Theory Polynomial_List

(*  Title:      HOL/Decision_Procs/Polynomial_List.thy
    Author:     Amine Chaieb
*)

section ‹Univariate Polynomials as lists›

theory Polynomial_List
  imports Complex_Main 

begin

text ‹Application of polynomial as a function.›

primrec (in semiring_0) poly :: "'a list  'a  'a"
where
  poly_Nil: "poly [] x = 0"
| poly_Cons: "poly (h # t) x = h + x * poly t x"


subsection ‹Arithmetic Operations on Polynomials›

text ‹Addition›
primrec (in semiring_0) padd :: "'a list  'a list  'a list"  (infixl +++ 65)
where
  padd_Nil: "[] +++ l2 = l2"
| padd_Cons: "(h # t) +++ l2 = (if l2 = [] then h # t else (h + hd l2) # (t +++ tl l2))"

text ‹Multiplication by a constant›
primrec (in semiring_0) cmult :: "'a  'a list  'a list"  (infixl %* 70) where
  cmult_Nil: "c %* [] = []"
| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"

text ‹Multiplication by a polynomial›
primrec (in semiring_0) pmult :: "'a list  'a list  'a list"  (infixl *** 70)
where
  pmult_Nil: "[] *** l2 = []"
| pmult_Cons: "(h # t) *** l2 = (if t = [] then h %* l2 else (h %* l2) +++ (0 # (t *** l2)))"

text ‹Repeated multiplication by a polynomial›
primrec (in semiring_0) mulexp :: "nat  'a list  'a  list  'a list"
where
  mulexp_zero: "mulexp 0 p q = q"
| mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"

text ‹Exponential›
primrec (in semiring_1) pexp :: "'a list  nat  'a list"  (infixl %^ 80)
where
  pexp_0: "p %^ 0 = [1]"
| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"

text ‹Quotient related value of dividing a polynomial by x + a.
  Useful for divisor properties in inductive proofs.›
primrec (in field) "pquot" :: "'a list  'a  'a list"
where
  pquot_Nil: "pquot [] a = []"
| pquot_Cons: "pquot (h # t) a =
    (if t = [] then [h] else (inverse a * (h - hd( pquot t a))) # pquot t a)"

text ‹Normalization of polynomials (remove extra 0 coeff).›
primrec (in semiring_0) pnormalize :: "'a list  'a list"
where
  pnormalize_Nil: "pnormalize [] = []"
| pnormalize_Cons: "pnormalize (h # p) =
    (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"

definition (in semiring_0) "pnormal p  pnormalize p = p  p  []"
definition (in semiring_0) "nonconstant p  pnormal p  (x. p  [x])"

text ‹Other definitions.›

definition (in ring_1) poly_minus :: "'a list  'a list" (-- _› [80] 80)
  where "-- p = (- 1) %* p"

definition (in semiring_0) divides :: "'a list  'a list  bool"  (infixl divides 70)
  where "p1 divides p2  (q. poly p2 = poly(p1 *** q))"

lemma (in semiring_0) dividesI: "poly p2 = poly (p1 *** q)  p1 divides p2"
  by (auto simp add: divides_def)

lemma (in semiring_0) dividesE:
  assumes "p1 divides p2"
  obtains q where "poly p2 = poly (p1 *** q)"
  using assms by (auto simp add: divides_def)

― ‹order of a polynomial›
definition (in ring_1) order :: "'a  'a list  nat"
  where "order a p = (SOME n. ([-a, 1] %^ n) divides p  ¬ (([-a, 1] %^ (Suc n)) divides p))"

― ‹degree of a polynomial›
definition (in semiring_0) degree :: "'a list  nat"
  where "degree p = length (pnormalize p) - 1"

― ‹squarefree polynomials --- NB with respect to real roots only›
definition (in ring_1) rsquarefree :: "'a list  bool"
  where "rsquarefree p  poly p  poly []  (a. order a p = 0  order a p = 1)"

context semiring_0
begin

lemma padd_Nil2[simp]: "p +++ [] = p"
  by (induct p) auto

lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
  by auto

lemma pminus_Nil: "-- [] = []"
  by (simp add: poly_minus_def)

lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp

end

lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t"
  by (induct t) auto

lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ (0 # t) = a # t"
  by simp


text ‹Handy general properties.›

lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
proof (induct b arbitrary: a)
  case Nil
  then show ?case
    by auto
next
  case (Cons b bs a)
  then show ?case
    by (cases a) (simp_all add: add.commute)
qed

lemma (in comm_semiring_0) padd_assoc: "(a +++ b) +++ c = a +++ (b +++ c)"
proof (induct a arbitrary: b c)
  case Nil
  then show ?case
    by simp
next
  case Cons
  then show ?case
    by (cases b) (simp_all add: ac_simps)
qed

lemma (in semiring_0) poly_cmult_distr: "a %* (p +++ q) = a %* p +++ a %* q"
proof (induct p arbitrary: q)
  case Nil
  then show ?case
    by simp
next
  case Cons
  then show ?case
    by (cases q) (simp_all add: distrib_left)
qed

lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = 0 # t"
proof (induct t)
  case Nil
  then show ?case
    by simp
next
  case (Cons a t)
  then show ?case
    by (cases t) (auto simp add: padd_commut)
qed

text ‹Properties of evaluation of polynomials.›

lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
proof (induct p1 arbitrary: p2)
  case Nil
  then show ?case
    by simp
next
  case (Cons a as p2)
  then show ?case
    by (cases p2) (simp_all add: ac_simps distrib_left)
qed

lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
proof (induct p)
  case Nil
  then show ?case
    by simp
next
  case Cons
  then show ?case
    by (cases "x = zero") (auto simp add: distrib_left ac_simps)
qed

lemma (in comm_semiring_0) poly_cmult_map: "poly (map ((*) c) p) x = c * poly p x"
  by (induct p) (auto simp add: distrib_left ac_simps)

lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
  by (simp add: poly_minus_def) (auto simp add: poly_cmult)

lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
proof (induct p1 arbitrary: p2)
  case Nil
  then show ?case
    by simp
next
  case (Cons a as)
  then show ?case
    by (cases as) (simp_all add: poly_cmult poly_add distrib_right distrib_left ac_simps)
qed

class idom_char_0 = idom + ring_char_0

subclass (in field_char_0) idom_char_0 ..

lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
  by (induct n) (auto simp add: poly_cmult poly_mult)


text ‹More Polynomial Evaluation lemmas.›

lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
  by simp

lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
  by (simp add: poly_mult mult.assoc)

lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
  by (induct p) auto

lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly (p %^ n *** p %^ d) x"
  by (induct n) (auto simp add: poly_mult mult.assoc)


subsection ‹Key Property: if termf a = 0 then term(x - a) divides termp(x).›

lemma (in comm_ring_1) lemma_poly_linear_rem: "q r. h#t = [r] +++ [-a, 1] *** q"
proof (induct t arbitrary: h)
  case Nil
  have "[h] = [h] +++ [- a, 1] *** []" by simp
  then show ?case by blast
next
  case (Cons  x xs)
  have "q r. h # x # xs = [r] +++ [-a, 1] *** q"
  proof -
    from Cons obtain q r where qr: "x # xs = [r] +++ [- a, 1] *** q"
      by blast
    have "h # x # xs = [a * r + h] +++ [-a, 1] *** (r # q)"
      using qr by (cases q) (simp_all add: algebra_simps)
    then show ?thesis by blast
  qed
  then show ?case by blast
qed

lemma (in comm_ring_1) poly_linear_rem: "q r. h#t = [r] +++ [-a, 1] *** q"
  using lemma_poly_linear_rem [where t = t and a = a] by auto

lemma (in comm_ring_1) poly_linear_divides: "poly p a = 0  p = []  (q. p = [-a, 1] *** q)"
proof (cases p)
  case Nil
  then show ?thesis by simp
next
  case (Cons x xs)
  have "poly p a = 0" if "p = [-a, 1] *** q" for q
    using that by (simp add: poly_add poly_cmult)
  moreover
  have "q. p = [- a, 1] *** q" if p0: "poly p a = 0"
  proof -
    from poly_linear_rem[of x xs a] obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q"
      by blast
    have "r = 0"
      using p0 by (simp only: Cons qr poly_mult poly_add) simp
    with Cons qr have "p = [- a, 1] *** q"
      by (simp add: local.padd_commut)
    then show ?thesis ..
  qed
  ultimately show ?thesis using Cons by blast
qed

lemma (in semiring_0) lemma_poly_length_mult[simp]:
  "length (k %* p +++  (h # (a %* p))) = Suc (length p)"
  by (induct p arbitrary: h k a) auto

lemma (in semiring_0) lemma_poly_length_mult2[simp]:
  "length (k %* p +++  (h # p)) = Suc (length p)"
  by (induct p arbitrary: h k) auto

lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
  by auto


subsection ‹Polynomial length›

lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
  by (induct p) auto

lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
  by (induct p1 arbitrary: p2) auto

lemma (in semiring_0) poly_root_mult_length[simp]: "length ([a, b] *** p) = Suc (length p)"
  by (simp add: poly_add_length)

lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
  "poly (p *** q) x  poly [] x  poly p x  poly [] x  poly q x  poly [] x"
  by (auto simp add: poly_mult)

lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0  poly p x = 0  poly q x = 0"
  by (auto simp add: poly_mult)


text ‹Normalisation Properties.›

lemma (in semiring_0) poly_normalized_nil: "pnormalize p = []  poly p x = 0"
  by (induct p) auto

text ‹A nontrivial polynomial of degree n has no more than n roots.›
lemma (in idom) poly_roots_index_lemma:
  assumes "poly p x  poly [] x"
    and "length p = n"
  shows "i. x. poly p x = 0  (mn. x = i m)"
  using assms
proof (induct n arbitrary: p x)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have False if C: "i. x. poly p x = 0  (mSuc n. x  i m)"
  proof -
    from Suc.prems have p0: "poly p x  0" "p  []"
      by auto
    from p0(1)[unfolded poly_linear_divides[of p x]]
    have "q. p  [- x, 1] *** q"
      by blast
    from C obtain a where a: "poly p a = 0"
      by blast
    from a[unfolded poly_linear_divides[of p a]] p0(2) obtain q where q: "p = [-a, 1] *** q"
      by blast
    have lg: "length q = n"
      using q Suc.prems(2) by simp
    from q p0 have qx: "poly q x  poly [] x"
      by (auto simp add: poly_mult poly_add poly_cmult)
    from Suc.hyps[OF qx lg] obtain i where i: "x. poly q x = 0  (mn. x = i m)"
      by blast
    let ?i = "λm. if m = Suc n then a else i m"
    from C[of ?i] obtain y where y: "poly p y = 0" "m Suc n. y  ?i m"
      by blast
    from y have "y = a  poly q y = 0"
      by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
    with i[of y] y show ?thesis
      using le_Suc_eq by auto
  qed
  then show ?case by blast
qed


lemma (in idom) poly_roots_index_length:
  "poly p x  poly [] x  i. x. poly p x = 0  (n. n  length p  x = i n)"
  by (blast intro: poly_roots_index_lemma)

lemma (in idom) poly_roots_finite_lemma1:
  "poly p x  poly [] x  N i. x. poly p x = 0  (n::nat. n < N  x = i n)"
  by (metis le_imp_less_Suc poly_roots_index_length)

lemma (in idom) idom_finite_lemma:
  assumes "x. P x  (n. n < length j  x = j!n)"
  shows "finite {x. P x}"
proof -
  from assms have "{x. P x}  set j"
    by auto
  then show ?thesis
    using finite_subset by auto
qed

lemma (in idom) poly_roots_finite_lemma2:
  "poly p x  poly [] x  i. x. poly p x = 0  x  set i"
  using poly_roots_index_length atMost_iff atMost_upto imageI set_map
  by metis

lemma (in ring_char_0) UNIV_ring_char_0_infinte: "¬ finite (UNIV :: 'a set)"
proof
  assume F: "finite (UNIV :: 'a set)"
  have "finite (UNIV :: nat set)"
  proof (rule finite_imageD)
    have "of_nat ` UNIV  UNIV"
      by simp
    then show "finite (of_nat ` UNIV :: 'a set)"
      using F by (rule finite_subset)
    show "inj (of_nat :: nat  'a)"
      by (simp add: inj_on_def)
  qed
  with infinite_UNIV_nat show False ..
qed

lemma (in idom_char_0) poly_roots_finite: "poly p  poly []  finite {x. poly p x = 0}"
  (is "?lhs  ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    have False if  F: "¬ finite {x. poly p x = 0}"
      and P: "x. poly p x = 0  x  set i" for  i
      by (smt (verit, del_insts) in_set_conv_nth local.idom_finite_lemma that)
    with that show ?thesis
      using local.poly_roots_finite_lemma2 by blast
  qed
  show ?lhs if ?rhs
    using UNIV_ring_char_0_infinte that by auto
qed


text ‹Entirety and Cancellation for polynomials›

lemma (in idom_char_0) poly_entire_lemma2:
  assumes p0: "poly p  poly []"
    and q0: "poly q  poly []"
  shows "poly (p***q)  poly []"
proof -
  let ?S = "λp. {x. poly p x = 0}"
  have "?S (p *** q) = ?S p  ?S q"
    by (auto simp add: poly_mult)
  with p0 q0 show ?thesis
    unfolding poly_roots_finite by auto
qed

lemma (in idom_char_0) poly_entire:
  "poly (p *** q) = poly []  poly p = poly []  poly q = poly []"
  using poly_entire_lemma2[of p q]
  by (auto simp add: fun_eq_iff poly_mult)

lemma (in idom_char_0) poly_entire_neg:
  "poly (p *** q)  poly []  poly p  poly []  poly q  poly []"
  by (simp add: poly_entire)

lemma (in comm_ring_1) poly_add_minus_zero_iff:
  "poly (p +++ -- q) = poly []  poly p = poly q"
  by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq_iff poly_cmult)

lemma (in comm_ring_1) poly_add_minus_mult_eq:
  "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
  by (auto simp add: poly_add poly_minus_def fun_eq_iff poly_mult poly_cmult algebra_simps)

subclass (in idom_char_0) comm_ring_1 ..

lemma (in idom_char_0) poly_mult_left_cancel:
  "poly (p *** q) = poly (p *** r)  poly p = poly []  poly q = poly r"
proof -
  have "poly (p *** q) = poly (p *** r)  poly (p *** q +++ -- (p *** r)) = poly []"
    by (simp only: poly_add_minus_zero_iff)
  also have "  poly p = poly []  poly q = poly r"
    by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
  finally show ?thesis .
qed

lemma (in idom) poly_exp_eq_zero[simp]: "poly (p %^ n) = poly []  poly p = poly []  n  0"
  by (simp add: local.poly_exp fun_eq_iff)

lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a, 1]  poly []"
proof -
  have "x. a + x  0"
    by (metis add_cancel_left_right zero_neq_one)
  then show ?thesis
    by (simp add: fun_eq_iff)
qed

lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n)  poly []"
  by auto


text ‹A more constructive notion of polynomials being trivial.›

lemma (in idom_char_0) poly_zero_lemma': 
  assumes "poly (h # t) = poly []" shows "h = 0  poly t = poly []"
proof -
  have "poly t x = 0" if H: "x. x = 0  poly t x = 0" and pnz: "poly t  poly []" for x
  proof -
    from H have "{x. poly t x = 0}  UNIV - {0}"
      by auto
    then show ?thesis
      using finite_subset local.poly_roots_finite pnz by fastforce
  qed
  with assms show ?thesis
    by (simp add: fun_eq_iff) (metis add_cancel_right_left mult_eq_0_iff)
qed

lemma (in idom_char_0) poly_zero: "poly p = poly []  (c  set p. c = 0)"
proof (induct p)
  case Nil
  then show ?case by simp
next
  case Cons
  then show ?case
    by (smt (verit) list.set_intros pmult_by_x poly_entire poly_zero_lemma' set_ConsD)
qed

lemma (in idom_char_0) poly_0: "c  set p. c = 0  poly p x = 0"
  unfolding poly_zero[symmetric] by simp


text ‹Basics of divisibility.›

lemma (in idom) poly_primes: "[a, 1] divides (p *** q)  [a, 1] divides p  [a, 1] divides q"
proof -
  have "q. x. poly p x = (a + x) * poly q x"
    if "poly p (uminus a) * poly q (uminus a) = (a + (uminus a)) * poly qa (uminus a)"
      and "qa. x. poly q x  (a + x) * poly qa x"
    for qa 
    using that   
    apply (simp add: poly_linear_divides poly_add)
    by (metis add_cancel_left_right combine_common_factor mult_eq_0_iff poly.poly_Cons poly.poly_Nil poly_add poly_cmult)
  moreover have "qb. x. (a + x) * poly qa x * poly q x = (a + x) * poly qb x" for qa
    by (metis local.poly_mult mult_assoc)
  moreover have "q. x. poly p x * ((a + x) * poly qa x) = (a + x) * poly q x" for qa 
    by (metis mult.left_commute local.poly_mult)
  ultimately show ?thesis
    by (auto simp: divides_def divisors_zero fun_eq_iff poly_mult poly_add poly_cmult simp flip: distrib_right)
qed

lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
proof -
  have "poly p = poly (p *** [1])"
    by (auto simp add: poly_mult fun_eq_iff)
  then show ?thesis
    using local.dividesI by blast
qed

lemma (in comm_semiring_1) poly_divides_trans: "p divides q  q divides r  p divides r"
  unfolding divides_def
  by (metis ext local.poly_mult local.poly_mult_assoc)

lemma (in comm_semiring_1) poly_divides_exp: "m  n  (p %^ m) divides (p %^ n)"
  by (auto simp: le_iff_add divides_def poly_exp_add fun_eq_iff)

lemma (in comm_semiring_1) poly_exp_divides: "(p %^ n) divides q  m  n  (p %^ m) divides q"
  by (blast intro: poly_divides_exp poly_divides_trans)

lemma (in comm_semiring_0) poly_divides_add:
  assumes "p divides q" and "p divides r" shows "p divides (q +++ r)"
proof -
  have "qa qb. poly q = poly (p *** qa); poly r = poly (p *** qb)
        poly (q +++ r) = poly (p *** (qa +++ qb))"
    by (auto simp add: poly_add fun_eq_iff poly_mult distrib_left)
  with assms show ?thesis
    by (auto simp add: divides_def)
qed

lemma (in comm_ring_1) poly_divides_diff:
  assumes "p divides q" and "p divides (q +++ r)"
  shows "p divides r"
proof -
  have "qa qb. poly q = poly (p *** qa); poly (q +++ r) = poly (p *** qb)
          poly r = poly (p *** (qb +++ -- qa))"
  by (auto simp add: poly_add fun_eq_iff poly_mult poly_minus algebra_simps)
  with assms show ?thesis
    by (auto simp add: divides_def)
qed

lemma (in comm_ring_1) poly_divides_diff2: "p divides r  p divides (q +++ r)  p divides q"
  by (metis local.padd_commut local.poly_divides_diff)

lemma (in semiring_0) poly_divides_zero: "poly p = poly []  q divides p"
  by (metis ext dividesI poly.poly_Nil poly_mult_Nil2)

lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
  using local.poly_divides_zero by force


text ‹At last, we can consider the order of a root.›

lemma (in idom_char_0) poly_order_exists_lemma:
  assumes "length p = d"
    and "poly p  poly []"
  shows "n q. p = mulexp n [-a, 1] q  poly q a  0"
  using assms
proof (induct d arbitrary: p)
  case 0
  then show ?case by simp
next
  case (Suc n p)
  show ?case
  proof (cases "poly p a = 0")
    case True
    from Suc.prems have h: "length p = Suc n" "poly p  poly []"
      by auto
    then have pN: "p  []"
      by auto
    from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
      by blast
    from q h True have qh: "length q = n" "poly q  poly []"
      using h(2) local.poly_entire q by fastforce+
    from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a  0"
      by blast
    from mr q have "p = mulexp (Suc m) [-a,1] r  poly r a  0"
      by simp
    then show ?thesis by blast
  next
    case False
    with Suc.prems show ?thesis
      by (smt (verit, best) local.mulexp.mulexp_zero)
  qed
qed


lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
  by (induct n) (auto simp add: poly_mult ac_simps)

lemma (in comm_semiring_1) divides_left_mult:
  assumes "(p *** q) divides r"
  shows "p divides r  q divides r"
proof-
  from assms obtain t where "poly r = poly (p *** q *** t)"
    unfolding divides_def by blast
  then have "poly r = poly (p *** (q *** t))" and "poly r = poly (q *** (p *** t))"
    by (auto simp add: fun_eq_iff poly_mult ac_simps)
  then show ?thesis
    unfolding divides_def by blast
qed


(* FIXME: Tidy up *)

lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
  by (induct n) simp_all

lemma (in idom_char_0) poly_order_exists:
  assumes "length p = d"
    and "poly p  poly []"
  shows "n. [- a, 1] %^ n divides p  ¬ [- a, 1] %^ Suc n divides p"
proof -
  from assms have "n q. p = mulexp n [- a, 1] q  poly q a  0"
    by (rule poly_order_exists_lemma)
  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a  0"
    by blast
  have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
  proof (rule dividesI)
    show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
      by (induct n) (simp_all add: poly_add poly_cmult poly_mult algebra_simps)
  qed
  moreover have "¬ [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
  proof
    assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
    then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"
      by (rule dividesE)
    moreover have "poly (mulexp n [- a, 1] q)  poly ([- a, 1] %^ Suc n *** m)"
    proof (induct n)
      case 0
      show ?case
      proof (rule ccontr)
        assume "¬ ?thesis"
        then have "poly q a = 0"
          by (simp add: poly_add poly_cmult)
        with poly q a  0 show False
          by simp
      qed
    next
      case (Suc n)
      show ?case
        by (rule pexp_Suc [THEN ssubst])
          (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
    qed
    ultimately show False by simp
  qed
  ultimately show ?thesis
    by (auto simp add: p)
qed

lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
  by (auto simp add: divides_def)

lemma (in idom_char_0) poly_order:
  "poly p  poly []  ∃!n. ([-a, 1] %^ n) divides p  ¬ (([-a, 1] %^ Suc n) divides p)"
  by (meson Suc_le_eq linorder_neqE_nat local.poly_exp_divides poly_order_exists)


text ‹Order›

lemma some1_equalityD: "n = (SOME n. P n)  ∃!n. P n  P n"
  by (blast intro: someI2)

lemma (in idom_char_0) order:
  "([-a, 1] %^ n) divides p  ¬ (([-a, 1] %^ Suc n) divides p) 
    n = order a p  poly p  poly []"
  unfolding order_def
  by (metis (no_types, lifting) local.poly_divides_zero local.poly_order someI)

lemma (in idom_char_0) order2:
  "poly p  poly [] 
    ([-a, 1] %^ (order a p)) divides p  ¬ ([-a, 1] %^ Suc (order a p)) divides p"
  by (simp add: order del: pexp_Suc)

lemma (in idom_char_0) order_unique:
  "poly p  poly []  ([-a, 1] %^ n) divides p  ¬ ([-a, 1] %^ (Suc n)) divides p 
    n = order a p"
  using order [of a n p] by auto

lemma (in idom_char_0) order_unique_lemma:
  "poly p  poly []  ([-a, 1] %^ n) divides p  ¬ ([-a, 1] %^ (Suc n)) divides p 
    n = order a p"
  by (blast intro: order_unique)

lemma (in ring_1) order_poly: "poly p = poly q  order a p = order a q"
  by (auto simp add: fun_eq_iff divides_def poly_mult order_def)

lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
  by (induct p) auto

lemma (in comm_ring_1) lemma_order_root:
  "0 < n  [- a, 1] %^ n divides p  ¬ [- a, 1] %^ (Suc n) divides p  poly p a = 0"
  by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)

lemma (in idom_char_0) order_root: "poly p a = 0  poly p = poly []  order a p  0"
proof (cases "poly p = poly []")
  case False
  then show ?thesis 
    by (metis (mono_tags, lifting) dividesI lemma_order_root order2 pexp_one poly_linear_divides neq0_conv)
qed auto

lemma (in idom_char_0) order_divides:
  "([-a, 1] %^ n) divides p  poly p = poly []  n  order a p"
proof (cases "poly p = poly []")
  case True
  then show ?thesis 
    using local.poly_divides_zero by force
next
  case False
  then show ?thesis 
    by (meson local.order2 local.poly_exp_divides not_less_eq_eq)
qed

lemma (in idom_char_0) order_decomp:
  assumes "poly p  poly []"
  shows "q. poly p = poly (([-a, 1] %^ order a p) *** q)  ¬ [-a, 1] divides q"
proof -
  obtain q where q: "poly p = poly ([- a, 1] %^ order a p *** q)"
    using assms local.order2 divides_def by blast
  have False if "poly q = poly ([- a, 1] *** qa)" for qa
  proof -
    have "poly p  poly ([- a, 1] %^ Suc (order a p) *** qa)"
      using assms local.divides_def local.order2 by blast
    with q that show False
      by (auto simp add: poly_mult ac_simps simp del: pmult_Cons)
  qed 
  with q show ?thesis
    unfolding divides_def by blast
qed

text ‹Important composition properties of orders.›
lemma order_mult:
  fixes a :: "'a::idom_char_0"
  assumes "poly (p *** q)  poly []"
  shows "order a (p *** q) = order a p + order a q"
proof -
  have p: "poly p  poly []" and q: "poly q  poly []"
    using assms poly_entire by auto
  obtain p' where p': 
          "x. poly p x = poly ([- a, 1] %^ order a p) x * poly p' x"
          "¬ [- a, 1] divides p'"
    by (metis order_decomp p poly_mult)
  obtain q' where q': 
          "x. poly q x = poly ([- a, 1] %^ order a q) x * poly q' x"
          "¬ [- a, 1] divides q'"
    by (metis order_decomp q poly_mult)
  have "[- a, 1] %^ (order a p + order a q) divides (p *** q)"
  proof -
    have *: "poly p x * poly q x =
          poly ([- a, 1] %^ order a p) x * poly ([- a, 1] %^ order a q) x * poly (p' *** q') x" for x
      using p' q' by (simp add: poly_mult)
    then show ?thesis
      unfolding divides_def  poly_exp_add poly_mult using * by blast
  qed
  moreover have False
    if pq: "order a (p *** q)  order a p + order a q"
      and dv: "[- a, 1] *** [- a, 1] %^ (order a p + order a q) divides (p *** q)"
  proof -
    obtain pq' :: "'a list"
      where pq': "poly (p *** q) = poly ([- a, 1] *** [- a, 1] %^ (order a p + order a q) *** pq')"
      using dv unfolding divides_def by auto
    have "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (p' *** q'))) =
          poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** pq')))"
      using p' q' pq pq'
      by (simp add: fun_eq_iff poly_exp_add poly_mult ac_simps del: pmult_Cons)
    then have "poly ([-a, 1] %^ (order a p) *** (p' *** q')) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** pq'))"
      by (simp add: poly_mult_left_cancel)
    then have "[-a, 1] divides (p' *** q')"
      unfolding divides_def by (meson poly_exp_prime_eq_zero poly_mult_left_cancel)
    with p' q' show ?thesis
      by (simp add: poly_primes)
  qed
  ultimately show ?thesis
    by (metis order pexp_Suc)
qed

lemma (in idom_char_0) order_root2: "poly p  poly []  poly p a = 0  order a p  0"
  using order_root by presburger

lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p"
  by auto

lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
  by (simp add: fun_eq_iff)

lemma (in idom_char_0) rsquarefree_decomp:
  assumes "rsquarefree p" and "poly p a = 0"
  shows "q. poly p = poly ([-a, 1] *** q)  poly q a  0"
proof -
  have "order a p = Suc 0"
    using assms local.order_root2 rsquarefree_def by force
  moreover
  obtain q where "poly p = poly ([- a, 1] %^ order a p *** q)" 
                 "¬ [- a, 1] divides q"
    using assms(1) order_decomp rsquarefree_def by blast
  ultimately show ?thesis
    using dividesI poly_linear_divides by auto
qed

text ‹Normalization of a polynomial.›

lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
  by (induct p) (auto simp add: fun_eq_iff)

text ‹The degree of a polynomial.›

lemma (in semiring_0) lemma_degree_zero: "(c  set p. c = 0)  pnormalize p = []"
  by (induct p) auto

lemma (in idom_char_0) degree_zero:
  assumes "poly p = poly []"
  shows "degree p = 0"
  using assms
  by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)

lemma (in semiring_0) pnormalize_sing: "pnormalize [x] = [x]  x  0"
  by simp

lemma (in semiring_0) pnormalize_pair: "y  0  pnormalize [x, y] = [x, y]"
  by simp

lemma (in semiring_0) pnormal_cons: "pnormal p  pnormal (c # p)"
  unfolding pnormal_def by simp

lemma (in semiring_0) pnormal_tail: "p  []  pnormal (c # p)  pnormal p"
  unfolding pnormal_def by (auto split: if_split_asm)

lemma (in semiring_0) pnormal_last_nonzero: "pnormal p  last p  0"
  by (induct p) (simp_all add: pnormal_def split: if_split_asm)

lemma (in semiring_0) pnormal_length: "pnormal p  0 < length p"
  unfolding pnormal_def length_greater_0_conv by blast

lemma (in semiring_0) pnormal_last_length: "0 < length p  last p  0  pnormal p"
  by (induct p) (auto simp: pnormal_def  split: if_split_asm)

lemma (in semiring_0) pnormal_id: "pnormal p  0 < length p  last p  0"
  using pnormal_last_length pnormal_length pnormal_last_nonzero by blast

lemma (in idom_char_0) poly_Cons_eq: "poly (c # cs) = poly (d # ds)  c = d  poly cs = poly ds"
  (is "?lhs  ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    from that have "poly ((c # cs) +++ -- (d # ds)) x = 0" for x
      by (simp only: poly_minus poly_add algebra_simps) (simp add: algebra_simps)
    then have "poly ((c # cs) +++ -- (d # ds)) = poly []"
      by (simp add: fun_eq_iff)
    then have "c = d" and "x  set (cs +++ -- ds). x = 0"
      unfolding poly_zero by (simp_all add: poly_minus_def algebra_simps)
    from this(2) have "poly (cs +++ -- ds) x = 0" for x
      unfolding poly_zero[symmetric] by simp
    with c = d show ?thesis
      by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
  qed
  show ?lhs if ?rhs
    using that by (simp add:fun_eq_iff)
qed

lemma (in idom_char_0) pnormalize_unique: "poly p = poly q  pnormalize p = pnormalize q"
proof (induct q arbitrary: p)
  case Nil
  then show ?case
    by (simp only: poly_zero lemma_degree_zero) simp
next
  case (Cons c cs p)
  then show ?case
  proof (induct p)
    case Nil
    then show ?case
      by (metis local.poly_zero_lemma')
  next
    case (Cons d ds)
    then show ?case
      by (metis pnormalize.pnormalize_Cons local.poly_Cons_eq)
  qed
qed

lemma (in idom_char_0) degree_unique:
  assumes pq: "poly p = poly q"
  shows "degree p = degree q"
  using pnormalize_unique[OF pq] unfolding degree_def by simp

lemma (in semiring_0) pnormalize_length: "length (pnormalize p)  length p"
  by (induct p) auto

lemma (in semiring_0) last_linear_mul_lemma:
  "last ((a %* p) +++ (x # (b %* p))) = (if p = [] then x else b * last p)"
proof (induct p arbitrary: a x b)
  case Nil
  then show ?case by auto
next
  case (Cons a p c x b)
  then have "padd (cmult c p) (times b a # cmult b p)  []"
    by (metis local.padd.padd_Nil local.padd_Cons_Cons neq_Nil_conv)
  then show ?case
    by (simp add: local.Cons)
qed

lemma (in semiring_1) last_linear_mul:
  assumes p: "p  []"
  shows "last ([a, 1] *** p) = last p"
proof -
  from p obtain c cs where cs: "p = c # cs"
    by (cases p) auto
  from cs have eq: "[a, 1] *** p = (a %* (c # cs)) +++ (0 # (1 %* (c # cs)))"
    by (simp add: poly_cmult_distr)
  show ?thesis
    using cs unfolding eq last_linear_mul_lemma by simp
qed

lemma (in semiring_0) pnormalize_eq: "last p  0  pnormalize p = p"
  by (induct p) (auto split: if_split_asm)

lemma (in semiring_0) last_pnormalize: "pnormalize p  []  last (pnormalize p)  0"
  by (induct p) auto

lemma (in semiring_0) pnormal_degree: "last p  0  degree p = length p - 1"
  using pnormalize_eq[of p] unfolding degree_def by simp

lemma (in semiring_0) poly_Nil_ext: "poly [] = (λx. 0)"
  by auto

lemma (in idom_char_0) linear_mul_degree:
  assumes p: "poly p  poly []"
  shows "degree ([a, 1] *** p) = degree p + 1"
proof -
  from p have pnz: "pnormalize p  []"
    unfolding poly_zero lemma_degree_zero .

  from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
  have l0: "last ([a, 1] *** pnormalize p)  0" by simp

  from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
    pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
  have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
    by simp

  have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
    by (rule ext) (simp add: poly_mult poly_add poly_cmult)
  from degree_unique[OF eqs] th show ?thesis
    by (simp add: degree_unique[OF poly_normalize])
qed

lemma (in idom_char_0) linear_pow_mul_degree:
  "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
proof (induct n arbitrary: a p)
  case (0 a p)
  show ?case
  proof (cases "poly p = poly []")
    case True
    then show ?thesis
      using degree_unique[OF True] by (simp add: degree_def)
  qed (auto simp add: poly_Nil_ext)
next
  case (Suc n a p)
  have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1] %^ n *** ([a, 1] *** p))"
    by (force simp add: poly_mult poly_add poly_cmult ac_simps distrib_left)
  note deq = degree_unique[OF eq]
  show ?case
  proof (cases "poly p = poly []")
    case True
    with eq have eq': "poly ([a, 1] %^(Suc n) *** p) = poly []"
      by (auto simp add: poly_mult poly_cmult poly_add)
    from degree_unique[OF eq'] True show ?thesis
      by (simp add: degree_def)
  next
    case False
    then have ap: "poly ([a,1] *** p)  poly []"
      using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
    have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1]%^n *** ([a, 1] *** p))"
      by (auto simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
    from ap have ap': "poly ([a, 1] *** p) = poly []  False"
      by blast
    have th0: "degree ([a, 1]%^n *** ([a, 1] *** p)) = degree ([a, 1] *** p) + n"
      unfolding Suc.hyps[of a "pmult [a,one] p"] ap' by simp
    from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
    show ?thesis
      by (auto simp del: poly.simps)
  qed
qed

lemma (in idom_char_0) order_degree:
  assumes p0: "poly p  poly []"
  shows "order a p  degree p"
proof -
  from order2[OF p0, unfolded divides_def]
  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)"
    by blast
  with q p0 have "poly q  poly []"
    by (simp add: poly_mult poly_entire)
  with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
    by auto
qed


text ‹Tidier versions of finiteness of roots.›
lemma (in idom_char_0) poly_roots_finite_set:
  "poly p  poly []  finite {x. poly p x = 0}"
  unfolding poly_roots_finite .


text ‹Bound for polynomial.›
lemma poly_mono:
  fixes x :: "'a::linordered_idom"
  shows "¦x¦  k  ¦poly p x¦  poly (map abs p) k"
proof (induct p)
  case Nil
  then show ?case by simp
next
  case (Cons a p)
  have "¦a + x * poly p x¦  ¦a¦ + ¦x * poly p x¦"
    using abs_triangle_ineq by blast
  also have "  ¦a¦ + k * poly (map abs p) k"
    by (simp add: Cons.hyps Cons.prems abs_mult mult_mono')
  finally show ?case
    using Cons by auto
qed

lemma (in semiring_0) poly_Sing: "poly [c] x = c"
  by simp

end