# 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›

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"

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: "-- [] = []"

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
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
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)"

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
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"

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 \<^term>‹f a = 0› then \<^term>‹(x - a)› divides \<^term>‹p(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
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"
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)"

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"

lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 ⟷ poly p x = 0 ∨ poly q x = 0"

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 ⟶ (∃m≤n. 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 ∧ (∀m≤Suc 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"
from Suc.hyps[OF qx lg] obtain i where i: "⋀x. poly q x = 0 ⟶ (∃m≤n. 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"
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)"
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"
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 []"

"poly (p +++ -- q) = poly [] ⟷ poly p = poly q"

"poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"

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 []"
also have "… ⟷ poly p = poly [] ∨ poly q = poly r"
finally show ?thesis .
qed

lemma (in idom) poly_exp_eq_zero[simp]: "poly (p %^ n) = poly [] ⟷ poly p = poly [] ∧ n ≠ 0"

lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a, 1] ≠ poly []"
proof -
have "∃x. a + x ≠ 0"
then show ?thesis
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
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
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)"

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)

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))"
with assms show ?thesis
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))"
with assms show ?thesis
qed

lemma (in comm_ring_1) poly_divides_diff2: "p divides r ⟹ p divides (q +++ r) ⟹ p divides q"

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)"
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"
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
qed

lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"

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'
then have "poly ([-a, 1] %^ (order a p) *** (p' *** q')) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** pq'))"
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
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]"

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
then have "poly ((c # cs) +++ -- (d # ds)) = poly []"
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
qed
show ?lhs if ?rhs
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) ≠ []"
then show ?case
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)))"
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)"
from degree_unique[OF eqs] th show ?thesis
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)
next
case (Suc n a p)
have eq: "poly ([a, 1] %^(Suc n) *** p) = poly ([a, 1] %^ n *** ([a, 1] *** p))"
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 []"
from degree_unique[OF eq'] True show ?thesis
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))"
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 []"
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
```