src/HOL/Computational_Algebra/Polynomial_Factorial.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 65965 088c79b40156
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(*  Title:      HOL/Computational_Algebra/Polynomial_Factorial.thy
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    Author:     Brian Huffman
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    Author:     Clemens Ballarin
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    Author:     Amine Chaieb
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    Author:     Florian Haftmann
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    Author:     Manuel Eberl
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*)
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theory Polynomial_Factorial
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imports 
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  Complex_Main
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  Polynomial
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  Normalized_Fraction
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  Field_as_Ring
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begin
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subsection \<open>Various facts about polynomials\<close>
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lemma prod_mset_const_poly: " (\<Prod>x\<in>#A. [:f x:]) = [:prod_mset (image_mset f A):]"
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  by (induct A) (simp_all add: ac_simps)
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lemma irreducible_const_poly_iff:
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  fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}"
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  shows "irreducible [:c:] \<longleftrightarrow> irreducible c"
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proof
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  assume A: "irreducible c"
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  show "irreducible [:c:]"
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  proof (rule irreducibleI)
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    fix a b assume ab: "[:c:] = a * b"
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    hence "degree [:c:] = degree (a * b)" by (simp only: )
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    also from A ab have "a \<noteq> 0" "b \<noteq> 0" by auto
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    hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq)
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    finally have "degree a = 0" "degree b = 0" by auto
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    then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE)
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    from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: )
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    hence "c = a' * b'" by (simp add: ab' mult_ac)
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    from A and this have "a' dvd 1 \<or> b' dvd 1" by (rule irreducibleD)
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    with ab' show "a dvd 1 \<or> b dvd 1"
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      by (auto simp add: is_unit_const_poly_iff)
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  qed (insert A, auto simp: irreducible_def is_unit_poly_iff)
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next
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  assume A: "irreducible [:c:]"
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  then have "c \<noteq> 0" and "\<not> c dvd 1"
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    by (auto simp add: irreducible_def is_unit_const_poly_iff)
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  then show "irreducible c"
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  proof (rule irreducibleI)
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    fix a b assume ab: "c = a * b"
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    hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac)
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    from A and this have "[:a:] dvd 1 \<or> [:b:] dvd 1" by (rule irreducibleD)
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    then show "a dvd 1 \<or> b dvd 1"
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      by (auto simp add: is_unit_const_poly_iff)
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  qed
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qed
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subsection \<open>Lifting elements into the field of fractions\<close>
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definition to_fract :: "'a :: idom \<Rightarrow> 'a fract" where "to_fract x = Fract x 1"
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  \<comment> \<open>FIXME: name \<open>of_idom\<close>, abbreviation\<close>
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lemma to_fract_0 [simp]: "to_fract 0 = 0"
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  by (simp add: to_fract_def eq_fract Zero_fract_def)
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lemma to_fract_1 [simp]: "to_fract 1 = 1"
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  by (simp add: to_fract_def eq_fract One_fract_def)
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lemma to_fract_add [simp]: "to_fract (x + y) = to_fract x + to_fract y"
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  by (simp add: to_fract_def)
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lemma to_fract_diff [simp]: "to_fract (x - y) = to_fract x - to_fract y"
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  by (simp add: to_fract_def)
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lemma to_fract_uminus [simp]: "to_fract (-x) = -to_fract x"
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  by (simp add: to_fract_def)
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lemma to_fract_mult [simp]: "to_fract (x * y) = to_fract x * to_fract y"
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  by (simp add: to_fract_def)
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lemma to_fract_eq_iff [simp]: "to_fract x = to_fract y \<longleftrightarrow> x = y"
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  by (simp add: to_fract_def eq_fract)
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lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 \<longleftrightarrow> x = 0"
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  by (simp add: to_fract_def Zero_fract_def eq_fract)
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lemma snd_quot_of_fract_nonzero [simp]: "snd (quot_of_fract x) \<noteq> 0"
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  by transfer simp
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lemma Fract_quot_of_fract [simp]: "Fract (fst (quot_of_fract x)) (snd (quot_of_fract x)) = x"
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  by transfer (simp del: fractrel_iff, subst fractrel_normalize_quot_left, simp)
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lemma to_fract_quot_of_fract:
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  assumes "snd (quot_of_fract x) = 1"
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  shows   "to_fract (fst (quot_of_fract x)) = x"
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proof -
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  have "x = Fract (fst (quot_of_fract x)) (snd (quot_of_fract x))" by simp
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  also note assms
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  finally show ?thesis by (simp add: to_fract_def)
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qed
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lemma snd_quot_of_fract_Fract_whole:
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  assumes "y dvd x"
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  shows   "snd (quot_of_fract (Fract x y)) = 1"
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  using assms by transfer (auto simp: normalize_quot_def Let_def gcd_proj2_if_dvd)
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lemma Fract_conv_to_fract: "Fract a b = to_fract a / to_fract b"
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  by (simp add: to_fract_def)
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lemma quot_of_fract_to_fract [simp]: "quot_of_fract (to_fract x) = (x, 1)"
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  unfolding to_fract_def by transfer (simp add: normalize_quot_def)
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lemma fst_quot_of_fract_eq_0_iff [simp]: "fst (quot_of_fract x) = 0 \<longleftrightarrow> x = 0"
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  by transfer simp
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lemma snd_quot_of_fract_to_fract [simp]: "snd (quot_of_fract (to_fract x)) = 1"
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  unfolding to_fract_def by (rule snd_quot_of_fract_Fract_whole) simp_all
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lemma coprime_quot_of_fract:
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  "coprime (fst (quot_of_fract x)) (snd (quot_of_fract x))"
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  by transfer (simp add: coprime_normalize_quot)
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lemma unit_factor_snd_quot_of_fract: "unit_factor (snd (quot_of_fract x)) = 1"
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  using quot_of_fract_in_normalized_fracts[of x] 
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  by (simp add: normalized_fracts_def case_prod_unfold)  
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lemma unit_factor_1_imp_normalized: "unit_factor x = 1 \<Longrightarrow> normalize x = x"
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  by (subst (2) normalize_mult_unit_factor [symmetric, of x])
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     (simp del: normalize_mult_unit_factor)
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lemma normalize_snd_quot_of_fract: "normalize (snd (quot_of_fract x)) = snd (quot_of_fract x)"
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  by (intro unit_factor_1_imp_normalized unit_factor_snd_quot_of_fract)
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subsection \<open>Lifting polynomial coefficients to the field of fractions\<close>
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abbreviation (input) fract_poly 
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  where "fract_poly \<equiv> map_poly to_fract"
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abbreviation (input) unfract_poly 
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  where "unfract_poly \<equiv> map_poly (fst \<circ> quot_of_fract)"
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lemma fract_poly_smult [simp]: "fract_poly (smult c p) = smult (to_fract c) (fract_poly p)"
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  by (simp add: smult_conv_map_poly map_poly_map_poly o_def)
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lemma fract_poly_0 [simp]: "fract_poly 0 = 0"
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  by (simp add: poly_eqI coeff_map_poly)
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lemma fract_poly_1 [simp]: "fract_poly 1 = 1"
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  by (simp add: map_poly_pCons)
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lemma fract_poly_add [simp]:
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  "fract_poly (p + q) = fract_poly p + fract_poly q"
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  by (intro poly_eqI) (simp_all add: coeff_map_poly)
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lemma fract_poly_diff [simp]:
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  "fract_poly (p - q) = fract_poly p - fract_poly q"
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  by (intro poly_eqI) (simp_all add: coeff_map_poly)
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lemma to_fract_sum [simp]: "to_fract (sum f A) = sum (\<lambda>x. to_fract (f x)) A"
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  by (cases "finite A", induction A rule: finite_induct) simp_all 
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lemma fract_poly_mult [simp]:
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  "fract_poly (p * q) = fract_poly p * fract_poly q"
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  by (intro poly_eqI) (simp_all add: coeff_map_poly coeff_mult)
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lemma fract_poly_eq_iff [simp]: "fract_poly p = fract_poly q \<longleftrightarrow> p = q"
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  by (auto simp: poly_eq_iff coeff_map_poly)
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lemma fract_poly_eq_0_iff [simp]: "fract_poly p = 0 \<longleftrightarrow> p = 0"
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  using fract_poly_eq_iff[of p 0] by (simp del: fract_poly_eq_iff)
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lemma fract_poly_dvd: "p dvd q \<Longrightarrow> fract_poly p dvd fract_poly q"
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  by (auto elim!: dvdE)
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lemma prod_mset_fract_poly: 
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  "(\<Prod>x\<in>#A. map_poly to_fract (f x)) = fract_poly (prod_mset (image_mset f A))"
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  by (induct A) (simp_all add: ac_simps)
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lemma is_unit_fract_poly_iff:
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  "p dvd 1 \<longleftrightarrow> fract_poly p dvd 1 \<and> content p = 1"
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proof safe
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  assume A: "p dvd 1"
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  with fract_poly_dvd [of p 1] show "is_unit (fract_poly p)"
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    by simp
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  from A show "content p = 1"
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    by (auto simp: is_unit_poly_iff normalize_1_iff)
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next
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  assume A: "fract_poly p dvd 1" and B: "content p = 1"
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  from A obtain c where c: "fract_poly p = [:c:]" by (auto simp: is_unit_poly_iff)
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  {
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    fix n :: nat assume "n > 0"
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    have "to_fract (coeff p n) = coeff (fract_poly p) n" by (simp add: coeff_map_poly)
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    also note c
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    also from \<open>n > 0\<close> have "coeff [:c:] n = 0" by (simp add: coeff_pCons split: nat.splits)
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    finally have "coeff p n = 0" by simp
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  }
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  hence "degree p \<le> 0" by (intro degree_le) simp_all
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  with B show "p dvd 1" by (auto simp: is_unit_poly_iff normalize_1_iff elim!: degree_eq_zeroE)
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qed
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lemma fract_poly_is_unit: "p dvd 1 \<Longrightarrow> fract_poly p dvd 1"
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  using fract_poly_dvd[of p 1] by simp
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lemma fract_poly_smult_eqE:
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  fixes c :: "'a :: {idom_divide,ring_gcd} fract"
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  assumes "fract_poly p = smult c (fract_poly q)"
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  obtains a b 
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    where "c = to_fract b / to_fract a" "smult a p = smult b q" "coprime a b" "normalize a = a"
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proof -
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  define a b where "a = fst (quot_of_fract c)" and "b = snd (quot_of_fract c)"
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  have "smult (to_fract a) (fract_poly q) = smult (to_fract b) (fract_poly p)"
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    by (subst smult_eq_iff) (simp_all add: a_def b_def Fract_conv_to_fract [symmetric] assms)
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  hence "fract_poly (smult a q) = fract_poly (smult b p)" by (simp del: fract_poly_eq_iff)
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  hence "smult b p = smult a q" by (simp only: fract_poly_eq_iff)
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  moreover have "c = to_fract a / to_fract b" "coprime b a" "normalize b = b"
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    by (simp_all add: a_def b_def coprime_quot_of_fract gcd.commute
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          normalize_snd_quot_of_fract Fract_conv_to_fract [symmetric])
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  ultimately show ?thesis by (intro that[of a b])
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qed
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subsection \<open>Fractional content\<close>
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abbreviation (input) Lcm_coeff_denoms 
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    :: "'a :: {semiring_Gcd,idom_divide,ring_gcd} fract poly \<Rightarrow> 'a"
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  where "Lcm_coeff_denoms p \<equiv> Lcm (snd ` quot_of_fract ` set (coeffs p))"
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definition fract_content :: 
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      "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a fract" where
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  "fract_content p = 
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     (let d = Lcm_coeff_denoms p in Fract (content (unfract_poly (smult (to_fract d) p))) d)" 
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definition primitive_part_fract :: 
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      "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> 'a poly" where
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  "primitive_part_fract p = 
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     primitive_part (unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p))"
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lemma primitive_part_fract_0 [simp]: "primitive_part_fract 0 = 0"
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  by (simp add: primitive_part_fract_def)
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lemma fract_content_eq_0_iff [simp]:
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  "fract_content p = 0 \<longleftrightarrow> p = 0"
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  unfolding fract_content_def Let_def Zero_fract_def
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  by (subst eq_fract) (auto simp: Lcm_0_iff map_poly_eq_0_iff)
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lemma content_primitive_part_fract [simp]: "p \<noteq> 0 \<Longrightarrow> content (primitive_part_fract p) = 1"
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  unfolding primitive_part_fract_def
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  by (rule content_primitive_part)
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     (auto simp: primitive_part_fract_def map_poly_eq_0_iff Lcm_0_iff)  
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lemma content_times_primitive_part_fract:
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  "smult (fract_content p) (fract_poly (primitive_part_fract p)) = p"
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proof -
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  define p' where "p' = unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p)"
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  have "fract_poly p' = 
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          map_poly (to_fract \<circ> fst \<circ> quot_of_fract) (smult (to_fract (Lcm_coeff_denoms p)) p)"
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    unfolding primitive_part_fract_def p'_def 
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    by (subst map_poly_map_poly) (simp_all add: o_assoc)
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  also have "\<dots> = smult (to_fract (Lcm_coeff_denoms p)) p"
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  proof (intro map_poly_idI, unfold o_apply)
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    fix c assume "c \<in> set (coeffs (smult (to_fract (Lcm_coeff_denoms p)) p))"
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    then obtain c' where c: "c' \<in> set (coeffs p)" "c = to_fract (Lcm_coeff_denoms p) * c'"
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      by (auto simp add: Lcm_0_iff coeffs_smult split: if_splits)
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    note c(2)
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    also have "c' = Fract (fst (quot_of_fract c')) (snd (quot_of_fract c'))"
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      by simp
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    also have "to_fract (Lcm_coeff_denoms p) * \<dots> = 
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                 Fract (Lcm_coeff_denoms p * fst (quot_of_fract c')) (snd (quot_of_fract c'))"
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      unfolding to_fract_def by (subst mult_fract) simp_all
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    also have "snd (quot_of_fract \<dots>) = 1"
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      by (intro snd_quot_of_fract_Fract_whole dvd_mult2 dvd_Lcm) (insert c(1), auto)
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    finally show "to_fract (fst (quot_of_fract c)) = c"
eberlm@63498
   272
      by (rule to_fract_quot_of_fract)
eberlm@63498
   273
  qed
eberlm@63498
   274
  also have "p' = smult (content p') (primitive_part p')" 
eberlm@63498
   275
    by (rule content_times_primitive_part [symmetric])
eberlm@63498
   276
  also have "primitive_part p' = primitive_part_fract p"
eberlm@63498
   277
    by (simp add: primitive_part_fract_def p'_def)
eberlm@63498
   278
  also have "fract_poly (smult (content p') (primitive_part_fract p)) = 
eberlm@63498
   279
               smult (to_fract (content p')) (fract_poly (primitive_part_fract p))" by simp
eberlm@63498
   280
  finally have "smult (to_fract (content p')) (fract_poly (primitive_part_fract p)) =
eberlm@63498
   281
                      smult (to_fract (Lcm_coeff_denoms p)) p" .
eberlm@63498
   282
  thus ?thesis
eberlm@63498
   283
    by (subst (asm) smult_eq_iff)
eberlm@63498
   284
       (auto simp add: Let_def p'_def Fract_conv_to_fract field_simps Lcm_0_iff fract_content_def)
eberlm@63498
   285
qed
eberlm@63498
   286
eberlm@63498
   287
lemma fract_content_fract_poly [simp]: "fract_content (fract_poly p) = to_fract (content p)"
eberlm@63498
   288
proof -
eberlm@63498
   289
  have "Lcm_coeff_denoms (fract_poly p) = 1"
haftmann@63905
   290
    by (auto simp: set_coeffs_map_poly)
eberlm@63498
   291
  hence "fract_content (fract_poly p) = 
eberlm@63498
   292
           to_fract (content (map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p))"
eberlm@63498
   293
    by (simp add: fract_content_def to_fract_def fract_collapse map_poly_map_poly del: Lcm_1_iff)
eberlm@63498
   294
  also have "map_poly (fst \<circ> quot_of_fract \<circ> to_fract) p = p"
eberlm@63498
   295
    by (intro map_poly_idI) simp_all
eberlm@63498
   296
  finally show ?thesis .
eberlm@63498
   297
qed
eberlm@63498
   298
eberlm@63498
   299
lemma content_decompose_fract:
eberlm@63498
   300
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly"
eberlm@63498
   301
  obtains c p' where "p = smult c (map_poly to_fract p')" "content p' = 1"
eberlm@63498
   302
proof (cases "p = 0")
eberlm@63498
   303
  case True
eberlm@63498
   304
  hence "p = smult 0 (map_poly to_fract 1)" "content 1 = 1" by simp_all
eberlm@63498
   305
  thus ?thesis ..
eberlm@63498
   306
next
eberlm@63498
   307
  case False
eberlm@63498
   308
  thus ?thesis
eberlm@63498
   309
    by (rule that[OF content_times_primitive_part_fract [symmetric] content_primitive_part_fract])
eberlm@63498
   310
qed
eberlm@63498
   311
eberlm@63498
   312
eberlm@63498
   313
subsection \<open>More properties of content and primitive part\<close>
eberlm@63498
   314
eberlm@63498
   315
lemma lift_prime_elem_poly:
eberlm@63633
   316
  assumes "prime_elem (c :: 'a :: semidom)"
eberlm@63633
   317
  shows   "prime_elem [:c:]"
eberlm@63633
   318
proof (rule prime_elemI)
eberlm@63498
   319
  fix a b assume *: "[:c:] dvd a * b"
eberlm@63498
   320
  from * have dvd: "c dvd coeff (a * b) n" for n
eberlm@63498
   321
    by (subst (asm) const_poly_dvd_iff) blast
eberlm@63498
   322
  {
eberlm@63498
   323
    define m where "m = (GREATEST m. \<not>c dvd coeff b m)"
eberlm@63498
   324
    assume "\<not>[:c:] dvd b"
eberlm@63498
   325
    hence A: "\<exists>i. \<not>c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
nipkow@65963
   326
    have B: "\<forall>i. \<not>c dvd coeff b i \<longrightarrow> i \<le> degree b"
nipkow@65963
   327
      by (auto intro: le_degree)
nipkow@65965
   328
    have coeff_m: "\<not>c dvd coeff b m" unfolding m_def by (rule GreatestI_ex_nat[OF A B])
eberlm@63498
   329
    have "i \<le> m" if "\<not>c dvd coeff b i" for i
nipkow@65965
   330
      unfolding m_def by (rule Greatest_le_nat[OF that B])
eberlm@63498
   331
    hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force
eberlm@63498
   332
eberlm@63498
   333
    have "c dvd coeff a i" for i
eberlm@63498
   334
    proof (induction i rule: nat_descend_induct[of "degree a"])
eberlm@63498
   335
      case (base i)
eberlm@63498
   336
      thus ?case by (simp add: coeff_eq_0)
eberlm@63498
   337
    next
eberlm@63498
   338
      case (descend i)
eberlm@63498
   339
      let ?A = "{..i+m} - {i}"
eberlm@63498
   340
      have "c dvd coeff (a * b) (i + m)" by (rule dvd)
eberlm@63498
   341
      also have "coeff (a * b) (i + m) = (\<Sum>k\<le>i + m. coeff a k * coeff b (i + m - k))"
eberlm@63498
   342
        by (simp add: coeff_mult)
eberlm@63498
   343
      also have "{..i+m} = insert i ?A" by auto
eberlm@63498
   344
      also have "(\<Sum>k\<in>\<dots>. coeff a k * coeff b (i + m - k)) =
eberlm@63498
   345
                   coeff a i * coeff b m + (\<Sum>k\<in>?A. coeff a k * coeff b (i + m - k))"
eberlm@63498
   346
        (is "_ = _ + ?S")
nipkow@64267
   347
        by (subst sum.insert) simp_all
eberlm@63498
   348
      finally have eq: "c dvd coeff a i * coeff b m + ?S" .
eberlm@63498
   349
      moreover have "c dvd ?S"
nipkow@64267
   350
      proof (rule dvd_sum)
eberlm@63498
   351
        fix k assume k: "k \<in> {..i+m} - {i}"
eberlm@63498
   352
        show "c dvd coeff a k * coeff b (i + m - k)"
eberlm@63498
   353
        proof (cases "k < i")
eberlm@63498
   354
          case False
eberlm@63498
   355
          with k have "c dvd coeff a k" by (intro descend.IH) simp
eberlm@63498
   356
          thus ?thesis by simp
eberlm@63498
   357
        next
eberlm@63498
   358
          case True
eberlm@63498
   359
          hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp
eberlm@63498
   360
          thus ?thesis by simp
eberlm@63498
   361
        qed
eberlm@63498
   362
      qed
eberlm@63498
   363
      ultimately have "c dvd coeff a i * coeff b m"
eberlm@63498
   364
        by (simp add: dvd_add_left_iff)
eberlm@63498
   365
      with assms coeff_m show "c dvd coeff a i"
eberlm@63633
   366
        by (simp add: prime_elem_dvd_mult_iff)
eberlm@63498
   367
    qed
eberlm@63498
   368
    hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast
eberlm@63498
   369
  }
haftmann@65486
   370
  then show "[:c:] dvd a \<or> [:c:] dvd b" by blast
haftmann@65486
   371
next
haftmann@65486
   372
  from assms show "[:c:] \<noteq> 0" and "\<not> [:c:] dvd 1"
haftmann@65486
   373
    by (simp_all add: prime_elem_def is_unit_const_poly_iff)
haftmann@65486
   374
qed
eberlm@63498
   375
eberlm@63498
   376
lemma prime_elem_const_poly_iff:
eberlm@63498
   377
  fixes c :: "'a :: semidom"
eberlm@63633
   378
  shows   "prime_elem [:c:] \<longleftrightarrow> prime_elem c"
eberlm@63498
   379
proof
eberlm@63633
   380
  assume A: "prime_elem [:c:]"
eberlm@63633
   381
  show "prime_elem c"
eberlm@63633
   382
  proof (rule prime_elemI)
eberlm@63498
   383
    fix a b assume "c dvd a * b"
eberlm@63498
   384
    hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac)
eberlm@63633
   385
    from A and this have "[:c:] dvd [:a:] \<or> [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
eberlm@63498
   386
    thus "c dvd a \<or> c dvd b" by simp
eberlm@63633
   387
  qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
eberlm@63498
   388
qed (auto intro: lift_prime_elem_poly)
eberlm@63498
   389
eberlm@63498
   390
context
eberlm@63498
   391
begin
eberlm@63498
   392
eberlm@63498
   393
private lemma content_1_mult:
eberlm@63498
   394
  fixes f g :: "'a :: {semiring_Gcd,factorial_semiring} poly"
eberlm@63498
   395
  assumes "content f = 1" "content g = 1"
eberlm@63498
   396
  shows   "content (f * g) = 1"
eberlm@63498
   397
proof (cases "f * g = 0")
eberlm@63498
   398
  case False
eberlm@63498
   399
  from assms have "f \<noteq> 0" "g \<noteq> 0" by auto
eberlm@63498
   400
eberlm@63498
   401
  hence "f * g \<noteq> 0" by auto
eberlm@63498
   402
  {
eberlm@63498
   403
    assume "\<not>is_unit (content (f * g))"
eberlm@63633
   404
    with False have "\<exists>p. p dvd content (f * g) \<and> prime p"
eberlm@63498
   405
      by (intro prime_divisor_exists) simp_all
eberlm@63633
   406
    then obtain p where "p dvd content (f * g)" "prime p" by blast
eberlm@63498
   407
    from \<open>p dvd content (f * g)\<close> have "[:p:] dvd f * g"
eberlm@63498
   408
      by (simp add: const_poly_dvd_iff_dvd_content)
eberlm@63633
   409
    moreover from \<open>prime p\<close> have "prime_elem [:p:]" by (simp add: lift_prime_elem_poly)
eberlm@63498
   410
    ultimately have "[:p:] dvd f \<or> [:p:] dvd g"
eberlm@63633
   411
      by (simp add: prime_elem_dvd_mult_iff)
eberlm@63498
   412
    with assms have "is_unit p" by (simp add: const_poly_dvd_iff_dvd_content)
eberlm@63633
   413
    with \<open>prime p\<close> have False by simp
eberlm@63498
   414
  }
eberlm@63498
   415
  hence "is_unit (content (f * g))" by blast
eberlm@63498
   416
  hence "normalize (content (f * g)) = 1" by (simp add: is_unit_normalize del: normalize_content)
eberlm@63498
   417
  thus ?thesis by simp
eberlm@63498
   418
qed (insert assms, auto)
eberlm@63498
   419
eberlm@63498
   420
lemma content_mult:
eberlm@63498
   421
  fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
eberlm@63498
   422
  shows "content (p * q) = content p * content q"
eberlm@63498
   423
proof -
eberlm@63498
   424
  from content_decompose[of p] guess p' . note p = this
eberlm@63498
   425
  from content_decompose[of q] guess q' . note q = this
eberlm@63498
   426
  have "content (p * q) = content p * content q * content (p' * q')"
eberlm@63498
   427
    by (subst p, subst q) (simp add: mult_ac normalize_mult)
eberlm@63498
   428
  also from p q have "content (p' * q') = 1" by (intro content_1_mult)
eberlm@63498
   429
  finally show ?thesis by simp
eberlm@63498
   430
qed
eberlm@63498
   431
eberlm@63498
   432
lemma primitive_part_mult:
eberlm@63498
   433
  fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
eberlm@63498
   434
  shows "primitive_part (p * q) = primitive_part p * primitive_part q"
eberlm@63498
   435
proof -
eberlm@63498
   436
  have "primitive_part (p * q) = p * q div [:content (p * q):]"
eberlm@63498
   437
    by (simp add: primitive_part_def div_const_poly_conv_map_poly)
eberlm@63498
   438
  also have "\<dots> = (p div [:content p:]) * (q div [:content q:])"
eberlm@63498
   439
    by (subst div_mult_div_if_dvd) (simp_all add: content_mult mult_ac)
eberlm@63498
   440
  also have "\<dots> = primitive_part p * primitive_part q"
eberlm@63498
   441
    by (simp add: primitive_part_def div_const_poly_conv_map_poly)
eberlm@63498
   442
  finally show ?thesis .
eberlm@63498
   443
qed
eberlm@63498
   444
eberlm@63498
   445
lemma primitive_part_smult:
eberlm@63498
   446
  fixes p :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
eberlm@63498
   447
  shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
eberlm@63498
   448
proof -
eberlm@63498
   449
  have "smult a p = [:a:] * p" by simp
eberlm@63498
   450
  also have "primitive_part \<dots> = smult (unit_factor a) (primitive_part p)"
eberlm@63498
   451
    by (subst primitive_part_mult) simp_all
eberlm@63498
   452
  finally show ?thesis .
eberlm@63498
   453
qed  
eberlm@63498
   454
eberlm@63498
   455
lemma primitive_part_dvd_primitive_partI [intro]:
eberlm@63498
   456
  fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
eberlm@63498
   457
  shows "p dvd q \<Longrightarrow> primitive_part p dvd primitive_part q"
eberlm@63498
   458
  by (auto elim!: dvdE simp: primitive_part_mult)
eberlm@63498
   459
nipkow@63830
   460
lemma content_prod_mset: 
eberlm@63498
   461
  fixes A :: "'a :: {factorial_semiring, semiring_Gcd} poly multiset"
nipkow@63830
   462
  shows "content (prod_mset A) = prod_mset (image_mset content A)"
eberlm@63498
   463
  by (induction A) (simp_all add: content_mult mult_ac)
eberlm@63498
   464
eberlm@63498
   465
lemma fract_poly_dvdD:
eberlm@63498
   466
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
eberlm@63498
   467
  assumes "fract_poly p dvd fract_poly q" "content p = 1"
eberlm@63498
   468
  shows   "p dvd q"
eberlm@63498
   469
proof -
eberlm@63498
   470
  from assms(1) obtain r where r: "fract_poly q = fract_poly p * r" by (erule dvdE)
eberlm@63498
   471
  from content_decompose_fract[of r] guess c r' . note r' = this
eberlm@63498
   472
  from r r' have eq: "fract_poly q = smult c (fract_poly (p * r'))" by simp  
eberlm@63498
   473
  from fract_poly_smult_eqE[OF this] guess a b . note ab = this
eberlm@63498
   474
  have "content (smult a q) = content (smult b (p * r'))" by (simp only: ab(2))
eberlm@63498
   475
  hence eq': "normalize b = a * content q" by (simp add: assms content_mult r' ab(4))
eberlm@63498
   476
  have "1 = gcd a (normalize b)" by (simp add: ab)
eberlm@63498
   477
  also note eq'
eberlm@63498
   478
  also have "gcd a (a * content q) = a" by (simp add: gcd_proj1_if_dvd ab(4))
eberlm@63498
   479
  finally have [simp]: "a = 1" by simp
eberlm@63498
   480
  from eq ab have "q = p * ([:b:] * r')" by simp
eberlm@63498
   481
  thus ?thesis by (rule dvdI)
eberlm@63498
   482
qed
eberlm@63498
   483
eberlm@63498
   484
lemma content_prod_eq_1_iff: 
eberlm@63498
   485
  fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
eberlm@63498
   486
  shows "content (p * q) = 1 \<longleftrightarrow> content p = 1 \<and> content q = 1"
eberlm@63498
   487
proof safe
eberlm@63498
   488
  assume A: "content (p * q) = 1"
eberlm@63498
   489
  {
eberlm@63498
   490
    fix p q :: "'a poly" assume "content p * content q = 1"
eberlm@63498
   491
    hence "1 = content p * content q" by simp
eberlm@63498
   492
    hence "content p dvd 1" by (rule dvdI)
eberlm@63498
   493
    hence "content p = 1" by simp
eberlm@63498
   494
  } note B = this
eberlm@63498
   495
  from A B[of p q] B [of q p] show "content p = 1" "content q = 1" 
eberlm@63498
   496
    by (simp_all add: content_mult mult_ac)
eberlm@63498
   497
qed (auto simp: content_mult)
eberlm@63498
   498
eberlm@63498
   499
end
eberlm@63498
   500
eberlm@63498
   501
eberlm@63498
   502
subsection \<open>Polynomials over a field are a Euclidean ring\<close>
eberlm@63498
   503
eberlm@63722
   504
definition unit_factor_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
eberlm@63498
   505
  "unit_factor_field_poly p = [:lead_coeff p:]"
eberlm@63498
   506
eberlm@63722
   507
definition normalize_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
eberlm@63498
   508
  "normalize_field_poly p = smult (inverse (lead_coeff p)) p"
eberlm@63498
   509
eberlm@63722
   510
definition euclidean_size_field_poly :: "'a :: field poly \<Rightarrow> nat" where
eberlm@63498
   511
  "euclidean_size_field_poly p = (if p = 0 then 0 else 2 ^ degree p)" 
eberlm@63498
   512
eberlm@63722
   513
lemma dvd_field_poly: "dvd.dvd (op * :: 'a :: field poly \<Rightarrow> _) = op dvd"
haftmann@64784
   514
  by (intro ext) (simp_all add: dvd.dvd_def dvd_def)
eberlm@63498
   515
eberlm@63498
   516
interpretation field_poly: 
haftmann@64784
   517
  unique_euclidean_ring where zero = "0 :: 'a :: field poly"
haftmann@64164
   518
    and one = 1 and plus = plus and uminus = uminus and minus = minus
haftmann@64164
   519
    and times = times
haftmann@64164
   520
    and normalize = normalize_field_poly and unit_factor = unit_factor_field_poly
haftmann@64164
   521
    and euclidean_size = euclidean_size_field_poly
haftmann@64784
   522
    and uniqueness_constraint = top
haftmann@64164
   523
    and divide = divide and modulo = modulo
eberlm@63498
   524
proof (standard, unfold dvd_field_poly)
eberlm@63498
   525
  fix p :: "'a poly"
eberlm@63498
   526
  show "unit_factor_field_poly p * normalize_field_poly p = p"
eberlm@63498
   527
    by (cases "p = 0") 
haftmann@64794
   528
       (simp_all add: unit_factor_field_poly_def normalize_field_poly_def)
eberlm@63498
   529
next
eberlm@63498
   530
  fix p :: "'a poly" assume "is_unit p"
haftmann@64848
   531
  then show "unit_factor_field_poly p = p"
haftmann@64848
   532
    by (elim is_unit_polyE) (auto simp: unit_factor_field_poly_def monom_0 one_poly_def field_simps)
eberlm@63498
   533
next
eberlm@63498
   534
  fix p :: "'a poly" assume "p \<noteq> 0"
eberlm@63498
   535
  thus "is_unit (unit_factor_field_poly p)"
haftmann@64794
   536
    by (simp add: unit_factor_field_poly_def is_unit_pCons_iff)
haftmann@64784
   537
next
haftmann@64784
   538
  fix p q s :: "'a poly" assume "s \<noteq> 0"
haftmann@64784
   539
  moreover assume "euclidean_size_field_poly p < euclidean_size_field_poly q"
haftmann@64784
   540
  ultimately show "euclidean_size_field_poly (p * s) < euclidean_size_field_poly (q * s)"
haftmann@64784
   541
    by (auto simp add: euclidean_size_field_poly_def degree_mult_eq)
haftmann@64784
   542
next
haftmann@64784
   543
  fix p q r :: "'a poly" assume "p \<noteq> 0"
haftmann@64784
   544
  moreover assume "euclidean_size_field_poly r < euclidean_size_field_poly p"
haftmann@64784
   545
  ultimately show "(q * p + r) div p = q"
haftmann@64784
   546
    by (cases "r = 0")
haftmann@64784
   547
      (auto simp add: unit_factor_field_poly_def euclidean_size_field_poly_def div_poly_less)
eberlm@63498
   548
qed (auto simp: unit_factor_field_poly_def normalize_field_poly_def lead_coeff_mult 
haftmann@64242
   549
       euclidean_size_field_poly_def Rings.div_mult_mod_eq intro!: degree_mod_less' degree_mult_right_le)
eberlm@63498
   550
eberlm@63722
   551
lemma field_poly_irreducible_imp_prime:
eberlm@63498
   552
  assumes "irreducible (p :: 'a :: field poly)"
eberlm@63633
   553
  shows   "prime_elem p"
eberlm@63498
   554
proof -
eberlm@63498
   555
  have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
eberlm@63633
   556
  from field_poly.irreducible_imp_prime_elem[of p] assms
eberlm@63633
   557
    show ?thesis unfolding irreducible_def prime_elem_def dvd_field_poly
eberlm@63633
   558
      comm_semiring_1.irreducible_def[OF A] comm_semiring_1.prime_elem_def[OF A] by blast
eberlm@63498
   559
qed
eberlm@63498
   560
nipkow@63830
   561
lemma field_poly_prod_mset_prime_factorization:
eberlm@63498
   562
  assumes "(x :: 'a :: field poly) \<noteq> 0"
nipkow@63830
   563
  shows   "prod_mset (field_poly.prime_factorization x) = normalize_field_poly x"
eberlm@63498
   564
proof -
eberlm@63498
   565
  have A: "class.comm_monoid_mult op * (1 :: 'a poly)" ..
nipkow@63830
   566
  have "comm_monoid_mult.prod_mset op * (1 :: 'a poly) = prod_mset"
nipkow@63830
   567
    by (intro ext) (simp add: comm_monoid_mult.prod_mset_def[OF A] prod_mset_def)
nipkow@63830
   568
  with field_poly.prod_mset_prime_factorization[OF assms] show ?thesis by simp
eberlm@63498
   569
qed
eberlm@63498
   570
eberlm@63722
   571
lemma field_poly_in_prime_factorization_imp_prime:
eberlm@63498
   572
  assumes "(p :: 'a :: field poly) \<in># field_poly.prime_factorization x"
eberlm@63633
   573
  shows   "prime_elem p"
eberlm@63498
   574
proof -
eberlm@63498
   575
  have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
eberlm@63498
   576
  have B: "class.normalization_semidom op div op + op - (0 :: 'a poly) op * 1 
haftmann@64848
   577
             unit_factor_field_poly normalize_field_poly" ..
haftmann@63905
   578
  from field_poly.in_prime_factors_imp_prime [of p x] assms
eberlm@63633
   579
    show ?thesis unfolding prime_elem_def dvd_field_poly
eberlm@63633
   580
      comm_semiring_1.prime_elem_def[OF A] normalization_semidom.prime_def[OF B] by blast
eberlm@63498
   581
qed
eberlm@63498
   582
eberlm@63498
   583
eberlm@63498
   584
subsection \<open>Primality and irreducibility in polynomial rings\<close>
eberlm@63498
   585
eberlm@63498
   586
lemma nonconst_poly_irreducible_iff:
eberlm@63498
   587
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
eberlm@63498
   588
  assumes "degree p \<noteq> 0"
eberlm@63498
   589
  shows   "irreducible p \<longleftrightarrow> irreducible (fract_poly p) \<and> content p = 1"
eberlm@63498
   590
proof safe
eberlm@63498
   591
  assume p: "irreducible p"
eberlm@63498
   592
eberlm@63498
   593
  from content_decompose[of p] guess p' . note p' = this
eberlm@63498
   594
  hence "p = [:content p:] * p'" by simp
eberlm@63498
   595
  from p this have "[:content p:] dvd 1 \<or> p' dvd 1" by (rule irreducibleD)
eberlm@63498
   596
  moreover have "\<not>p' dvd 1"
eberlm@63498
   597
  proof
eberlm@63498
   598
    assume "p' dvd 1"
eberlm@63498
   599
    hence "degree p = 0" by (subst p') (auto simp: is_unit_poly_iff)
eberlm@63498
   600
    with assms show False by contradiction
eberlm@63498
   601
  qed
eberlm@63498
   602
  ultimately show [simp]: "content p = 1" by (simp add: is_unit_const_poly_iff)
eberlm@63498
   603
  
eberlm@63498
   604
  show "irreducible (map_poly to_fract p)"
eberlm@63498
   605
  proof (rule irreducibleI)
eberlm@63498
   606
    have "fract_poly p = 0 \<longleftrightarrow> p = 0" by (intro map_poly_eq_0_iff) auto
eberlm@63498
   607
    with assms show "map_poly to_fract p \<noteq> 0" by auto
eberlm@63498
   608
  next
eberlm@63498
   609
    show "\<not>is_unit (fract_poly p)"
eberlm@63498
   610
    proof
eberlm@63498
   611
      assume "is_unit (map_poly to_fract p)"
eberlm@63498
   612
      hence "degree (map_poly to_fract p) = 0"
eberlm@63498
   613
        by (auto simp: is_unit_poly_iff)
eberlm@63498
   614
      hence "degree p = 0" by (simp add: degree_map_poly)
eberlm@63498
   615
      with assms show False by contradiction
eberlm@63498
   616
   qed
eberlm@63498
   617
 next
eberlm@63498
   618
   fix q r assume qr: "fract_poly p = q * r"
eberlm@63498
   619
   from content_decompose_fract[of q] guess cg q' . note q = this
eberlm@63498
   620
   from content_decompose_fract[of r] guess cr r' . note r = this
eberlm@63498
   621
   from qr q r p have nz: "cg \<noteq> 0" "cr \<noteq> 0" by auto
eberlm@63498
   622
   from qr have eq: "fract_poly p = smult (cr * cg) (fract_poly (q' * r'))"
eberlm@63498
   623
     by (simp add: q r)
eberlm@63498
   624
   from fract_poly_smult_eqE[OF this] guess a b . note ab = this
eberlm@63498
   625
   hence "content (smult a p) = content (smult b (q' * r'))" by (simp only:)
eberlm@63498
   626
   with ab(4) have a: "a = normalize b" by (simp add: content_mult q r)
eberlm@63498
   627
   hence "normalize b = gcd a b" by simp
eberlm@63498
   628
   also from ab(3) have "\<dots> = 1" .
eberlm@63498
   629
   finally have "a = 1" "is_unit b" by (simp_all add: a normalize_1_iff)
eberlm@63498
   630
   
eberlm@63498
   631
   note eq
eberlm@63498
   632
   also from ab(1) \<open>a = 1\<close> have "cr * cg = to_fract b" by simp
eberlm@63498
   633
   also have "smult \<dots> (fract_poly (q' * r')) = fract_poly (smult b (q' * r'))" by simp
eberlm@63498
   634
   finally have "p = ([:b:] * q') * r'" by (simp del: fract_poly_smult)
eberlm@63498
   635
   from p and this have "([:b:] * q') dvd 1 \<or> r' dvd 1" by (rule irreducibleD)
eberlm@63498
   636
   hence "q' dvd 1 \<or> r' dvd 1" by (auto dest: dvd_mult_right simp del: mult_pCons_left)
eberlm@63498
   637
   hence "fract_poly q' dvd 1 \<or> fract_poly r' dvd 1" by (auto simp: fract_poly_is_unit)
eberlm@63498
   638
   with q r show "is_unit q \<or> is_unit r"
eberlm@63498
   639
     by (auto simp add: is_unit_smult_iff dvd_field_iff nz)
eberlm@63498
   640
 qed
eberlm@63498
   641
eberlm@63498
   642
next
eberlm@63498
   643
eberlm@63498
   644
  assume irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
eberlm@63498
   645
  show "irreducible p"
eberlm@63498
   646
  proof (rule irreducibleI)
eberlm@63498
   647
    from irred show "p \<noteq> 0" by auto
eberlm@63498
   648
  next
eberlm@63498
   649
    from irred show "\<not>p dvd 1"
eberlm@63498
   650
      by (auto simp: irreducible_def dest: fract_poly_is_unit)
eberlm@63498
   651
  next
eberlm@63498
   652
    fix q r assume qr: "p = q * r"
eberlm@63498
   653
    hence "fract_poly p = fract_poly q * fract_poly r" by simp
eberlm@63498
   654
    from irred and this have "fract_poly q dvd 1 \<or> fract_poly r dvd 1" 
eberlm@63498
   655
      by (rule irreducibleD)
eberlm@63498
   656
    with primitive qr show "q dvd 1 \<or> r dvd 1"
eberlm@63498
   657
      by (auto simp:  content_prod_eq_1_iff is_unit_fract_poly_iff)
eberlm@63498
   658
  qed
eberlm@63498
   659
qed
eberlm@63498
   660
eberlm@63722
   661
context
eberlm@63722
   662
begin
eberlm@63722
   663
eberlm@63498
   664
private lemma irreducible_imp_prime_poly:
eberlm@63498
   665
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
eberlm@63498
   666
  assumes "irreducible p"
eberlm@63633
   667
  shows   "prime_elem p"
eberlm@63498
   668
proof (cases "degree p = 0")
eberlm@63498
   669
  case True
eberlm@63498
   670
  with assms show ?thesis
eberlm@63498
   671
    by (auto simp: prime_elem_const_poly_iff irreducible_const_poly_iff
eberlm@63633
   672
             intro!: irreducible_imp_prime_elem elim!: degree_eq_zeroE)
eberlm@63498
   673
next
eberlm@63498
   674
  case False
eberlm@63498
   675
  from assms False have irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
eberlm@63498
   676
    by (simp_all add: nonconst_poly_irreducible_iff)
eberlm@63633
   677
  from irred have prime: "prime_elem (fract_poly p)" by (rule field_poly_irreducible_imp_prime)
eberlm@63498
   678
  show ?thesis
eberlm@63633
   679
  proof (rule prime_elemI)
eberlm@63498
   680
    fix q r assume "p dvd q * r"
eberlm@63498
   681
    hence "fract_poly p dvd fract_poly (q * r)" by (rule fract_poly_dvd)
eberlm@63498
   682
    hence "fract_poly p dvd fract_poly q * fract_poly r" by simp
eberlm@63498
   683
    from prime and this have "fract_poly p dvd fract_poly q \<or> fract_poly p dvd fract_poly r"
eberlm@63633
   684
      by (rule prime_elem_dvd_multD)
eberlm@63498
   685
    with primitive show "p dvd q \<or> p dvd r" by (auto dest: fract_poly_dvdD)
eberlm@63498
   686
  qed (insert assms, auto simp: irreducible_def)
eberlm@63498
   687
qed
eberlm@63498
   688
eberlm@63498
   689
eberlm@63498
   690
lemma degree_primitive_part_fract [simp]:
eberlm@63498
   691
  "degree (primitive_part_fract p) = degree p"
eberlm@63498
   692
proof -
eberlm@63498
   693
  have "p = smult (fract_content p) (fract_poly (primitive_part_fract p))"
eberlm@63498
   694
    by (simp add: content_times_primitive_part_fract)
eberlm@63498
   695
  also have "degree \<dots> = degree (primitive_part_fract p)"
eberlm@63498
   696
    by (auto simp: degree_map_poly)
eberlm@63498
   697
  finally show ?thesis ..
eberlm@63498
   698
qed
eberlm@63498
   699
eberlm@63498
   700
lemma irreducible_primitive_part_fract:
eberlm@63498
   701
  fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
eberlm@63498
   702
  assumes "irreducible p"
eberlm@63498
   703
  shows   "irreducible (primitive_part_fract p)"
eberlm@63498
   704
proof -
eberlm@63498
   705
  from assms have deg: "degree (primitive_part_fract p) \<noteq> 0"
eberlm@63498
   706
    by (intro notI) 
eberlm@63498
   707
       (auto elim!: degree_eq_zeroE simp: irreducible_def is_unit_poly_iff dvd_field_iff)
eberlm@63498
   708
  hence [simp]: "p \<noteq> 0" by auto
eberlm@63498
   709
eberlm@63498
   710
  note \<open>irreducible p\<close>
eberlm@63498
   711
  also have "p = [:fract_content p:] * fract_poly (primitive_part_fract p)" 
eberlm@63498
   712
    by (simp add: content_times_primitive_part_fract)
eberlm@63498
   713
  also have "irreducible \<dots> \<longleftrightarrow> irreducible (fract_poly (primitive_part_fract p))"
eberlm@63498
   714
    by (intro irreducible_mult_unit_left) (simp_all add: is_unit_poly_iff dvd_field_iff)
eberlm@63498
   715
  finally show ?thesis using deg
eberlm@63498
   716
    by (simp add: nonconst_poly_irreducible_iff)
eberlm@63498
   717
qed
eberlm@63498
   718
eberlm@63633
   719
lemma prime_elem_primitive_part_fract:
eberlm@63498
   720
  fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd} fract poly"
eberlm@63633
   721
  shows "irreducible p \<Longrightarrow> prime_elem (primitive_part_fract p)"
eberlm@63498
   722
  by (intro irreducible_imp_prime_poly irreducible_primitive_part_fract)
eberlm@63498
   723
eberlm@63498
   724
lemma irreducible_linear_field_poly:
eberlm@63498
   725
  fixes a b :: "'a::field"
eberlm@63498
   726
  assumes "b \<noteq> 0"
eberlm@63498
   727
  shows "irreducible [:a,b:]"
eberlm@63498
   728
proof (rule irreducibleI)
eberlm@63498
   729
  fix p q assume pq: "[:a,b:] = p * q"
wenzelm@63539
   730
  also from pq assms have "degree \<dots> = degree p + degree q" 
eberlm@63498
   731
    by (intro degree_mult_eq) auto
eberlm@63498
   732
  finally have "degree p = 0 \<or> degree q = 0" using assms by auto
eberlm@63498
   733
  with assms pq show "is_unit p \<or> is_unit q"
eberlm@63498
   734
    by (auto simp: is_unit_const_poly_iff dvd_field_iff elim!: degree_eq_zeroE)
eberlm@63498
   735
qed (insert assms, auto simp: is_unit_poly_iff)
eberlm@63498
   736
eberlm@63633
   737
lemma prime_elem_linear_field_poly:
eberlm@63633
   738
  "(b :: 'a :: field) \<noteq> 0 \<Longrightarrow> prime_elem [:a,b:]"
eberlm@63498
   739
  by (rule field_poly_irreducible_imp_prime, rule irreducible_linear_field_poly)
eberlm@63498
   740
eberlm@63498
   741
lemma irreducible_linear_poly:
eberlm@63498
   742
  fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
eberlm@63498
   743
  shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> irreducible [:a,b:]"
eberlm@63498
   744
  by (auto intro!: irreducible_linear_field_poly 
eberlm@63498
   745
           simp:   nonconst_poly_irreducible_iff content_def map_poly_pCons)
eberlm@63498
   746
eberlm@63633
   747
lemma prime_elem_linear_poly:
eberlm@63498
   748
  fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd}"
eberlm@63633
   749
  shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> prime_elem [:a,b:]"
eberlm@63498
   750
  by (rule irreducible_imp_prime_poly, rule irreducible_linear_poly)
eberlm@63498
   751
eberlm@63722
   752
end
eberlm@63722
   753
haftmann@64591
   754
 
eberlm@63498
   755
subsection \<open>Prime factorisation of polynomials\<close>   
eberlm@63498
   756
eberlm@63722
   757
context
eberlm@63722
   758
begin 
eberlm@63722
   759
eberlm@63498
   760
private lemma poly_prime_factorization_exists_content_1:
eberlm@63498
   761
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
eberlm@63498
   762
  assumes "p \<noteq> 0" "content p = 1"
nipkow@63830
   763
  shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
eberlm@63498
   764
proof -
eberlm@63498
   765
  let ?P = "field_poly.prime_factorization (fract_poly p)"
nipkow@63830
   766
  define c where "c = prod_mset (image_mset fract_content ?P)"
eberlm@63498
   767
  define c' where "c' = c * to_fract (lead_coeff p)"
nipkow@63830
   768
  define e where "e = prod_mset (image_mset primitive_part_fract ?P)"
eberlm@63498
   769
  define A where "A = image_mset (normalize \<circ> primitive_part_fract) ?P"
eberlm@63498
   770
  have "content e = (\<Prod>x\<in>#field_poly.prime_factorization (map_poly to_fract p). 
eberlm@63498
   771
                      content (primitive_part_fract x))"
nipkow@63830
   772
    by (simp add: e_def content_prod_mset multiset.map_comp o_def)
eberlm@63498
   773
  also have "image_mset (\<lambda>x. content (primitive_part_fract x)) ?P = image_mset (\<lambda>_. 1) ?P"
eberlm@63498
   774
    by (intro image_mset_cong content_primitive_part_fract) auto
haftmann@64591
   775
  finally have content_e: "content e = 1"
haftmann@64591
   776
    by simp    
eberlm@63498
   777
  
eberlm@63498
   778
  have "fract_poly p = unit_factor_field_poly (fract_poly p) * 
eberlm@63498
   779
          normalize_field_poly (fract_poly p)" by simp
eberlm@63498
   780
  also have "unit_factor_field_poly (fract_poly p) = [:to_fract (lead_coeff p):]" 
haftmann@64794
   781
    by (simp add: unit_factor_field_poly_def monom_0 degree_map_poly coeff_map_poly)
nipkow@63830
   782
  also from assms have "normalize_field_poly (fract_poly p) = prod_mset ?P" 
nipkow@63830
   783
    by (subst field_poly_prod_mset_prime_factorization) simp_all
nipkow@63830
   784
  also have "\<dots> = prod_mset (image_mset id ?P)" by simp
eberlm@63498
   785
  also have "image_mset id ?P = 
eberlm@63498
   786
               image_mset (\<lambda>x. [:fract_content x:] * fract_poly (primitive_part_fract x)) ?P"
eberlm@63498
   787
    by (intro image_mset_cong) (auto simp: content_times_primitive_part_fract)
nipkow@63830
   788
  also have "prod_mset \<dots> = smult c (fract_poly e)"
haftmann@64591
   789
    by (subst prod_mset.distrib) (simp_all add: prod_mset_fract_poly prod_mset_const_poly c_def e_def)
eberlm@63498
   790
  also have "[:to_fract (lead_coeff p):] * \<dots> = smult c' (fract_poly e)"
eberlm@63498
   791
    by (simp add: c'_def)
eberlm@63498
   792
  finally have eq: "fract_poly p = smult c' (fract_poly e)" .
eberlm@63498
   793
  also obtain b where b: "c' = to_fract b" "is_unit b"
eberlm@63498
   794
  proof -
eberlm@63498
   795
    from fract_poly_smult_eqE[OF eq] guess a b . note ab = this
eberlm@63498
   796
    from ab(2) have "content (smult a p) = content (smult b e)" by (simp only: )
eberlm@63498
   797
    with assms content_e have "a = normalize b" by (simp add: ab(4))
eberlm@63498
   798
    with ab have ab': "a = 1" "is_unit b" by (simp_all add: normalize_1_iff)
eberlm@63498
   799
    with ab ab' have "c' = to_fract b" by auto
eberlm@63498
   800
    from this and \<open>is_unit b\<close> show ?thesis by (rule that)
eberlm@63498
   801
  qed
eberlm@63498
   802
  hence "smult c' (fract_poly e) = fract_poly (smult b e)" by simp
eberlm@63498
   803
  finally have "p = smult b e" by (simp only: fract_poly_eq_iff)
eberlm@63498
   804
  hence "p = [:b:] * e" by simp
eberlm@63498
   805
  with b have "normalize p = normalize e" 
eberlm@63498
   806
    by (simp only: normalize_mult) (simp add: is_unit_normalize is_unit_poly_iff)
nipkow@63830
   807
  also have "normalize e = prod_mset A"
nipkow@63830
   808
    by (simp add: multiset.map_comp e_def A_def normalize_prod_mset)
nipkow@63830
   809
  finally have "prod_mset A = normalize p" ..
eberlm@63498
   810
  
eberlm@63633
   811
  have "prime_elem p" if "p \<in># A" for p
eberlm@63633
   812
    using that by (auto simp: A_def prime_elem_primitive_part_fract prime_elem_imp_irreducible 
eberlm@63498
   813
                        dest!: field_poly_in_prime_factorization_imp_prime )
nipkow@63830
   814
  from this and \<open>prod_mset A = normalize p\<close> show ?thesis
eberlm@63498
   815
    by (intro exI[of _ A]) blast
eberlm@63498
   816
qed
eberlm@63498
   817
eberlm@63498
   818
lemma poly_prime_factorization_exists:
eberlm@63498
   819
  fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
eberlm@63498
   820
  assumes "p \<noteq> 0"
nipkow@63830
   821
  shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
eberlm@63498
   822
proof -
eberlm@63498
   823
  define B where "B = image_mset (\<lambda>x. [:x:]) (prime_factorization (content p))"
nipkow@63830
   824
  have "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize (primitive_part p)"
eberlm@63498
   825
    by (rule poly_prime_factorization_exists_content_1) (insert assms, simp_all)
eberlm@63498
   826
  then guess A by (elim exE conjE) note A = this
nipkow@63830
   827
  moreover from assms have "prod_mset B = [:content p:]"
nipkow@63830
   828
    by (simp add: B_def prod_mset_const_poly prod_mset_prime_factorization)
eberlm@63633
   829
  moreover have "\<forall>p. p \<in># B \<longrightarrow> prime_elem p"
haftmann@63905
   830
    by (auto simp: B_def intro!: lift_prime_elem_poly dest: in_prime_factors_imp_prime)
eberlm@63498
   831
  ultimately show ?thesis by (intro exI[of _ "B + A"]) auto
eberlm@63498
   832
qed
eberlm@63498
   833
eberlm@63498
   834
end
eberlm@63498
   835
eberlm@63498
   836
eberlm@63498
   837
subsection \<open>Typeclass instances\<close>
eberlm@63498
   838
eberlm@63498
   839
instance poly :: (factorial_ring_gcd) factorial_semiring
eberlm@63498
   840
  by standard (rule poly_prime_factorization_exists)  
eberlm@63498
   841
eberlm@63498
   842
instantiation poly :: (factorial_ring_gcd) factorial_ring_gcd
eberlm@63498
   843
begin
eberlm@63498
   844
eberlm@63498
   845
definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
eberlm@63498
   846
  [code del]: "gcd_poly = gcd_factorial"
eberlm@63498
   847
eberlm@63498
   848
definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
eberlm@63498
   849
  [code del]: "lcm_poly = lcm_factorial"
eberlm@63498
   850
  
eberlm@63498
   851
definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
eberlm@63498
   852
 [code del]: "Gcd_poly = Gcd_factorial"
eberlm@63498
   853
eberlm@63498
   854
definition Lcm_poly :: "'a poly set \<Rightarrow> 'a poly" where
eberlm@63498
   855
 [code del]: "Lcm_poly = Lcm_factorial"
eberlm@63498
   856
 
eberlm@63498
   857
instance by standard (simp_all add: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)
eberlm@63498
   858
eberlm@63498
   859
end
eberlm@63498
   860
haftmann@64784
   861
instantiation poly :: ("{field,factorial_ring_gcd}") unique_euclidean_ring
eberlm@63498
   862
begin
eberlm@63498
   863
haftmann@64784
   864
definition euclidean_size_poly :: "'a poly \<Rightarrow> nat"
haftmann@64784
   865
  where "euclidean_size_poly p = (if p = 0 then 0 else 2 ^ degree p)"
haftmann@64784
   866
haftmann@64784
   867
definition uniqueness_constraint_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
haftmann@64784
   868
  where [simp]: "uniqueness_constraint_poly = top"
eberlm@63498
   869
eberlm@63498
   870
instance 
haftmann@64784
   871
  by standard
haftmann@64784
   872
   (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
haftmann@64784
   873
    split: if_splits)
haftmann@64784
   874
eberlm@63498
   875
end
eberlm@63498
   876
eberlm@63498
   877
instance poly :: ("{field,factorial_ring_gcd}") euclidean_ring_gcd
haftmann@64786
   878
  by (rule euclidean_ring_gcd_class.intro, rule factorial_euclidean_semiring_gcdI)
haftmann@64786
   879
    standard
eberlm@63498
   880
eberlm@63498
   881
  
eberlm@63498
   882
subsection \<open>Polynomial GCD\<close>
eberlm@63498
   883
eberlm@63498
   884
lemma gcd_poly_decompose:
eberlm@63498
   885
  fixes p q :: "'a :: factorial_ring_gcd poly"
eberlm@63498
   886
  shows "gcd p q = 
eberlm@63498
   887
           smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
eberlm@63498
   888
proof (rule sym, rule gcdI)
eberlm@63498
   889
  have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
eberlm@63498
   890
          [:content p:] * primitive_part p" by (intro mult_dvd_mono) simp_all
eberlm@63498
   891
  thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd p"
eberlm@63498
   892
    by simp
eberlm@63498
   893
next
eberlm@63498
   894
  have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
eberlm@63498
   895
          [:content q:] * primitive_part q" by (intro mult_dvd_mono) simp_all
eberlm@63498
   896
  thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd q"
eberlm@63498
   897
    by simp
eberlm@63498
   898
next
eberlm@63498
   899
  fix d assume "d dvd p" "d dvd q"
eberlm@63498
   900
  hence "[:content d:] * primitive_part d dvd 
eberlm@63498
   901
           [:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q)"
eberlm@63498
   902
    by (intro mult_dvd_mono) auto
eberlm@63498
   903
  thus "d dvd smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
eberlm@63498
   904
    by simp
eberlm@63498
   905
qed (auto simp: normalize_smult)
eberlm@63498
   906
  
eberlm@63498
   907
eberlm@63498
   908
lemma gcd_poly_pseudo_mod:
eberlm@63498
   909
  fixes p q :: "'a :: factorial_ring_gcd poly"
eberlm@63498
   910
  assumes nz: "q \<noteq> 0" and prim: "content p = 1" "content q = 1"
eberlm@63498
   911
  shows   "gcd p q = gcd q (primitive_part (pseudo_mod p q))"
eberlm@63498
   912
proof -
eberlm@63498
   913
  define r s where "r = fst (pseudo_divmod p q)" and "s = snd (pseudo_divmod p q)"
eberlm@63498
   914
  define a where "a = [:coeff q (degree q) ^ (Suc (degree p) - degree q):]"
eberlm@63498
   915
  have [simp]: "primitive_part a = unit_factor a"
eberlm@63498
   916
    by (simp add: a_def unit_factor_poly_def unit_factor_power monom_0)
eberlm@63498
   917
  from nz have [simp]: "a \<noteq> 0" by (auto simp: a_def)
eberlm@63498
   918
  
eberlm@63498
   919
  have rs: "pseudo_divmod p q = (r, s)" by (simp add: r_def s_def)
eberlm@63498
   920
  have "gcd (q * r + s) q = gcd q s"
eberlm@63498
   921
    using gcd_add_mult[of q r s] by (simp add: gcd.commute add_ac mult_ac)
eberlm@63498
   922
  with pseudo_divmod(1)[OF nz rs]
eberlm@63498
   923
    have "gcd (p * a) q = gcd q s" by (simp add: a_def)
eberlm@63498
   924
  also from prim have "gcd (p * a) q = gcd p q"
eberlm@63498
   925
    by (subst gcd_poly_decompose)
eberlm@63498
   926
       (auto simp: primitive_part_mult gcd_mult_unit1 primitive_part_prim 
eberlm@63498
   927
             simp del: mult_pCons_right )
eberlm@63498
   928
  also from prim have "gcd q s = gcd q (primitive_part s)"
eberlm@63498
   929
    by (subst gcd_poly_decompose) (simp_all add: primitive_part_prim)
eberlm@63498
   930
  also have "s = pseudo_mod p q" by (simp add: s_def pseudo_mod_def)
eberlm@63498
   931
  finally show ?thesis .
eberlm@63498
   932
qed
eberlm@63498
   933
eberlm@63498
   934
lemma degree_pseudo_mod_less:
eberlm@63498
   935
  assumes "q \<noteq> 0" "pseudo_mod p q \<noteq> 0"
eberlm@63498
   936
  shows   "degree (pseudo_mod p q) < degree q"
eberlm@63498
   937
  using pseudo_mod(2)[of q p] assms by auto
eberlm@63498
   938
eberlm@63498
   939
function gcd_poly_code_aux :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
eberlm@63498
   940
  "gcd_poly_code_aux p q = 
eberlm@63498
   941
     (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))" 
eberlm@63498
   942
by auto
eberlm@63498
   943
termination
eberlm@63498
   944
  by (relation "measure ((\<lambda>p. if p = 0 then 0 else Suc (degree p)) \<circ> snd)")
haftmann@64164
   945
     (auto simp: degree_pseudo_mod_less)
eberlm@63498
   946
eberlm@63498
   947
declare gcd_poly_code_aux.simps [simp del]
eberlm@63498
   948
eberlm@63498
   949
lemma gcd_poly_code_aux_correct:
eberlm@63498
   950
  assumes "content p = 1" "q = 0 \<or> content q = 1"
eberlm@63498
   951
  shows   "gcd_poly_code_aux p q = gcd p q"
eberlm@63498
   952
  using assms
eberlm@63498
   953
proof (induction p q rule: gcd_poly_code_aux.induct)
eberlm@63498
   954
  case (1 p q)
eberlm@63498
   955
  show ?case
eberlm@63498
   956
  proof (cases "q = 0")
eberlm@63498
   957
    case True
eberlm@63498
   958
    thus ?thesis by (subst gcd_poly_code_aux.simps) auto
eberlm@63498
   959
  next
eberlm@63498
   960
    case False
eberlm@63498
   961
    hence "gcd_poly_code_aux p q = gcd_poly_code_aux q (primitive_part (pseudo_mod p q))"
eberlm@63498
   962
      by (subst gcd_poly_code_aux.simps) simp_all
eberlm@63498
   963
    also from "1.prems" False 
eberlm@63498
   964
      have "primitive_part (pseudo_mod p q) = 0 \<or> 
eberlm@63498
   965
              content (primitive_part (pseudo_mod p q)) = 1"
eberlm@63498
   966
      by (cases "pseudo_mod p q = 0") auto
eberlm@63498
   967
    with "1.prems" False 
eberlm@63498
   968
      have "gcd_poly_code_aux q (primitive_part (pseudo_mod p q)) = 
eberlm@63498
   969
              gcd q (primitive_part (pseudo_mod p q))"
eberlm@63498
   970
      by (intro 1) simp_all
eberlm@63498
   971
    also from "1.prems" False 
eberlm@63498
   972
      have "\<dots> = gcd p q" by (intro gcd_poly_pseudo_mod [symmetric]) auto
eberlm@63498
   973
    finally show ?thesis .
eberlm@63498
   974
  qed
eberlm@63498
   975
qed
eberlm@63498
   976
eberlm@63498
   977
definition gcd_poly_code 
eberlm@63498
   978
    :: "'a :: factorial_ring_gcd poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" 
eberlm@63498
   979
  where "gcd_poly_code p q = 
eberlm@63498
   980
           (if p = 0 then normalize q else if q = 0 then normalize p else
eberlm@63498
   981
              smult (gcd (content p) (content q)) 
eberlm@63498
   982
                (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
eberlm@63498
   983
haftmann@64591
   984
lemma gcd_poly_code [code]: "gcd p q = gcd_poly_code p q"
haftmann@64591
   985
  by (simp add: gcd_poly_code_def gcd_poly_code_aux_correct gcd_poly_decompose [symmetric])
haftmann@64591
   986
eberlm@63498
   987
lemma lcm_poly_code [code]: 
eberlm@63498
   988
  fixes p q :: "'a :: factorial_ring_gcd poly"
eberlm@63498
   989
  shows "lcm p q = normalize (p * q) div gcd p q"
haftmann@64591
   990
  by (fact lcm_gcd)
eberlm@63498
   991
haftmann@64850
   992
lemmas Gcd_poly_set_eq_fold [code] = Gcd_set_eq_fold [where ?'a = "'a :: factorial_ring_gcd poly"]
haftmann@64850
   993
lemmas Lcm_poly_set_eq_fold [code] = Lcm_set_eq_fold [where ?'a = "'a :: factorial_ring_gcd poly"]
haftmann@64860
   994
haftmann@64591
   995
text \<open>Example:
haftmann@64591
   996
  @{lemma "Lcm {[:1, 2, 3:], [:2, 3, 4:]} = [:[:2:], [:7:], [:16:], [:17:], [:12 :: int:]:]" by eval}
haftmann@64591
   997
\<close>
eberlm@63498
   998
  
wenzelm@63764
   999
end