly_is_unit)
```
```   643   next
```
```   644     fix q r assume qr: "p = q * r"
```
```   645     hence "fract_poly p = fract_poly q * fract_poly r" by simp
```
```   646     from irred and this have "fract_poly q dvd 1 \<or> fract_poly r dvd 1"
```
```   647       by (rule irreducibleD)
```
```   648     with primitive qr show "q dvd 1 \<or> r dvd 1"
```
```   649       by (auto simp:  content_prod_eq_1_iff is_unit_fract_poly_iff)
```
```   650   qed
```
```   651 qed
```
```   652
```
```   653 context
```
```   654 begin
```
```   655
```
```   656 private lemma irreducible_imp_prime_poly:
```
```   657   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
```
```   658   assumes "irreducible p"
```
```   659   shows   "prime_elem p"
```
```   660 proof (cases "degree p = 0")
```
```   661   case True
```
```   662   with assms show ?thesis
```
```   663     by (auto simp: prime_elem_const_poly_iff irreducible_const_poly_iff
```
```   664              intro!: irreducible_imp_prime_elem elim!: degree_eq_zeroE)
```
```   665 next
```
```   666   case False
```
```   667   from assms False have irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
```
```   668     by (simp_all add: nonconst_poly_irreducible_iff)
```
```   669   from irred have prime: "prime_elem (fract_poly p)" by (rule field_poly_irreducible_imp_prime)
```
```   670   show ?thesis
```
```   671   proof (rule prime_elemI)
```
```   672     fix q r assume "p dvd q * r"
```
```   673     hence "fract_poly p dvd fract_poly (q * r)" by (rule fract_poly_dvd)
```
```   674     hence "fract_poly p dvd fract_poly q * fract_poly r" by simp
```
```   675     from prime and this have "fract_poly p dvd fract_poly q \<or> fract_poly p dvd fract_poly r"
```
```   676       by (rule prime_elem_dvd_multD)
```
```   677     with primitive show "p dvd q \<or> p dvd r" by (auto dest: fract_poly_dvdD)
```
```   678   qed (insert assms, auto simp: irreducible_def)
```
```   679 qed
```
```   680
```
```   681
```
```   682 lemma degree_primitive_part_fract [simp]:
```
```   683   "degree (primitive_part_fract p) = degree p"
```
```   684 proof -
```
```   685   have "p = smult (fract_content p) (fract_poly (primitive_part_fract p))"
```
```   686     by (simp add: content_times_primitive_part_fract)
```
```   687   also have "degree \<dots> = degree (primitive_part_fract p)"
```
```   688     by (auto simp: degree_map_poly)
```
```   689   finally show ?thesis ..
```
```   690 qed
```
```   691
```
```   692 lemma irreducible_primitive_part_fract:
```
```   693   fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
```
```   694   assumes "irreducible p"
```
```   695   shows   "irreducible (primitive_part_fract p)"
```
```   696 proof -
```
```   697   from assms have deg: "degree (primitive_part_fract p) \<noteq> 0"
```
```   698     by (intro notI)
```
```   699        (auto elim!: degree_eq_zeroE simp: irreducible_def is_unit_poly_iff dvd_field_iff)
```
```   700   hence [simp]: "p \<noteq> 0" by auto
```
```   701
```
```   702   note \<open>irreducible p\<close>
```
```   703   also have "p = [:fract_content p:] * fract_poly (primitive_part_fract p)"
```
```   704     by (simp add: content_times_primitive_part_fract)
```
```   705   also have "irreducible \<dots> \<longleftrightarrow> irreducible (fract_poly (primitive_part_fract p))"
```
```   706     by (intro irreducible_mult_unit_left) (simp_all add: is_unit_poly_iff dvd_field_iff)
```
```   707   finally show ?thesis using deg
```
```   708     by (simp add: nonconst_poly_irreducible_iff)
```
```   709 qed
```
```   710
```
```   711 lemma prime_elem_primitive_part_fract:
```
```   712   fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
```
```   713   shows "irreducible p \<Longrightarrow> prime_elem (primitive_part_fract p)"
```
```   714   by (intro irreducible_imp_prime_poly irreducible_primitive_part_fract)
```
```   715
```
```   716 lemma irreducible_linear_field_poly:
```
```   717   fixes a b :: "'a::field"
```
```   718   assumes "b \<noteq> 0"
```
```   719   shows "irreducible [:a,b:]"
```
```   720 proof (rule irreducibleI)
```
```   721   fix p q assume pq: "[:a,b:] = p * q"
```
```   722   also from pq assms have "degree \<dots> = degree p + degree q"
```
```   723     by (intro degree_mult_eq) auto
```
```   724   finally have "degree p = 0 \<or> degree q = 0" using assms by auto
```
```   725   with assms pq show "is_unit p \<or> is_unit q"
```
```   726     by (auto simp: is_unit_const_poly_iff dvd_field_iff elim!: degree_eq_zeroE)
```
```   727 qed (insert assms, auto simp: is_unit_poly_iff)
```
```   728
```
```   729 lemma prime_elem_linear_field_poly:
```
```   730   "(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"
```
```   731   by (rule field_poly_irreducible_imp_prime, rule irreducible_linear_field_poly)
```
```   732
```
```   733 lemma irreducible_linear_poly:
```
```   734   fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
```
```   735   shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> irreducible [:a,b:]"
```
```   736   by (auto intro!: irreducible_linear_field_poly
```
```   737            simp:   nonconst_poly_irreducible_iff content_def map_poly_pCons)
```
```   738
```
```   739 lemma prime_elem_linear_poly:
```
```   740   fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
```
```   741   shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> prime_elem [:a,b:]"
```
```   742   by (rule irreducible_imp_prime_poly, rule irreducible_linear_poly)
```
```   743
```
```   744 end
```
```   745
```
```   746
```
```   747 subsection \<open>Prime factorisation of polynomials\<close>
```
```   748
```
```   749 context
```
```   750 begin
```
```   751
```
```   752 private lemma poly_prime_factorization_exists_content_1:
```
```   753   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
```
```   754   assumes "p \<noteq> 0" "content p = 1"
```
```   755   shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
```
```   756 proof -
```
```   757   let ?P = "field_poly.prime_factorization (fract_poly p)"
```
```   758   define c where "c = prod_mset (image_mset fract_content ?P)"
```
```   759   define c' where "c' = c * to_fract (lead_coeff p)"
```
```   760   define e where "e = prod_mset (image_mset primitive_part_fract ?P)"
```
```   761   define A where "A = image_mset (normalize \<circ> primitive_part_fract) ?P"
```
```   762   have "content e = (\<Prod>x\<in>#field_poly.prime_factorization (map_poly to_fract p).
```
```   763                       content (primitive_part_fract x))"
```
```   764     by (simp add: e_def content_prod_mset multiset.map_comp o_def)
```
```   765   also have "image_mset (\<lambda>x. content (primitive_part_fract x)) ?P = image_mset (\<lambda>_. 1) ?P"
```
```   766     by (intro image_mset_cong content_primitive_part_fract) auto
```
```   767   finally have content_e: "content e = 1"
```
```   768     by simp
```
```   769
```
```   770   have "fract_poly p = unit_factor_field_poly (fract_poly p) *
```
```   771           normalize_field_poly (fract_poly p)" by simp
```
```   772   also have "unit_factor_field_poly (fract_poly p) = [:to_fract (lead_coeff p):]"
```
```   773     by (simp add: unit_factor_field_poly_def monom_0 degree_map_poly coeff_map_poly)
```
```   774   also from assms have "normalize_field_poly (fract_poly p) = prod_mset ?P"
```
```   775     by (subst field_poly_prod_mset_prime_factorization) simp_all
```
```   776   also have "\<dots> = prod_mset (image_mset id ?P)" by simp
```
```   777   also have "image_mset id ?P =
```
```   778                image_mset (\<lambda>x. [:fract_content x:] * fract_poly (primitive_part_fract x)) ?P"
```
```   779     by (intro image_mset_cong) (auto simp: content_times_primitive_part_fract)
```
```   780   also have "prod_mset \<dots> = smult c (fract_poly e)"
```
```   781     by (subst prod_mset.distrib) (simp_all add: prod_mset_fract_poly prod_mset_const_poly c_def e_def)
```
```   782   also have "[:to_fract (lead_coeff p):] * \<dots> = smult c' (fract_poly e)"
```
```   783     by (simp add: c'_def)
```
```   784   finally have eq: "fract_poly p = smult c' (fract_poly e)" .
```
```   785   also obtain b where b: "c' = to_fract b" "is_unit b"
```
```   786   proof -
```
```   787     from fract_poly_smult_eqE[OF eq] guess a b . note ab = this
```
```   788     from ab(2) have "content (smult a p) = content (smult b e)" by (simp only: )
```
```   789     with assms content_e have "a = normalize b" by (simp add: ab(4))
```
```   790     with ab have ab': "a = 1" "is_unit b" by (simp_all add: normalize_1_iff)
```
```   791     with ab ab' have "c' = to_fract b" by auto
```
```   792     from this and \<open>is_unit b\<close> show ?thesis by (rule that)
```
```   793   qed
```
```   794   hence "smult c' (fract_poly e) = fract_poly (smult b e)" by simp
```
```   795   finally have "p = smult b e" by (simp only: fract_poly_eq_iff)
```
```   796   hence "p = [:b:] * e" by simp
```
```   797   with b have "normalize p = normalize e"
```
```   798     by (simp only: normalize_mult) (simp add: is_unit_normalize is_unit_poly_iff)
```
```   799   also have "normalize e = prod_mset A"
```
```   800     by (simp add: multiset.map_comp e_def A_def normalize_prod_mset)
```
```   801   finally have "prod_mset A = normalize p" ..
```
```   802
```
```   803   have "prime_elem p" if "p \<in># A" for p
```
```   804     using that by (auto simp: A_def prime_elem_primitive_part_fract prime_elem_imp_irreducible
```
```   805                         dest!: field_poly_in_prime_factorization_imp_prime )
```
```   806   from this and \<open>prod_mset A = normalize p\<close> show ?thesis
```
```   807     by (intro exI[of _ A]) blast
```
```   808 qed
```
```   809
```
```   810 lemma poly_prime_factorization_exists:
```
```   811   fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
```
```   812   assumes "p \<noteq> 0"
```
```   813   shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
```
```   814 proof -
```
```   815   define B where "B = image_mset (\<lambda>x. [:x:]) (prime_factorization (content p))"
```
```   816   have "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize (primitive_part p)"
```
```   817     by (rule poly_prime_factorization_exists_content_1) (insert assms, simp_all)
```
```   818   then guess A by (elim exE conjE) note A = this
```
```   819   moreover from assms have "prod_mset B = [:content p:]"
```
```   820     by (simp add: B_def prod_mset_const_poly prod_mset_prime_factorization)
```
```   821   moreover have "\<forall>p. p \<in># B \<longrightarrow> prime_elem p"
```
```   822     by (auto simp: B_def intro!: lift_prime_elem_poly dest: in_prime_factors_imp_prime)
```
```   823   ultimately show ?thesis by (intro exI[of _ "B + A"]) auto
```
```   824 qed
```
```   825
```
```   826 end
```
```   827
```
```   828
```
```   829 subsection \<open>Typeclass instances\<close>
```
```   830
```
```   831 instance poly :: (factorial_ring_gcd) factorial_semiring
```
```   832   by standard (rule poly_prime_factorization_exists)
```
```   833
```
```   834 instantiation poly :: (factorial_ring_gcd) factorial_ring_gcd
```
```   835 begin
```
```   836
```
```   837 definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   838   [code del]: "gcd_poly = gcd_factorial"
```
```   839
```
```   840 definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   841   [code del]: "lcm_poly = lcm_factorial"
```
```   842
```
```   843 definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
```
```   844  [code del]: "Gcd_poly = Gcd_factorial"
```
```   845
```
```   846 definition Lcm_poly :: "'a poly set \<Rightarrow> 'a poly" where
```
```   847  [code del]: "Lcm_poly = Lcm_factorial"
```
```   848
```
```   849 instance by standard (simp_all add: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)
```
```   850
```
```   851 end
```
```   852
```
```   853 instantiation poly :: ("{field,factorial_ring_gcd}") unique_euclidean_ring
```
```   854 begin
```
```   855
```
```   856 definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
```
```   857   where "euclidean_size_poly p = (if p = 0 then 0 else 2 ^ degree p)"
```
```   858
```
```   859 definition uniqueness_constraint_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
```
```   860   where [simp]: "uniqueness_constraint_poly = top"
```
```   861
```
```   862 instance
```
```   863   by standard
```
```   864    (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
```
```   865     split: if_splits)
```
```   866
```
```   867 end
```
```   868
```
```   869 instance poly :: ("{field,factorial_ring_gcd}") euclidean_ring_gcd
```
```   870   by (rule euclidean_ring_gcd_class.intro, rule factorial_euclidean_semiring_gcdI)
```
```   871     standard
```
```   872
```
```   873
```
```   874 subsection \<open>Polynomial GCD\<close>
```
```   875
```
```   876 lemma gcd_poly_decompose:
```
```   877   fixes p q :: "'a :: factorial_ring_gcd poly"
```
```   878   shows "gcd p q =
```
```   879            smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
```
```   880 proof (rule sym, rule gcdI)
```
```   881   have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
```
```   882           [:content p:] * primitive_part p" by (intro mult_dvd_mono) simp_all
```
```   883   thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd p"
```
```   884     by simp
```
```   885 next
```
```   886   have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
```
```   887           [:content q:] * primitive_part q" by (intro mult_dvd_mono) simp_all
```
```   888   thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd q"
```
```   889     by simp
```
```   890 next
```
```   891   fix d assume "d dvd p" "d dvd q"
```
```   892   hence "[:content d:] * primitive_part d dvd
```
```   893            [:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q)"
```
```   894     by (intro mult_dvd_mono) auto
```
```   895   thus "d dvd smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
```
```   896     by simp
```
```   897 qed (auto simp: normalize_smult)
```
```   898
```
```   899
```
```   900 lemma gcd_poly_pseudo_mod:
```
```   901   fixes p q :: "'a :: factorial_ring_gcd poly"
```
```   902   assumes nz: "q \<noteq> 0" and prim: "content p = 1" "content q = 1"
```
```   903   shows   "gcd p q = gcd q (primitive_part (pseudo_mod p q))"
```
```   904 proof -
```
```   905   define r s where "r = fst (pseudo_divmod p q)" and "s = snd (pseudo_divmod p q)"
```
```   906   define a where "a = [:coeff q (degree q) ^ (Suc (degree p) - degree q):]"
```
```   907   have [simp]: "primitive_part a = unit_factor a"
```
```   908     by (simp add: a_def unit_factor_poly_def unit_factor_power monom_0)
```
```   909   from nz have [simp]: "a \<noteq> 0" by (auto simp: a_def)
```
```   910
```
```   911   have rs: "pseudo_divmod p q = (r, s)" by (simp add: r_def s_def)
```
```   912   have "gcd (q * r + s) q = gcd q s"
```
```   913     using gcd_add_mult[of q r s] by (simp add: gcd.commute add_ac mult_ac)
```
```   914   with pseudo_divmod(1)[OF nz rs]
```
```   915     have "gcd (p * a) q = gcd q s" by (simp add: a_def)
```
```   916   also from prim have "gcd (p * a) q = gcd p q"
```
```   917     by (subst gcd_poly_decompose)
```
```   918        (auto simp: primitive_part_mult gcd_mult_unit1 primitive_part_prim
```
```   919              simp del: mult_pCons_right )
```
```   920   also from prim have "gcd q s = gcd q (primitive_part s)"
```
```   921     by (subst gcd_poly_decompose) (simp_all add: primitive_part_prim)
```
```   922   also have "s = pseudo_mod p q" by (simp add: s_def pseudo_mod_def)
```
```   923   finally show ?thesis .
```
```   924 qed
```
```   925
```
```   926 lemma degree_pseudo_mod_less:
```
```   927   assumes "q \<noteq> 0" "pseudo_mod p q \<noteq> 0"
```
```   928   shows   "degree (pseudo_mod p q) < degree q"
```
```   929   using pseudo_mod(2)[of q p] assms by auto
```
```   930
```
```   931 function gcd_poly_code_aux :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
```
```   932   "gcd_poly_code_aux p q =
```
```   933      (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
```
```   934 by auto
```
```   935 termination
```
```   936   by (relation "measure ((\<lambda>p. if p = 0 then 0 else Suc (degree p)) \<circ> snd)")
```
```   937      (auto simp: degree_pseudo_mod_less)
```
```   938
```
```   939 declare gcd_poly_code_aux.simps [simp del]
```
```   940
```
```   941 lemma gcd_poly_code_aux_correct:
```
```   942   assumes "content p = 1" "q = 0 \<or> content q = 1"
```
```   943   shows   "gcd_poly_code_aux p q = gcd p q"
```
```   944   using assms
```
```   945 proof (induction p q rule: gcd_poly_code_aux.induct)
```
```   946   case (1 p q)
```
```   947   show ?case
```
```   948   proof (cases "q = 0")
```
```   949     case True
```
```   950     thus ?thesis by (subst gcd_poly_code_aux.simps) auto
```
```   951   next
```
```   952     case False
```
```   953     hence "gcd_poly_code_aux p q = gcd_poly_code_aux q (primitive_part (pseudo_mod p q))"
```
```   954       by (subst gcd_poly_code_aux.simps) simp_all
```
```   955     also from "1.prems" False
```
```   956       have "primitive_part (pseudo_mod p q) = 0 \<or>
```
```   957               content (primitive_part (pseudo_mod p q)) = 1"
```
```   958       by (cases "pseudo_mod p q = 0") auto
```
```   959     with "1.prems" False
```
```   960       have "gcd_poly_code_aux q (primitive_part (pseudo_mod p q)) =
```
```   961               gcd q (primitive_part (pseudo_mod p q))"
```
```   962       by (intro 1) simp_all
```
```   963     also from "1.prems" False
```
```   964       have "\<dots> = gcd p q" by (intro gcd_poly_pseudo_mod [symmetric]) auto
```
```   965     finally show ?thesis .
```
```   966   qed
```
```   967 qed
```
```   968
```
```   969 definition gcd_poly_code
```
```   970     :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
```
```   971   where "gcd_poly_code p q =
```
```   972            (if p = 0 then normalize q else if q = 0 then normalize p else
```
```   973               smult (gcd (content p) (content q))
```
```   974                 (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
```
```   975
```
```   976 lemma gcd_poly_code [code]: "gcd p q = gcd_poly_code p q"
```
```   977   by (simp add: gcd_poly_code_def gcd_poly_code_aux_correct gcd_poly_decompose [symmetric])
```
```   978
```
```   979 lemma lcm_poly_code [code]:
```
```   980   fixes p q :: "'a :: factorial_ring_gcd poly"
```
```   981   shows "lcm p q = normalize (p * q) div gcd p q"
```
```   982   by (fact lcm_gcd)
```
```   983
```
```   984 lemmas Gcd_poly_set_eq_fold [code] = Gcd_set_eq_fold [where ?'a = "'a :: factorial_ring_gcd poly"]
```
```   985 lemmas Lcm_poly_set_eq_fold [code] = Lcm_set_eq_fold [where ?'a = "'a :: factorial_ring_gcd poly"]
```
```   986
```
```   987 text \<open>Example:
```
```   988   @{lemma "Lcm {[:1, 2, 3:], [:2, 3, 4:]} = [:[:2:], [:7:], [:16:], [:17:], [:12 :: int:]:]" by eval}
```
```   989 \<close>
```
```   990
```
```   991 end
```