Theory Missing_Polynomial

(*  
    Author:      René Thiemann 
                 Akihisa Yamada
                 Jose Divason
    License:     BSD
*)
section ‹Missing Polynomial›

text ‹The theory contains some basic results on polynomials which have not been detected in
  the distribution, especially on linear factors and degrees.›

theory Missing_Polynomial
imports 
  "HOL-Computational_Algebra.Polynomial_Factorial"
  Missing_Unsorted
begin

subsection ‹Basic Properties›

lemma degree_0_id: assumes "degree p = 0"
  shows "[: coeff p 0 :] = p" 
proof -
  have "⋀ x. 0 ≠ Suc x" by auto 
  thus ?thesis using assms
  by (metis coeff_pCons_0 degree_pCons_eq_if pCons_cases)
qed

lemma degree0_coeffs: "degree p = 0 ⟹
  ∃ a. p = [: a :]"
  by (metis degree_pCons_eq_if old.nat.distinct(2) pCons_cases)

lemma degree1_coeffs: "degree p = 1 ⟹
  ∃ a b. p = [: b, a :] ∧ a ≠ 0" 
  by (metis One_nat_def degree_pCons_eq_if nat.inject old.nat.distinct(2) pCons_0_0 pCons_cases)

lemma degree2_coeffs: "degree p = 2 ⟹
  ∃ a b c. p = [: c, b, a :] ∧ a ≠ 0"
  by (metis Suc_1 Suc_neq_Zero degree1_coeffs degree_pCons_eq_if nat.inject pCons_cases)

lemma poly_zero:
  fixes p :: "'a :: comm_ring_1 poly"
  assumes x: "poly p x = 0" shows "p = 0 ⟷ degree p = 0"
proof
  assume degp: "degree p = 0"
  hence "poly p x = coeff p (degree p)" by(subst degree_0_id[OF degp,symmetric], simp)
  hence "coeff p (degree p) = 0" using x by auto
  thus "p = 0" by auto
qed auto

lemma coeff_monom_Suc: "coeff (monom a (Suc d) * p) (Suc i) = coeff (monom a d * p) i"
  by (simp add: monom_Suc)

lemma coeff_sum_monom:
  assumes n: "n ≤ d"
  shows "coeff (∑i≤d. monom (f i) i) n = f n" (is "?l = _")
proof -
  have "?l = (∑i≤d. coeff (monom (f i) i) n)" (is "_ = sum ?cmf _")
    using coeff_sum.
  also have "{..d} = insert n ({..d}-{n})" using n by auto
    hence "sum ?cmf {..d} = sum ?cmf ..." by auto
  also have "... = sum ?cmf ({..d}-{n}) + ?cmf n" by (subst sum.insert,auto)
  also have "sum ?cmf ({..d}-{n}) = 0" by (subst sum.neutral, auto)
  finally show ?thesis by simp
qed

lemma linear_poly_root: "(a :: 'a :: comm_ring_1) ∈ set as ⟹ poly (∏ a ← as. [: - a, 1:]) a = 0"
proof (induct as)
  case (Cons b as)
  show ?case
  proof (cases "a = b")
    case False
    with Cons have "a ∈ set as" by auto
    from Cons(1)[OF this] show ?thesis by simp
  qed simp
qed simp

lemma degree_lcoeff_sum: assumes deg: "degree (f q) = n"
  and fin: "finite S" and q: "q ∈ S" and degle: "⋀ p . p ∈ S - {q} ⟹ degree (f p) < n"
  and cong: "coeff (f q) n = c"
  shows "degree (sum f S) = n ∧ coeff (sum f S) n = c"
proof (cases "S = {q}")
  case True
  thus ?thesis using deg cong by simp
next
  case False
  with q obtain p where "p ∈ S - {q}" by auto
  from degle[OF this] have n: "n > 0" by auto
  have "degree (sum f S) = degree (f q + sum f (S - {q}))"
    unfolding sum.remove[OF fin q] ..
  also have "… = degree (f q)"
  proof (rule degree_add_eq_left)
    have "degree (sum f (S - {q})) ≤ n - 1"
    proof (rule degree_sum_le)
      fix p
      show "p ∈ S - {q} ⟹ degree (f p) ≤ n - 1"
        using degle[of p] by auto
    qed (insert fin, auto)
    also have "… < n" using n by simp
    finally show "degree (sum f (S - {q})) < degree (f q)" unfolding deg .
  qed
  finally show ?thesis unfolding deg[symmetric] cong[symmetric]
  proof (rule conjI)
    have id: "(∑x∈S - {q}. coeff (f x) (degree (f q))) = 0"
      by (rule sum.neutral, rule ballI, rule coeff_eq_0[OF degle[folded deg]])
    show "coeff (sum f S) (degree (f q)) = coeff (f q) (degree (f q))"
      unfolding coeff_sum
      by (subst sum.remove[OF _ q], unfold id, insert fin, auto)
  qed
qed

lemma degree_sum_list_le: "(⋀ p . p ∈ set ps ⟹ degree p ≤ n)
  ⟹ degree (sum_list ps) ≤ n"
proof (induct ps)
  case (Cons p ps)
  hence "degree (sum_list ps) ≤ n" "degree p ≤ n" by auto
  thus ?case unfolding sum_list.Cons by (metis degree_add_le)
qed simp

lemma degree_prod_list_le: "degree (prod_list ps) ≤ sum_list (map degree ps)"
proof (induct ps)
  case (Cons p ps)
  show ?case unfolding prod_list.Cons
    by (rule order.trans[OF degree_mult_le], insert Cons, auto)
qed simp

lemma smult_sum: "smult (∑i ∈ S. f i) p = (∑i ∈ S. smult (f i) p)"
  by (induct S rule: infinite_finite_induct, auto simp: smult_add_left)


lemma range_coeff: "range (coeff p) = insert 0 (set (coeffs p))" 
  by (metis nth_default_coeffs_eq range_nth_default)

lemma smult_power: "(smult a p) ^ n = smult (a ^ n) (p ^ n)"
  by (induct n, auto simp: field_simps)

lemma poly_sum_list: "poly (sum_list ps) x = sum_list (map (λ p. poly p x) ps)"
  by (induct ps, auto)

lemma poly_prod_list: "poly (prod_list ps) x = prod_list (map (λ p. poly p x) ps)"
  by (induct ps, auto)

lemma sum_list_neutral: "(⋀ x. x ∈ set xs ⟹ x = 0) ⟹ sum_list xs = 0"
  by (induct xs, auto)

lemma prod_list_neutral: "(⋀ x. x ∈ set xs ⟹ x = 1) ⟹ prod_list xs = 1"
  by (induct xs, auto)

lemma (in comm_monoid_mult) prod_list_map_remove1:
  "x ∈ set xs ⟹ prod_list (map f xs) = f x * prod_list (map f (remove1 x xs))"
  by (induct xs) (auto simp add: ac_simps)

lemma poly_as_sum:
  fixes p :: "'a::comm_semiring_1 poly"
  shows "poly p x = (∑i≤degree p. x ^ i * coeff p i)"
  unfolding poly_altdef by (simp add: ac_simps)

lemma poly_prod_0: "finite ps ⟹ poly (prod f ps) x = (0 :: 'a :: field) ⟷ (∃ p ∈ ps. poly (f p) x = 0)"
  by (induct ps rule: finite_induct, auto)

lemma coeff_monom_mult:
  shows "coeff (monom a d * p) i =
    (if d ≤ i then a * coeff p (i-d) else 0)" (is "?l = ?r")
proof (cases "d ≤ i")
  case False thus ?thesis unfolding coeff_mult by simp
  next case True
    let ?f = "λj. coeff (monom a d) j * coeff p (i - j)"
    have "⋀j. j ∈ {0..i} - {d} ⟹ ?f j = 0" by auto
    hence "0 = (∑j ∈ {0..i} - {d}. ?f j)" by auto
    also have "... + ?f d = (∑j ∈ insert d ({0..i} - {d}). ?f j)"
      by(subst sum.insert, auto)
    also have "... = (∑j ∈ {0..i}. ?f j)" by (subst insert_Diff, insert True, auto)
    also have "... = (∑j≤i. ?f j)" by (rule sum.cong, auto)
    also have "... = ?l" unfolding coeff_mult ..
    finally show ?thesis using True by auto
qed

lemma poly_eqI2:
  assumes "degree p = degree q" and "⋀i. i ≤ degree p ⟹ coeff p i = coeff q i"
  shows "p = q"
  apply(rule poly_eqI) by (metis assms le_degree)

text {* A nice extension rule for polynomials. *}
lemma poly_ext[intro]:
  fixes p q :: "'a :: {ring_char_0, idom} poly"
  assumes "⋀x. poly p x = poly q x" shows "p = q"
  unfolding poly_eq_poly_eq_iff[symmetric]
  using assms by (rule ext)

text {* Copied from non-negative variants. *}
lemma coeff_linear_power_neg[simp]:
  fixes a :: "'a::comm_ring_1"
  shows "coeff ([:a, -1:] ^ n) n = (-1)^n"
apply (induct n, simp_all)
apply (subst coeff_eq_0)
apply (auto intro: le_less_trans degree_power_le)
done

lemma degree_linear_power_neg[simp]:
  fixes a :: "'a::{idom,comm_ring_1}"
  shows "degree ([:a, -1:] ^ n) = n"
apply (rule order_antisym)
apply (rule ord_le_eq_trans [OF degree_power_le], simp)
apply (rule le_degree)
unfolding coeff_linear_power_neg
apply (auto)
done


subsection ‹Polynomial Composition›

lemmas [simp] = pcompose_pCons

lemma pcompose_eq_0: fixes q :: "'a :: idom poly"
  assumes q: "degree q ≠ 0"
  shows "p ∘⇩p q = 0 ⟷ p = 0"
proof (induct p)
  case 0
  show ?case by auto
next
  case (pCons a p)
  have id: "(pCons a p) ∘⇩p q = [:a:] + q * (p ∘⇩p q)" by simp
  show ?case 
  proof (cases "p = 0")
    case True
    show ?thesis unfolding id unfolding True by simp
  next
    case False
    with pCons(2) have "p ∘⇩p q ≠ 0" by auto
    from degree_mult_eq[OF _ this, of q] q have "degree (q * (p ∘⇩p q)) ≠ 0" by force
    hence deg: "degree ([:a:] + q * (p ∘⇩p q)) ≠ 0"
      by (subst degree_add_eq_right, auto)
    show ?thesis unfolding id using False deg by auto
  qed
qed

declare degree_pcompose[simp]

subsection ‹Monic Polynomials›

abbreviation monic where "monic p ≡ coeff p (degree p) = 1"

lemma unit_factor_field [simp]: 
  "unit_factor (x :: 'a :: {field,normalization_semidom}) = x"
  by (cases "is_unit x") (auto simp: is_unit_unit_factor dvd_field_iff)

lemma poly_gcd_monic: 
  fixes p :: "'a :: {field,factorial_ring_gcd} poly"
  assumes "p ≠ 0 ∨ q ≠ 0"
  shows   "monic (gcd p q)"
proof -
  from assms have "1 = unit_factor (gcd p q)" by (auto simp: unit_factor_gcd)
  also have "… = [:lead_coeff (gcd p q):]" unfolding unit_factor_poly_def
    by (simp add: monom_0)
  finally show ?thesis
    by (metis coeff_pCons_0 degree_1 lead_coeff_1)
qed

lemma normalize_monic: "monic p ⟹ normalize p = p"
  by (simp add: normalize_poly_eq_map_poly is_unit_unit_factor)

lemma lcoeff_monic_mult: assumes monic: "monic (p :: 'a :: comm_semiring_1 poly)"
  shows "coeff (p * q) (degree p + degree q) = coeff q (degree q)"
proof -
  let ?pqi = "λ i. coeff p i * coeff q (degree p + degree q - i)" 
  have "coeff (p * q) (degree p + degree q) = 
    (∑i≤degree p + degree q. ?pqi i)"
    unfolding coeff_mult by simp
  also have "… = ?pqi (degree p) + (sum ?pqi ({.. degree p + degree q} - {degree p}))"
    by (subst sum.remove[of _ "degree p"], auto)
  also have "?pqi (degree p) = coeff q (degree q)" unfolding monic by simp
  also have "(sum ?pqi ({.. degree p + degree q} - {degree p})) = 0"
  proof (rule sum.neutral, intro ballI)
    fix d
    assume d: "d ∈ {.. degree p + degree q} - {degree p}"
    show "?pqi d = 0"
    proof (cases "d < degree p")
      case True
      hence "degree p + degree q - d > degree q" by auto
      hence "coeff q (degree p + degree q - d) = 0" by (rule coeff_eq_0)
      thus ?thesis by simp
    next
      case False
      with d have "d > degree p" by auto
      hence "coeff p d = 0" by (rule coeff_eq_0)
      thus ?thesis by simp
    qed
  qed
  finally show ?thesis by simp
qed

lemma degree_monic_mult: assumes monic: "monic (p :: 'a :: comm_semiring_1 poly)"
  and q: "q ≠ 0"
  shows "degree (p * q) = degree p + degree q"
proof -
  have "degree p + degree q ≥ degree (p * q)" by (rule degree_mult_le)
  also have "degree p + degree q ≤ degree (p * q)"
  proof -
    from q have cq: "coeff q (degree q) ≠ 0" by auto
    hence "coeff (p * q) (degree p + degree q) ≠ 0" unfolding lcoeff_monic_mult[OF monic] .
    thus "degree (p * q) ≥ degree p + degree q" by (rule le_degree)
  qed
  finally show ?thesis .
qed

lemma degree_prod_sum_monic: assumes
  S: "finite S"
  and nzd: "0 ∉ (degree o f) ` S"
  and monic: "(⋀ a . a ∈ S ⟹ monic (f a))"
  shows "degree (prod f S) = (sum (degree o f) S) ∧ coeff (prod f S) (sum (degree o f) S) = 1"
proof -
  from S nzd monic 
  have "degree (prod f S) = sum (degree ∘ f) S 
  ∧ (S ≠ {} ⟶ degree (prod f S) ≠ 0 ∧ prod f S ≠ 0) ∧ coeff (prod f S) (sum (degree o f) S) = 1"
  proof (induct S rule: finite_induct)
    case (insert a S)
    have IH1: "degree (prod f S) = sum (degree o f) S"
      using insert by auto
    have IH2: "coeff (prod f S) (degree (prod f S)) = 1"
      using insert by auto
    have id: "degree (prod f (insert a S)) = sum (degree ∘ f) (insert a S)
      ∧ coeff (prod f (insert a S)) (sum (degree o f) (insert a S)) = 1"
    proof (cases "S = {}")
      case False
      with insert have nz: "prod f S ≠ 0" by auto
      from insert have monic: "coeff (f a) (degree (f a)) = 1" by auto
      have id: "(degree ∘ f) a = degree (f a)" by simp
      show ?thesis unfolding prod.insert[OF insert(1-2)] sum.insert[OF insert(1-2)] id
        unfolding degree_monic_mult[OF monic nz] 
        unfolding IH1[symmetric]
        unfolding lcoeff_monic_mult[OF monic] IH2 by simp
    qed (insert insert, auto)
    show ?case using id unfolding sum.insert[OF insert(1-2)] using insert by auto
  qed simp
  thus ?thesis by auto
qed 

lemma degree_prod_monic: 
  assumes "⋀ i. i < n ⟹ degree (f i :: 'a :: comm_semiring_1 poly) = 1"
    and "⋀ i. i < n ⟹ coeff (f i) 1 = 1"
  shows "degree (prod f {0 ..< n}) = n ∧ coeff (prod f {0 ..< n}) n = 1"
proof -
  from degree_prod_sum_monic[of "{0 ..< n}" f] show ?thesis using assms by force
qed

lemma degree_prod_sum_lt_n: assumes "⋀ i. i < n ⟹ degree (f i :: 'a :: comm_semiring_1 poly) ≤ 1"
  and i: "i < n" and fi: "degree (f i) = 0"
  shows "degree (prod f {0 ..< n}) < n"
proof -
  have "degree (prod f {0 ..< n}) ≤ sum (degree o f) {0 ..< n}"
    by (rule degree_prod_sum_le, auto)
  also have "sum (degree o f) {0 ..< n} = (degree o f) i + sum (degree o f) ({0 ..< n} - {i})"
    by (rule sum.remove, insert i, auto)
  also have "(degree o f) i = 0" using fi by simp
  also have "sum (degree o f) ({0 ..< n} - {i}) ≤ sum (λ _. 1) ({0 ..< n} - {i})"
    by (rule sum_mono, insert assms, auto)
  also have "… = n - 1" using i by simp
  also have "… < n" using i by simp
  finally show ?thesis by simp
qed

lemma degree_linear_factors: "degree (∏ a ← as. [: f a, 1:]) = length as"
proof (induct as)
  case (Cons b as) note IH = this
  have id: "(∏a←b # as. [:f a, 1:]) = [:f b,1 :] * (∏a←as. [:f a, 1:])" by simp
  show ?case unfolding id
    by (subst degree_monic_mult, insert IH, auto)
qed simp

lemma monic_mult:
  fixes p q :: "'a :: idom poly"
  assumes "monic p" "monic q"
  shows "monic (p * q)"
proof -
  from assms have nz: "p ≠ 0" "q ≠ 0" by auto
  show ?thesis unfolding degree_mult_eq[OF nz] coeff_mult_degree_sum
    using assms by simp
qed

lemma monic_factor:
  fixes p q :: "'a :: idom poly"
  assumes "monic (p * q)" "monic p"
  shows "monic q"
proof -
  from assms have nz: "p ≠ 0" "q ≠ 0" by auto
  from assms[unfolded degree_mult_eq[OF nz] coeff_mult_degree_sum `monic p`]
  show ?thesis by simp
qed

lemma monic_prod:
  fixes f :: "'a ⇒ 'b :: idom poly"
  assumes "⋀ a. a ∈ as ⟹ monic (f a)"
  shows "monic (prod f as)" using assms
proof (induct as rule: infinite_finite_induct)
  case (insert a as)
  hence id: "prod f (insert a as) = f a * prod f as" 
    and *: "monic (f a)" "monic (prod f as)" by auto
  show ?case unfolding id by (rule monic_mult[OF *])
qed auto

lemma monic_prod_list:
  fixes as :: "'a :: idom poly list"
  assumes "⋀ a. a ∈ set as ⟹ monic a"
  shows "monic (prod_list as)" using assms
  by (induct as, auto intro: monic_mult)

lemma monic_power:
  assumes "monic (p :: 'a :: idom poly)"
  shows "monic (p ^ n)"
  by (induct n, insert assms, auto intro: monic_mult)

lemma monic_prod_list_pow: "monic (∏(x::'a::idom, i)←xis. [:- x, 1:] ^ Suc i)"
proof (rule monic_prod_list, goal_cases)
  case (1 a)
  then obtain x i where a: "a = [:-x, 1:]^Suc i" by force
  show "monic a" unfolding a
    by (rule monic_power, auto)
qed

lemma monic_degree_0: "monic p ⟹ (degree p = 0) = (p = 1)"
  using le_degree poly_eq_iff by force

subsection ‹Roots›

text ‹The following proof structure is completely similar to the one
  of @{thm poly_roots_finite}.›

lemma poly_roots_degree:
  fixes p :: "'a::idom poly"
  shows "p ≠ 0 ⟹ card {x. poly p x = 0} ≤ degree p"
proof (induct n ≡ "degree p" arbitrary: p)
  case (0 p)
  then obtain a where "a ≠ 0" and "p = [:a:]"
    by (cases p, simp split: if_splits)
  then show ?case by simp
next
  case (Suc n p)
  show ?case
  proof (cases "∃x. poly p x = 0")
    case True
    then obtain a where a: "poly p a = 0" ..
    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
    then obtain k where k: "p = [:-a, 1:] * k" ..
    with `p ≠ 0` have "k ≠ 0" by auto
    with k have "degree p = Suc (degree k)"
      by (simp add: degree_mult_eq del: mult_pCons_left)
    with `Suc n = degree p` have "n = degree k" by simp
    from Suc.hyps(1)[OF this `k ≠ 0`]
    have le: "card {x. poly k x = 0} ≤ degree k" .
    have "card {x. poly p x = 0} = card {x. poly ([:-a, 1:] * k) x = 0}" unfolding k ..
    also have "{x. poly ([:-a, 1:] * k) x = 0} = insert a {x. poly k x = 0}"
      by auto
    also have "card … ≤ Suc (card {x. poly k x = 0})" 
      unfolding card_insert_if[OF poly_roots_finite[OF `k ≠ 0`]] by simp
    also have "… ≤ Suc (degree k)" using le by auto
    finally show ?thesis using `degree p = Suc (degree k)` by simp
  qed simp
qed

lemma poly_root_factor: "(poly ([: r, 1:] * q) (k :: 'a :: idom) = 0) = (k = -r ∨ poly q k = 0)" (is ?one)
  "(poly (q * [: r, 1:]) k = 0) = (k = -r ∨ poly q k = 0)" (is ?two)
  "(poly [: r, 1 :] k = 0) = (k = -r)" (is ?three)
proof -
  have [simp]: "r + k = 0 ⟹ k = - r" by (simp add: minus_unique)
  show ?one unfolding poly_mult by auto
  show ?two unfolding poly_mult by auto
  show ?three by auto
qed

lemma poly_root_constant: "c ≠ 0 ⟹ (poly (p * [:c:]) (k :: 'a :: idom) = 0) = (poly p k = 0)"
  unfolding poly_mult by auto


lemma poly_linear_exp_linear_factors_rev: 
  "([:b,1:])^(length (filter ((=) b) as)) dvd (∏ (a :: 'a :: comm_ring_1) ← as. [: a, 1:])"
proof (induct as)
  case (Cons a as)
  let ?ls = "length (filter ((=) b) (a # as))"
  let ?l = "length (filter ((=) b) as)"
  have prod: "(∏ a ← Cons a as. [: a, 1:]) = [: a, 1 :] * (∏ a ← as. [: a, 1:])" by simp
  show ?case
  proof (cases "a = b")
    case False
    hence len: "?ls = ?l" by simp
    show ?thesis unfolding prod len using Cons by (rule dvd_mult)
  next
    case True
    hence len: "[: b, 1 :] ^ ?ls = [: a, 1 :] * [: b, 1 :] ^ ?l" by simp
    show ?thesis unfolding prod len using Cons using dvd_refl mult_dvd_mono by blast
  qed
qed simp

lemma order_max: assumes dvd: "[: -a, 1 :] ^ k dvd p" and p: "p ≠ 0"
  shows "k ≤ order a p"
proof (rule ccontr)
  assume "¬ ?thesis"
  hence "∃ j. k = Suc (order a p + j)" by arith
  then obtain j where k: "k = Suc (order a p + j)" by auto
  have "[: -a, 1 :] ^ Suc (order a p) dvd p"
    by (rule power_le_dvd[OF dvd[unfolded k]], simp)
  with order_2[OF p, of a] show False by blast
qed


subsection ‹Divisibility›

context
  assumes "SORT_CONSTRAINT('a :: idom)"
begin
lemma poly_linear_linear_factor: assumes 
  dvd: "[:b,1:] dvd (∏ (a :: 'a) ← as. [: a, 1:])"
  shows "b ∈ set as"
proof -
  let ?p = "λ as. (∏ a ← as. [: a, 1:])"
  let ?b = "[:b,1:]"
  from assms[unfolded dvd_def] obtain p where id: "?p as = ?b * p" ..
  from arg_cong[OF id, of "λ p. poly p (-b)"]
  have "poly (?p as) (-b) = 0" by simp
  thus ?thesis
  proof (induct as)
    case (Cons a as)
    have "?p (a # as) = [:a,1:] * ?p as" by simp
    from Cons(2)[unfolded this] have "poly (?p as) (-b) = 0 ∨ (a - b) = 0" by simp
    with Cons(1) show ?case by auto
  qed simp
qed

lemma poly_linear_exp_linear_factors: 
  assumes dvd: "([:b,1:])^n dvd (∏ (a :: 'a) ← as. [: a, 1:])"
  shows "length (filter ((=) b) as) ≥ n"
proof -
  let ?p = "λ as. (∏ a ← as. [: a, 1:])"
  let ?b = "[:b,1:]"
  from dvd show ?thesis
  proof (induct n arbitrary: as)
    case (Suc n as)
    have bs: "?b ^ Suc n = ?b * ?b ^ n" by simp
    from poly_linear_linear_factor[OF dvd_mult_left[OF Suc(2)[unfolded bs]], 
      unfolded in_set_conv_decomp]
    obtain as1 as2 where as: "as = as1 @ b # as2" by auto
    have "?p as = [:b,1:] * ?p (as1 @ as2)" unfolding as
    proof (induct as1)
      case (Cons a as1)
      have "?p (a # as1 @ b # as2) = [:a,1:] * ?p (as1 @ b # as2)" by simp
      also have "?p (as1 @ b # as2) = [:b,1:] * ?p (as1 @ as2)" unfolding Cons by simp
      also have "[:a,1:] * … = [:b,1:] * ([:a,1:] * ?p (as1 @ as2))" 
        by (metis (no_types, lifting) mult.left_commute)
      finally show ?case by simp
    qed simp
    from Suc(2)[unfolded bs this dvd_mult_cancel_left]
    have "?b ^ n dvd ?p (as1 @ as2)" by simp
    from Suc(1)[OF this] show ?case unfolding as by simp
  qed simp    
qed
end

lemma const_poly_dvd: "([:a:] dvd [:b:]) = (a dvd b)"
proof
  assume "a dvd b"
  then obtain c where "b = a * c" unfolding dvd_def by auto
  hence "[:b:] = [:a:] * [: c:]" by (auto simp: ac_simps)
  thus "[:a:] dvd [:b:]" unfolding dvd_def by blast
next
  assume "[:a:] dvd [:b:]"
  then obtain pc where "[:b:] =  [:a:] * pc" unfolding dvd_def by blast
  from arg_cong[OF this, of "λ p. coeff p 0", unfolded coeff_mult]
  have "b = a * coeff pc 0" by auto
  thus "a dvd b" unfolding dvd_def by blast
qed

lemma const_poly_dvd_1 [simp]:
  "[:a:] dvd 1 ⟷ a dvd 1"
  by (metis const_poly_dvd one_poly_eq_simps(2))

lemma poly_dvd_1:
  fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
  shows "p dvd 1 ⟷ degree p = 0 ∧ coeff p 0 dvd 1"
proof (cases "degree p = 0")
  case False
  with divides_degree[of p 1] show ?thesis by auto
next
  case True
  from degree0_coeffs[OF this] obtain a where p: "p = [:a:]" by auto
  show ?thesis unfolding p by auto
qed

text ‹Degree based version of irreducibility.›

definition irreducible⇩d :: "'a :: comm_semiring_1 poly ⇒ bool" where
  "irreducible⇩d p = (degree p > 0 ∧ (∀ q r. degree q < degree p ⟶ degree r < degree p ⟶ p ≠ q * r))"

lemma irreducible⇩dI [intro]:
  assumes 1: "degree p > 0"
    and 2: "⋀q r. degree q > 0 ⟹ degree q < degree p ⟹ degree r > 0 ⟹ degree r < degree p ⟹ p = q * r ⟹ False"
  shows "irreducible⇩d p"
proof (unfold irreducible⇩d_def, intro conjI allI impI notI 1)
  fix q r
  assume "degree q < degree p" and "degree r < degree p" and "p = q * r"
  with degree_mult_le[of q r]
  show False by (intro 2, auto)
qed

lemma irreducible⇩dI2:
  fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes deg: "degree p > 0" and ndvd: "⋀ q. degree q > 0 ⟹ degree q ≤ degree p div 2 ⟹ ¬ q dvd p"
  shows "irreducible⇩d p"
proof (rule ccontr)
  assume "¬ ?thesis"
  from this[unfolded irreducible⇩d_def] deg obtain q r where dq: "degree q < degree p" and dr: "degree r < degree p"
    and p: "p = q * r" by auto
  from deg have p0: "p ≠ 0" by auto
  with p have "q ≠ 0" "r ≠ 0" by auto
  from degree_mult_eq[OF this] p have dp: "degree p = degree q + degree r" by simp
  show False
  proof (cases "degree q ≤ degree p div 2")
    case True
    from ndvd[OF _ True] dq dr dp p show False by auto
  next
    case False
    with dp have dr: "degree r ≤ degree p div 2" by auto
    from p have dvd: "r dvd p" by auto
    from ndvd[OF _ dr] dvd dp dq show False by auto
  qed
qed

lemma reducible⇩dI:
  assumes "degree p > 0 ⟹ ∃q r. degree q < degree p ∧ degree r < degree p ∧ p = q * r"
  shows "¬ irreducible⇩d p"
  using assms by (auto simp: irreducible⇩d_def)

lemma irreducible⇩dE [elim]:
  assumes "irreducible⇩d p"
    and "degree p > 0 ⟹ (⋀q r. degree q < degree p ⟹ degree r < degree p ⟹ p ≠ q * r) ⟹ thesis"
  shows thesis
  using assms by (auto simp: irreducible⇩d_def)

lemma reducible⇩dE [elim]:
  assumes red: "¬ irreducible⇩d p"
    and 1: "degree p = 0 ⟹ thesis"
    and 2: "⋀q r. degree q > 0 ⟹ degree q < degree p ⟹ degree r > 0 ⟹ degree r < degree p ⟹ p = q * r ⟹ thesis"
  shows thesis
  using red[unfolded irreducible⇩d_def de_Morgan_conj not_not not_all not_imp]
proof (elim disjE exE conjE)
  show "¬degree p > 0 ⟹ thesis" using 1 by auto
next
  fix q r
  assume "degree q < degree p" and "degree r < degree p" and "p = q * r"
  with degree_mult_le[of q r]
  show thesis by (intro 2, auto)
qed

lemma irreducible⇩dD:
  assumes "irreducible⇩d p"
  shows "degree p > 0" "⋀q r. degree q < degree p ⟹ degree r < degree p ⟹ p ≠ q * r"
  using assms unfolding irreducible⇩d_def by auto

theorem irreducible⇩d_factorization_exists:
  assumes "degree p > 0"
  shows "∃fs. fs ≠ [] ∧ (∀f ∈ set fs. irreducible⇩d f ∧ degree f ≤ degree p) ∧ p = prod_list fs"
    and "¬irreducible⇩d p ⟹ ∃fs. length fs > 1 ∧ (∀f ∈ set fs. irreducible⇩d f ∧ degree f < degree p) ∧ p = prod_list fs"
proof (atomize(full), insert assms, induct "degree p" arbitrary:p rule: less_induct)
  case less
  then have deg_f: "degree p > 0" by auto
  show ?case
  proof (cases "irreducible⇩d p")
    case True
    then have "set [p] ⊆ Collect irreducible⇩d" "p = prod_list [p]" by auto
    with True show ?thesis by (auto intro: exI[of _ "[p]"])
  next
    case False
    with deg_f obtain g h
    where deg_g: "degree g < degree p" "degree g > 0"
      and deg_h: "degree h < degree p" "degree h > 0"
      and f_gh: "p = g * h" by auto
    from less.hyps[OF deg_g] less.hyps[OF deg_h]
    obtain gs hs
    where emp: "length gs > 0" "length hs > 0"
      and "∀f ∈ set gs. irreducible⇩d f ∧ degree f ≤ degree g" "g = prod_list gs"
      and "∀f ∈ set hs. irreducible⇩d f ∧ degree f ≤ degree h" "h = prod_list hs" by auto
    with f_gh deg_g deg_h
    have len: "length (gs@hs) > 1"
     and mem: "∀f ∈ set (gs@hs). irreducible⇩d f ∧ degree f < degree p"
     and p: "p = prod_list (gs@hs)" by (auto simp del: length_greater_0_conv)
    with False show ?thesis by (auto intro!: exI[of _ "gs@hs"] simp: less_imp_le)
  qed
qed

lemma irreducible⇩d_factor:
  fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes "degree p > 0"
  shows "∃ q r. irreducible⇩d q ∧ p = q * r ∧ degree r < degree p" using assms
proof (induct "degree p" arbitrary: p rule: less_induct)
  case (less p)
  show ?case
  proof (cases "irreducible⇩d p")
    case False
    with less(2) obtain q r
    where q: "degree q < degree p" "degree q > 0"
      and r: "degree r < degree p" "degree r > 0"
      and p: "p = q * r"
      by auto
    from less(1)[OF q] obtain s t where IH: "irreducible⇩d s" "q = s * t" by auto
    from p have p: "p = s * (t * r)" unfolding IH by (simp add: ac_simps)
    from less(2) have "p ≠ 0" by auto
    hence "degree p = degree s + (degree (t * r))" unfolding p 
      by (subst degree_mult_eq, insert p, auto)
    with irreducible⇩dD[OF IH(1)] have "degree p > degree (t * r)" by auto
    with p IH show ?thesis by auto
  next
    case True
    show ?thesis
      by (rule exI[of _ p], rule exI[of _ 1], insert True less(2), auto)
  qed
qed

context mult_zero begin (* least class with times and zero *)

definition zero_divisor where "zero_divisor a ≡ ∃b. b ≠ 0 ∧ a * b = 0"

lemma zero_divisorI[intro]:
  assumes "b ≠ 0" and "a * b = 0" shows "zero_divisor a"
  using assms by (auto simp: zero_divisor_def)

lemma zero_divisorE[elim]:
  assumes "zero_divisor a"
    and "⋀b. b ≠ 0 ⟹ a * b = 0 ⟹ thesis"
  shows thesis
  using assms by (auto simp: zero_divisor_def)

end

lemma zero_divisor_0[simp]:
  "zero_divisor (0::'a::{mult_zero,zero_neq_one})" (* No need for one! *)
  by (auto intro!: zero_divisorI[of 1])

lemma not_zero_divisor_1:
  "¬ zero_divisor (1 :: 'a :: {monoid_mult,mult_zero})" (* No need for associativity! *)
  by auto

lemma zero_divisor_iff_eq_0[simp]:
  fixes a :: "'a :: {semiring_no_zero_divisors, zero_neq_one}"
  shows "zero_divisor a ⟷ a = 0" by auto

lemma mult_eq_0_not_zero_divisor_left[simp]:
  fixes a b :: "'a :: mult_zero"
  assumes "¬ zero_divisor a"
  shows "a * b = 0 ⟷ b = 0"
  using assms unfolding zero_divisor_def by force

lemma mult_eq_0_not_zero_divisor_right[simp]:
  fixes a b :: "'a :: {ab_semigroup_mult,mult_zero}" (* No need for associativity! *)
  assumes "¬ zero_divisor b"
  shows "a * b = 0 ⟷ a = 0"
  using assms unfolding zero_divisor_def by (force simp: ac_simps)

lemma degree_smult_not_zero_divisor_left[simp]:
  assumes "¬ zero_divisor c"
  shows "degree (smult c p) = degree p"
proof(cases "p = 0")
  case False
  then have "coeff (smult c p) (degree p) ≠ 0" using assms by auto
  from le_degree[OF this] degree_smult_le[of c p]
  show ?thesis by auto
qed auto

lemma degree_smult_not_zero_divisor_right[simp]:
  assumes "¬ zero_divisor (lead_coeff p)"
  shows "degree (smult c p) = (if c = 0 then 0 else degree p)"
proof(cases "c = 0")
  case False
  then have "coeff (smult c p) (degree p) ≠ 0" using assms by auto
  from le_degree[OF this] degree_smult_le[of c p]
  show ?thesis by auto
qed auto


lemma irreducible⇩d_smult_not_zero_divisor_left:
  assumes c0: "¬ zero_divisor c"
  assumes L: "irreducible⇩d (smult c p)"
  shows "irreducible⇩d p"
proof (intro irreducible⇩dI)
  from L have "degree (smult c p) > 0" by auto
  also note degree_smult_le
  finally show "degree p > 0" by auto
  fix q r
  assume deg_q: "degree q < degree p"
    and deg_r: "degree r < degree p"
    and p_qr: "p = q * r"
  then have 1: "smult c p = smult c q * r" by auto
  note degree_smult_le[of c q]
  also note deg_q
  finally have 2: "degree (smult c q) < degree (smult c p)" using c0 by auto
  from deg_r have 3: "degree r < …" using c0 by auto
  from irreducible⇩dD(2)[OF L 2 3] 1 show False by auto
qed

lemmas irreducible⇩d_smultI =
  irreducible⇩d_smult_not_zero_divisor_left
  [where 'a = "'a :: {comm_semiring_1,semiring_no_zero_divisors}", simplified]

lemma irreducible⇩d_smult_not_zero_divisor_right:
  assumes p0: "¬ zero_divisor (lead_coeff p)" and L: "irreducible⇩d (smult c p)"
  shows "irreducible⇩d p"
proof-
  from L have "c ≠ 0" by auto
  with p0 have [simp]: "degree (smult c p) = degree p" by simp
  show "irreducible⇩d p"
  proof (intro iffI irreducible⇩dI conjI)
    from L show "degree p > 0" by auto
    fix q r
    assume deg_q: "degree q < degree p"
      and deg_r: "degree r < degree p"
      and p_qr: "p = q * r"
    then have 1: "smult c p = smult c q * r" by auto
    note degree_smult_le[of c q]
    also note deg_q
    finally have 2: "degree (smult c q) < degree (smult c p)" by simp
    from deg_r have 3: "degree r < …" by simp
    from irreducible⇩dD(2)[OF L 2 3] 1 show False by auto
  qed
qed

lemma zero_divisor_mult_left:
  fixes a b :: "'a :: {ab_semigroup_mult, mult_zero}"
  assumes "zero_divisor a"
  shows "zero_divisor (a * b)"
proof-
  from assms obtain c where c0: "c ≠ 0" and [simp]: "a * c = 0" by auto
  have "a * b * c = a * c * b" by (simp only: ac_simps)
  with c0 show ?thesis by auto
qed

lemma zero_divisor_mult_right:
  fixes a b :: "'a :: {semigroup_mult, mult_zero}"
  assumes "zero_divisor b"
  shows "zero_divisor (a * b)"
proof-
  from assms obtain c where c0: "c ≠ 0" and [simp]: "b * c = 0" by auto
  have "a * b * c = a * (b * c)" by (simp only: ac_simps)
  with c0 show ?thesis by auto
qed

lemma not_zero_divisor_mult:
  fixes a b :: "'a :: {ab_semigroup_mult, mult_zero}"
  assumes "¬ zero_divisor (a * b)"
  shows "¬ zero_divisor a" and "¬ zero_divisor b"
  using assms by (auto dest: zero_divisor_mult_right zero_divisor_mult_left)

lemma zero_divisor_smult_left:
  assumes "zero_divisor a"
  shows "zero_divisor (smult a f)"
proof-
  from assms obtain b where b0: "b ≠ 0" and "a * b = 0" by auto
  then have "smult a f * [:b:] = 0" by (simp add: ac_simps)
  with b0 show ?thesis by (auto intro!: zero_divisorI[of "[:b:]"])
qed

lemma unit_not_zero_divisor:
  fixes a :: "'a :: {comm_monoid_mult, mult_zero}"
  assumes "a dvd 1"
  shows "¬zero_divisor a"
proof
  from assms obtain b where ab: "1 = a * b" by (elim dvdE)
  assume "zero_divisor a"
  then have "zero_divisor (1::'a)" by (unfold ab, intro zero_divisor_mult_left)
  then show False by auto
qed


lemma linear_irreducible⇩d: assumes "degree p = 1"
  shows "irreducible⇩d p"
  by (rule irreducible⇩dI, insert assms, auto)

lemma irreducible⇩d_dvd_smult:
  fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes "degree p > 0" "irreducible⇩d q" "p dvd q"
  shows "∃ c. c ≠ 0 ∧ q = smult c p"
proof -
  from assms obtain r where q: "q = p * r" by (elim dvdE, auto)
  from degree_mult_eq[of p r] assms(1) q
  have "¬ degree p < degree q" and nz: "p ≠ 0" "q ≠ 0"
    apply (metis assms(2) degree_mult_eq_0 gr_implies_not_zero irreducible⇩dD(2) less_add_same_cancel2)
    using assms by auto
  hence deg: "degree p ≥ degree q" by auto
  from `p dvd q` obtain k where q: "q = k * p" unfolding dvd_def by (auto simp: ac_simps)
  with nz have "k ≠ 0" by auto
  from deg[unfolded q degree_mult_eq[OF `k ≠ 0` `p ≠ 0` ]] have "degree k = 0" 
    unfolding q by auto 
  then obtain c where k: "k = [: c :]" by (metis degree_0_id)
  with `k ≠ 0` have "c ≠ 0" by auto
  have "q = smult c p" unfolding q k by simp
  with `c ≠ 0` show ?thesis by auto
qed

subsection ‹Map over Polynomial Coefficients›
lemma map_poly_simps:
  shows "map_poly f (pCons c p) =
    (if c = 0 ∧ p = 0 then 0 else pCons (f c) (map_poly f p))"
proof (cases "c = 0")
  case True note c0 = this show ?thesis
    proof (cases "p = 0")
      case True thus ?thesis using c0 unfolding map_poly_def by simp
      next case False thus ?thesis
        unfolding map_poly_def by auto
    qed
  next case False thus ?thesis
    unfolding map_poly_def by auto
qed

lemma map_poly_pCons[simp]:
  assumes "c ≠ 0 ∨ p ≠ 0"
  shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
  unfolding map_poly_simps using assms by auto

lemma map_poly_map_poly:
  assumes f0: "f 0 = 0"
  shows "map_poly f (map_poly g p) = map_poly (f ∘ g) p"
proof (induct p)
  case (pCons a p) show ?case
  proof(cases "g a ≠ 0 ∨ map_poly g p ≠ 0")
    case True show ?thesis
      unfolding map_poly_pCons[OF pCons(1)]
      unfolding map_poly_pCons[OF True]
      unfolding pCons(2)
      by simp
  next
    case False then show ?thesis
      unfolding map_poly_pCons[OF pCons(1)]
      unfolding pCons(2)[symmetric]
      by (simp add: f0)
  qed
qed simp

lemma map_poly_zero:
  assumes f: "∀c. f c = 0 ⟶ c = 0"
  shows [simp]: "map_poly f p = 0 ⟷ p = 0"
  by (induct p; auto simp: map_poly_simps f)

lemma map_poly_add:
  assumes h0: "h 0 = 0"
      and h_add: "∀p q. h (p + q) = h p + h q"
  shows "map_poly h (p + q) = map_poly h p + map_poly h q"
proof (induct p arbitrary: q)
  case (pCons a p) note pIH = this
    show ?case
    proof(induct "q")
      case (pCons b q) note qIH = this
        show ?case
          unfolding map_poly_pCons[OF qIH(1)]
          unfolding map_poly_pCons[OF pIH(1)]
          unfolding add_pCons
          unfolding pIH(2)[symmetric]
          unfolding h_add[rule_format,symmetric]
          unfolding map_poly_simps using h0 by auto
    qed auto
qed auto

subsection ‹Morphismic properties of @{term "pCons 0"}›

lemma monom_pCons_0_monom:
  "monom (pCons 0 (monom a n)) d = map_poly (pCons 0) (monom (monom a n) d)"
  apply (induct d)
  unfolding monom_0 unfolding map_poly_simps apply simp
  unfolding monom_Suc map_poly_simps by auto

lemma pCons_0_add: "pCons 0 (p + q) = pCons 0 p + pCons 0 q" by auto

lemma sum_pCons_0_commute:
  "sum (λi. pCons 0 (f i)) S = pCons 0 (sum f S)"
  by(induct S rule: infinite_finite_induct;simp)

lemma pCons_0_as_mult:
  fixes p:: "'a :: comm_semiring_1 poly"
  shows "pCons 0 p = [:0,1:] * p" by auto



subsection ‹Misc›

fun expand_powers :: "(nat × 'a)list ⇒ 'a list" where
  "expand_powers [] = []"
| "expand_powers ((Suc n, a) # ps) = a # expand_powers ((n,a) # ps)"
| "expand_powers ((0,a) # ps) = expand_powers ps"

lemma expand_powers: fixes f :: "'a ⇒ 'b :: comm_ring_1"
  shows "(∏ (n,a) ← n_as. f a ^ n) = (∏ a ← expand_powers n_as. f a)"
  by (rule sym, induct n_as rule: expand_powers.induct, auto)

lemma poly_smult_zero_iff: fixes x :: "'a :: idom" 
  shows "(poly (smult a p) x = 0) = (a = 0 ∨ poly p x = 0)"
  by simp

lemma poly_prod_list_zero_iff: fixes x :: "'a :: idom" 
  shows "(poly (prod_list ps) x = 0) = (∃ p ∈ set ps. poly p x = 0)"
  by (induct ps, auto)

lemma poly_mult_zero_iff: fixes x :: "'a :: idom" 
  shows "(poly (p * q) x = 0) = (poly p x = 0 ∨ poly q x = 0)"
  by simp

lemma poly_power_zero_iff: fixes x :: "'a :: idom" 
  shows "(poly (p^n) x = 0) = (n ≠ 0 ∧ poly p x = 0)"
  by (cases n, auto)


lemma sum_monom_0_iff: assumes fin: "finite S"
  and g: "⋀ i j. g i = g j ⟹ i = j"
  shows "sum (λ i. monom (f i) (g i)) S = 0 ⟷ (∀ i ∈ S. f i = 0)" (is "?l = ?r")
proof -
  {
    assume "¬ ?r"
    then obtain i where i: "i ∈ S" and fi: "f i ≠ 0" by auto
    let ?g = "λ i. monom (f i) (g i)"
    have "coeff (sum ?g S) (g i) = f i + sum (λ j. coeff (?g j) (g i)) (S - {i})"
      by (unfold sum.remove[OF fin i], simp add: coeff_sum)
    also have "sum (λ j. coeff (?g j) (g i)) (S - {i}) = 0"
      by (rule sum.neutral, insert g, auto)
    finally have "coeff (sum ?g S) (g i) ≠ 0" using fi by auto
    hence "¬ ?l" by auto
  }
  thus ?thesis by auto
qed

lemma degree_prod_list_eq: assumes "⋀ p. p ∈ set ps ⟹ (p :: 'a :: idom poly) ≠ 0"
  shows "degree (prod_list ps) = sum_list (map degree ps)" using assms
proof (induct ps)
  case (Cons p ps)
  show ?case unfolding prod_list.Cons
    by (subst degree_mult_eq, insert Cons, auto simp: prod_list_zero_iff)
qed simp

lemma degree_power_eq: assumes p: "p ≠ 0"
  shows "degree (p ^ n) = degree (p :: 'a :: idom poly) * n"
proof (induct n)
  case (Suc n)
  from p have pn: "p ^ n ≠ 0" by auto
  show ?case using degree_mult_eq[OF p pn] Suc by auto
qed simp

lemma coeff_Poly: "coeff (Poly xs) i = (nth_default 0 xs i)"
  unfolding nth_default_coeffs_eq[of "Poly xs", symmetric] coeffs_Poly by simp

lemma rsquarefree_def': "rsquarefree p = (p ≠ 0 ∧ (∀a. order a p ≤ 1))"
proof -
  have "⋀ a. order a p ≤ 1 ⟷ order a p = 0 ∨ order a p = 1" by linarith
  thus ?thesis unfolding rsquarefree_def by auto
qed

lemma order_prod_list: "(⋀ p. p ∈ set ps ⟹ p ≠ 0) ⟹ order x (prod_list ps) = sum_list (map (order x) ps)"
  by (induct ps, auto, subst order_mult, auto simp: prod_list_zero_iff)

lemma irreducible⇩d_dvd_eq:
  fixes a b :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes "irreducible⇩d a" and "irreducible⇩d b"
    and "a dvd b"
    and "monic a" and "monic b" 
  shows "a = b"
  using assms
  by (metis (no_types, lifting) coeff_smult degree_smult_eq irreducible⇩dD(1) irreducible⇩d_dvd_smult 
    mult.right_neutral smult_1_left)

lemma monic_gcd_dvd:
  assumes fg: "f dvd g" and mon: "monic f" and gcd: "gcd g h ∈ {1, g}"
  shows "gcd f h ∈ {1, f}"
proof (cases "coprime g h")
  case True
  with dvd_refl have "coprime f h"
    using fg by (blast intro: coprime_divisors)
  then show ?thesis
    by simp
next
  case False
  with gcd have gcd: "gcd g h = g"
    by (simp add: coprime_iff_gcd_eq_1)
  with fg have "f dvd gcd g h"
    by simp
  then have "f dvd h"
    by simp
  then have "gcd f h = normalize f"
    by (simp add: gcd_proj1_iff)
  also have "normalize f = f"
    using mon by (rule normalize_monic)
  finally show ?thesis
    by simp
qed

lemma monom_power: "(monom a b)^n = monom (a^n) (b*n)" 
  by (induct n, auto simp add: mult_monom)

lemma poly_const_pow: "[:a:]^b = [:a^b:]"
  by (metis Groups.mult_ac(2) monom_0 monom_power mult_zero_right)

lemma degree_pderiv_le: "degree (pderiv f) ≤ degree f - 1" 
proof (rule ccontr)
  assume "¬ ?thesis"
  hence ge: "degree (pderiv f) ≥ Suc (degree f - 1)" by auto
  hence "pderiv f ≠ 0" by auto
  hence "coeff (pderiv f) (degree (pderiv f)) ≠ 0" by auto
  from this[unfolded coeff_pderiv]
  have "coeff f (Suc (degree (pderiv f))) ≠ 0" by auto
  moreover have "Suc (degree (pderiv f)) > degree f" using ge by auto
  ultimately show False by (simp add: coeff_eq_0)
qed

lemma map_div_is_smult_inverse: "map_poly (λx. x / (a :: 'a :: field)) p = smult (inverse a) p" 
  unfolding smult_conv_map_poly
  by (simp add: divide_inverse_commute)

lemma normalize_poly_old_def:
  "normalize (f :: 'a :: {normalization_semidom,field} poly) = smult (inverse (unit_factor (lead_coeff f))) f"
  by (simp add: normalize_poly_eq_map_poly map_div_is_smult_inverse)

(* was in Euclidean_Algorithm in Number_Theory before, but has been removed *)
lemma poly_dvd_antisym:
  fixes p q :: "'b::idom poly"
  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
proof (cases "p = 0")
  case True with coeff show "p = q" by simp
next
  case False with coeff have "q ≠ 0" by auto
  have degree: "degree p = degree q"
    using ‹p dvd q› ‹q dvd p› ‹p ≠ 0› ‹q ≠ 0›
    by (intro order_antisym dvd_imp_degree_le)

  from ‹p dvd q› obtain a where a: "q = p * a" ..
  with ‹q ≠ 0› have "a ≠ 0" by auto
  with degree a ‹p ≠ 0› have "degree a = 0"
    by (simp add: degree_mult_eq)
  with coeff a show "p = q"
    by (cases a, auto split: if_splits)
qed

lemma coeff_f_0_code[code_unfold]: "coeff f 0 = (case coeffs f of [] ⇒ 0 | x # _ ⇒ x)" 
  by (cases f, auto simp: cCons_def)

lemma poly_compare_0_code[code_unfold]: "(f = 0) = (case coeffs f of [] ⇒ True | _ ⇒ False)" 
  using coeffs_eq_Nil list.disc_eq_case(1) by blast

text ‹Getting more efficient code for abbreviation @{term lead_coeff}"›

definition leading_coeff
  where [code_abbrev, simp]: "leading_coeff = lead_coeff" 

lemma leading_coeff_code [code]:
  "leading_coeff f = (let xs = coeffs f in if xs = [] then 0 else last xs)"
  by (simp add: last_coeffs_eq_coeff_degree)

lemma nth_coeffs_coeff: "i < length (coeffs f) ⟹ coeffs f ! i = coeff f i"
  by (metis nth_default_coeffs_eq nth_default_def)

lemma degree_prod_eq_sum_degree:
fixes A :: "'a set"
and f :: "'a ⇒ 'b::field poly"
assumes f0: "∀i∈A. f i ≠ 0"
shows "degree (∏i∈A. (f i)) = (∑i∈A. degree (f i))"
using f0
proof (induct A rule: infinite_finite_induct)
  case (insert x A)
  have "(∑i∈insert x A. degree (f i)) = degree (f x) + (∑i∈A. degree (f i))"
    by (simp add: insert.hyps(1) insert.hyps(2))
  also have "... = degree (f x) + degree (∏i∈A. (f i))"
    by (simp add: insert.hyps insert.prems)
  also have "... = degree (f x * (∏i∈A. (f i)))"
  proof (rule degree_mult_eq[symmetric])
    show "f x ≠ 0" using insert.prems by auto
    show "prod f A ≠ 0" by (simp add: insert.hyps(1) insert.prems)
  qed
  also have "... = degree (∏i∈insert x A. (f i))"
    by (simp add: insert.hyps)
  finally show ?case ..
qed auto

definition monom_mult :: "nat ⇒ 'a :: comm_semiring_1 poly ⇒ 'a poly"
  where "monom_mult n f = monom 1 n * f" 

lemma monom_mult_unfold [code_unfold]:
  "monom 1 n * f = monom_mult n f"
  "f * monom 1 n = monom_mult n f" 
  by (auto simp: monom_mult_def ac_simps)

lemma monom_mult_code [code abstract]:
  "coeffs (monom_mult n f) = (let xs = coeffs f in
    if xs = [] then xs else replicate n 0 @ xs)" 
  by (rule coeffs_eqI)
    (auto simp add: Let_def monom_mult_def coeff_monom_mult nth_default_append nth_default_coeffs_eq)

lemma coeff_pcompose_monom: fixes f :: "'a :: comm_ring_1 poly" 
  assumes n: "j < n" 
  shows "coeff (f ∘⇩p monom 1 n) (n * i + j) = (if j = 0 then coeff f i else 0)"     
proof (induct f arbitrary: i)
  case (pCons a f i)
  note d = pcompose_pCons coeff_add coeff_monom_mult coeff_pCons
  show ?case 
  proof (cases i)
    case 0
    show ?thesis unfolding d 0 using n by (cases j, auto)
  next
    case (Suc ii)
    have id: "n * Suc ii + j - n = n * ii + j" using n by (simp add: diff_mult_distrib2)
    have id1: "(n ≤ n * Suc ii + j) = True" by auto
    have id2: "(case n * Suc ii + j of 0 ⇒ a | Suc x ⇒ coeff 0 x) = 0" using n
      by (cases "n * Suc ii + j", auto)
    show ?thesis unfolding d Suc id id1 id2 pCons(2) if_True by auto
  qed
qed auto

lemma coeff_pcompose_x_pow_n: fixes f :: "'a :: comm_ring_1 poly" 
  assumes n: "n ≠ 0" 
  shows "coeff (f ∘⇩p monom 1 n) (n * i) = coeff f i"     
  using coeff_pcompose_monom[of 0 n f i] n by auto
        
lemma dvd_dvd_smult: "a dvd b ⟹ f dvd g ⟹ smult a f dvd smult b g"
  unfolding dvd_def by (metis mult_smult_left mult_smult_right smult_smult)

definition sdiv_poly :: "'a :: idom_divide poly ⇒ 'a ⇒ 'a poly" where
  "sdiv_poly p a = (map_poly (λ c. c div a) p)"  

lemma smult_map_poly: "smult a = map_poly (( * ) a)"
  by (rule ext, rule poly_eqI, subst coeff_map_poly, auto)
  
lemma smult_exact_sdiv_poly: assumes "⋀ c. c ∈ set (coeffs p) ⟹ a dvd c"
  shows "smult a (sdiv_poly p a) = p" 
  unfolding smult_map_poly sdiv_poly_def
  by (subst map_poly_map_poly,simp,rule map_poly_idI, insert assms, auto)

lemma coeff_sdiv_poly: "coeff (sdiv_poly f a) n = coeff f n div a" 
  unfolding sdiv_poly_def by (rule coeff_map_poly, auto)    

lemma poly_pinfty_ge:
  fixes p :: "real poly"
  assumes "lead_coeff p > 0" "degree p ≠ 0" 
  shows "∃n. ∀ x ≥ n. poly p x ≥ b"
proof -
  let ?p = "p - [:b - lead_coeff p :]" 
  have id: "lead_coeff ?p = lead_coeff p" using assms(2)
    by (cases p, auto)
  with assms(1) have "lead_coeff ?p > 0" by auto
  from poly_pinfty_gt_lc[OF this, unfolded id] obtain n
    where "⋀ x. x ≥ n ⟹ 0 ≤ poly p x - b" by auto
  thus ?thesis by auto
qed

lemma pderiv_sum: "pderiv (sum f I) = sum (λ i. (pderiv (f i))) I" 
  by (induct I rule: infinite_finite_induct, auto simp: pderiv_add)

lemma smult_sum2: "smult m (∑i ∈ S. f i) = (∑i ∈ S. smult m (f i))"
  by (induct S rule: infinite_finite_induct, auto simp add: smult_add_right)

lemma degree_mult_not_eq:
  "degree (f * g) ≠ degree f + degree g ⟹ lead_coeff f * lead_coeff g = 0"
  by (rule ccontr, auto simp: coeff_mult_degree_sum degree_mult_le le_antisym le_degree)

lemma irreducible⇩d_multD:
  fixes a b :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes l: "irreducible⇩d (a*b)"
  shows "degree a = 0 ∧ a ≠ 0 ∧ irreducible⇩d b ∨ degree b = 0 ∧ b ≠ 0 ∧ irreducible⇩d a"
proof-
  from l have a0: "a ≠ 0" and b0: "b ≠ 0" by auto
  note [simp] = degree_mult_eq[OF this]
  from l have "degree a = 0 ∨ degree b = 0" apply (unfold irreducible⇩d_def) by force
  then show ?thesis
  proof(elim disjE)
    assume a: "degree a = 0"
    with l a0 have "irreducible⇩d b"
      unfolding irreducible⇩d_def
      by (smt add.left_neutral degree_mult_not_eq leading_coeff_0_iff sign_simps(4) mult_eq_0_iff)
    with a a0 show ?thesis by auto
  next
    assume b: "degree b = 0"
    with l b0 have "irreducible⇩d a"
      unfolding irreducible⇩d_def
      by (smt add_cancel_left_right degree_mult_eq degree_mult_eq_0 neq0_conv semiring_normalization_rules(16))
    with b b0 show ?thesis by auto
  qed
qed

lemma irreducible_connect_field[simp]:
  fixes f :: "'a :: field poly"
  shows "irreducible⇩d f = irreducible f" (is "?l = ?r")
proof
  show "?r ⟹ ?l"
    apply (intro irreducible⇩dI, force simp:is_unit_iff_degree)
    by (auto dest!: irreducible_multD simp: poly_dvd_1)
next
  assume l: ?l
  show ?r
  proof (rule irreducibleI)
    from l show "f ≠ 0" "¬ is_unit f" by (auto simp: poly_dvd_1)
    fix a b assume "f = a * b"
    from l[unfolded this]
    show "a dvd 1 ∨ b dvd 1" by (auto dest!: irreducible⇩d_multD simp:is_unit_iff_degree)
  qed
qed

lemma is_unit_field_poly[simp]:
  fixes p :: "'a::field poly"
  shows "is_unit p ⟷ p ≠ 0 ∧ degree p = 0"
  by (cases "p=0", auto simp: is_unit_iff_degree)

lemma irreducible_smult_field[simp]:
  fixes c :: "'a :: field"
  shows "irreducible (smult c p) ⟷ c ≠ 0 ∧ irreducible p" (is "?L ⟷ ?R")
proof (intro iffI conjI irreducible⇩d_smult_not_zero_divisor_left[of c p, simplified])
  assume "irreducible (smult c p)"
  then show "c ≠ 0" by auto
next
  assume ?R
  then have c0: "c ≠ 0" and irr: "irreducible p" by auto
  show ?L
  proof (fold irreducible_connect_field, intro irreducible⇩dI, unfold degree_smult_eq if_not_P[OF c0])
    show "degree p > 0" using irr by auto
    fix q r
    from c0 have "p = smult (1/c) (smult c p)" by simp
    also assume "smult c p = q * r"
    finally have [simp]: "p = smult (1/c) …".
    assume main: "degree q < degree p" "degree r < degree p"
    have "¬irreducible⇩d p" by (rule reducible⇩dI, rule exI[of _ "smult (1/c) q"], rule exI[of _ r], insert irr c0 main, simp)
    with irr show False by auto
  qed
qed auto

lemma irreducible_monic_factor: fixes p :: "'a :: field poly" 
  assumes "degree p > 0" 
  shows "∃ q r. irreducible q ∧ p = q * r ∧ monic q"
proof -
  from irreducible⇩d_factorization_exists[OF assms]
  obtain fs where "fs ≠ []" and "set fs ⊆ Collect irreducible" and "p = prod_list fs" by auto
  then have q: "irreducible (hd fs)" and p: "p = hd fs * prod_list (tl fs)" by (atomize(full), cases fs, auto)
  define c where "c = coeff (hd fs) (degree (hd fs))"
  from q have c: "c ≠ 0" unfolding c_def irreducible⇩d_def by auto
  show ?thesis
    by (rule exI[of _ "smult (1/c) (hd fs)"], rule exI[of _ "smult c (prod_list (tl fs))"], unfold p,
    insert q c, auto simp: c_def)
qed

lemma monic_irreducible_factorization: fixes p :: "'a :: field poly" 
  shows "monic p ⟹ 
  ∃ as f. finite as ∧ p = prod (λ a. a ^ Suc (f a)) as ∧ as ⊆ {q. irreducible q ∧ monic q}"
proof (induct "degree p" arbitrary: p rule: less_induct)
  case (less p)
  show ?case
  proof (cases "degree p > 0")
    case False
    with less(2) have "p = 1" by (simp add: coeff_eq_0 poly_eq_iff)
    thus ?thesis by (intro exI[of _ "{}"], auto)
  next
    case True
    from irreducible⇩d_factor[OF this] obtain q r where p: "p = q * r"
      and q: "irreducible q" and deg: "degree r < degree p" by auto
    hence q0: "q ≠ 0" by auto
    define c where "c = coeff q (degree q)"
    let ?q = "smult (1/c) q"
    let ?r = "smult c r"
    from q0 have c: "c ≠ 0" "1 / c ≠ 0" unfolding c_def by auto
    hence p: "p = ?q * ?r" unfolding p by auto
    have deg: "degree ?r < degree p" using c deg by auto
    let ?Q = "{q. irreducible q ∧ monic (q :: 'a poly)}"
    have mon: "monic ?q" unfolding c_def using q0 by auto
    from monic_factor[OF `monic p`[unfolded p] this] have "monic ?r" .
    from less(1)[OF deg this] obtain f as
      where as: "finite as" "?r = (∏ a ∈as. a ^ Suc (f a))"
        "as ⊆ ?Q" by blast
    from q c have irred: "irreducible ?q" by simp
    show ?thesis
    proof (cases "?q ∈ as")
      case False
      let ?as = "insert ?q as"
      let ?f = "λ a. if a = ?q then 0 else f a"
      have "p = ?q * (∏ a ∈as. a ^ Suc (f a))" unfolding p as by simp
      also have "(∏ a ∈as. a ^ Suc (f a)) = (∏ a ∈as. a ^ Suc (?f a))"
        by (rule prod.cong, insert False, auto)
      also have "?q * … = (∏ a ∈ ?as. a ^ Suc (?f a))"
        by (subst prod.insert, insert as False, auto)
      finally have p: "p = (∏ a ∈ ?as. a ^ Suc (?f a))" .
      from as(1) have fin: "finite ?as" by auto
      from as mon irred have Q: "?as ⊆ ?Q" by auto
      from fin p Q show ?thesis 
        by(intro exI[of _ ?as] exI[of _ ?f], auto)
    next
      case True
      let ?f = "λ a. if a = ?q then Suc (f a) else f a"
      have "p = ?q * (∏ a ∈as. a ^ Suc (f a))" unfolding p as by simp
      also have "(∏ a ∈as. a ^ Suc (f a)) = ?q ^ Suc (f ?q) * (∏ a ∈(as - {?q}). a ^ Suc (f a))"
        by (subst prod.remove[OF _ True], insert as, auto)
      also have "(∏ a ∈(as - {?q}). a ^ Suc (f a)) = (∏ a ∈(as - {?q}). a ^ Suc (?f a))"
        by (rule prod.cong, auto)
      also have "?q * (?q ^ Suc (f ?q) * … ) = ?q ^ Suc (?f ?q) * …"
        by (simp add: ac_simps)
      also have "… = (∏ a ∈ as. a ^ Suc (?f a))"
        by (subst prod.remove[OF _ True], insert as, auto)
      finally have "p = (∏ a ∈ as. a ^ Suc (?f a))" .
      with as show ?thesis 
        by (intro exI[of _ as] exI[of _ ?f], auto)
    qed
  qed
qed

lemma monic_irreducible_gcd: 
  "monic (f::'a::{field,euclidean_ring_gcd} poly) ⟹ irreducible f ⟹ gcd f u ∈ {1,f}"
  by (metis gcd_dvd1 irreducible_altdef insertCI is_unit_gcd_iff poly_dvd_antisym poly_gcd_monic)
end