# Theory Algebraic_Numbers_Prelim

```(*
Author:      René Thiemann
Sebastiaan Joosten
*)
section ‹Algebraic Numbers -- Excluding Addition and Multiplication›

text ‹This theory contains basic definition and results on algebraic numbers, namely that
algebraic numbers are closed under negation, inversion, \$n\$-th roots, and
that every rational number is algebraic. For all of these closure properties, corresponding
polynomial witnesses are available.

Moreover, this theory contains the uniqueness result,
that for every algebraic number there is exactly one content-free irreducible polynomial with
positive leading coefficient for it.
This result is stronger than similar ones which you find in many textbooks.
The reason is that here we do not require a least degree construction.

This is essential, since given some content-free irreducible polynomial for x,
how should we check whether the degree is optimal. In the formalized result, this is
not required. The result is proven via GCDs, and that the GCD does not change
when executed on the rational numbers or on the reals or complex numbers, and that
the GCD of a rational polynomial can be expressed via the GCD of integer polynomials.›

text ‹Many results are taken from the textbook \cite[pages 317ff]{AlgNumbers}.›

theory Algebraic_Numbers_Prelim
imports
"HOL-Computational_Algebra.Fundamental_Theorem_Algebra"
Polynomial_Interpolation.Newton_Interpolation (* for lemma smult_1 *)
Polynomial_Factorization.Gauss_Lemma
Berlekamp_Zassenhaus.Unique_Factorization_Poly
Polynomial_Factorization.Square_Free_Factorization
begin

lemma primitive_imp_unit_iff:
fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
assumes pr: "primitive p"
shows "p dvd 1 ⟷ degree p = 0"
proof
assume "degree p = 0"
from degree0_coeffs[OF this] obtain p0 where p: "p = [:p0:]" by auto
then have "∀c ∈ set (coeffs p). p0 dvd c" by (simp add: cCons_def)
with pr have "p0 dvd 1" by (auto dest: primitiveD)
with p show "p dvd 1" by auto
next
assume "p dvd 1"
then show "degree p = 0" by (auto simp: poly_dvd_1)
qed

lemma dvd_all_coeffs_imp_dvd:
assumes "∀a ∈ set (coeffs p). c dvd a" shows "[:c:] dvd p"
proof (insert assms, induct p)
case 0
then show ?case by simp
next
case (pCons a p)
have "pCons a p = [:a:] + pCons 0 p" by simp
also have "[:c:] dvd ..."
from pCons show "[:c:] dvd [:a:]" by (auto simp: cCons_def)
from pCons have "[:c:] dvd p" by auto
from Rings.dvd_mult[OF this]
show "[:c:] dvd pCons 0 p" by (subst pCons_0_as_mult)
qed
finally show ?case.
qed

lemma irreducible_content:
fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
assumes "irreducible p" shows "degree p = 0 ∨ primitive p"
proof(rule ccontr)
assume not: "¬?thesis"
then obtain c where c1: "¬c dvd 1" and "∀a ∈ set (coeffs p). c dvd a" by (auto elim: not_primitiveE)
from dvd_all_coeffs_imp_dvd[OF this(2)]
obtain r where p: "p = r * [:c:]" by (elim dvdE, auto)
from irreducibleD[OF assms this] have "r dvd 1 ∨ [:c:] dvd 1" by auto
with c1 have "r dvd 1" unfolding const_poly_dvd_1 by auto
then have "degree r = 0" unfolding poly_dvd_1 by auto
with p have "degree p = 0" by auto
with not show False by auto
qed

(* TODO: move *)
lemma linear_irreducible_field:
fixes p :: "'a :: field poly"
assumes deg: "degree p = 1" shows "irreducible p"
proof (intro irreducibleI)
from deg show p0: "p ≠ 0" by auto
from deg show "¬ p dvd 1" by (auto simp: poly_dvd_1)
fix a b assume p: "p = a * b"
with p0 have a0: "a ≠ 0" and b0: "b ≠ 0" by auto
from degree_mult_eq[OF this, folded p] assms
consider "degree a = 1" "degree b = 0" | "degree a = 0" "degree b = 1" by force
then show "a dvd 1 ∨ b dvd 1"
by (cases; insert a0 b0, auto simp: primitive_imp_unit_iff)
qed

(* TODO: move *)
lemma linear_irreducible_int:
fixes p :: "int poly"
assumes deg: "degree p = 1" and cp: "content p dvd 1"
shows "irreducible p"
proof (intro irreducibleI)
from deg show p0: "p ≠ 0" by auto
from deg show "¬ p dvd 1" by (auto simp: poly_dvd_1)
fix a b assume p: "p = a * b"
note * = cp[unfolded p is_unit_content_iff, unfolded content_mult]
have a1: "content a dvd 1" and b1: "content b dvd 1"
using content_ge_0_int[of a] pos_zmult_eq_1_iff_lemma[OF *] * by (auto simp: abs_mult)
with p0 have a0: "a ≠ 0" and b0: "b ≠ 0" by auto
from degree_mult_eq[OF this, folded p] assms
consider "degree a = 1" "degree b = 0" | "degree a = 0" "degree b = 1" by force
then show "a dvd 1 ∨ b dvd 1"
by (cases; insert a1 b1, auto simp: primitive_imp_unit_iff)
qed

lemma irreducible_connect_rev:
fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
assumes irr: "irreducible p" and deg: "degree p > 0"
shows "irreducible⇩d p"
proof(intro irreducible⇩dI deg)
fix q r
assume degq: "degree q > 0" and diff: "degree q < degree p" and p: "p = q * r"
from degq have nu: "¬ q dvd 1" by (auto simp: poly_dvd_1)
from irreducibleD[OF irr p] nu have "r dvd 1" by auto
then have "degree r = 0" by (auto simp: poly_dvd_1)
with degq diff show False unfolding p using degree_mult_le[of q r] by auto
qed

subsection ‹Polynomial Evaluation of Integer and Rational Polynomials in Fields.›

abbreviation ipoly where "ipoly f x ≡ poly (of_int_poly f) x"

lemma poly_map_poly_code[code_unfold]: "poly (map_poly h p) x = fold_coeffs (λ a b. h a + x * b) p 0"
by (induct p, auto)

abbreviation real_of_int_poly :: "int poly ⇒ real poly" where
"real_of_int_poly ≡ of_int_poly"

abbreviation real_of_rat_poly :: "rat poly ⇒ real poly" where
"real_of_rat_poly ≡ map_poly of_rat"

lemma of_rat_of_int[simp]: "of_rat ∘ of_int = of_int" by auto

lemma ipoly_of_rat[simp]: "ipoly p (of_rat y) = of_rat (ipoly p y)"
proof-
have id: "of_int = of_rat o of_int" unfolding comp_def by auto
show ?thesis by (subst id, subst map_poly_map_poly[symmetric], auto)
qed

lemma ipoly_of_real[simp]:
"ipoly p (of_real x :: 'a :: {field,real_algebra_1}) = of_real (ipoly p x)"
proof -
have id: "of_int = of_real o of_int" unfolding comp_def by auto
show ?thesis by (subst id, subst map_poly_map_poly[symmetric], auto)
qed

lemma finite_ipoly_roots: assumes "p ≠ 0"
shows "finite {x :: real. ipoly p x = 0}"
proof -
let ?p = "real_of_int_poly p"
from assms have "?p ≠ 0" by auto
thus ?thesis by (rule poly_roots_finite)
qed

subsection ‹Algebraic Numbers -- Definition, Inverse, and Roots›

text ‹A number @{term "x :: 'a :: field"} is algebraic iff it is the root of an integer polynomial.
Whereas the Isabelle distribution this is defined via the embedding
of integers in an field via @{const Ints}, we work with integer polynomials
of type @{type int} and then use @{const ipoly} for evaluating the polynomial at
a real or complex point.›

lemma algebraic_altdef_ipoly:
shows "algebraic x ⟷ (∃p. ipoly p x = 0 ∧ p ≠ 0)"
unfolding algebraic_def
proof (safe, goal_cases)
case (1 p)
define the_int where "the_int = (λx::'a. THE r. x = of_int r)"
define p' where "p' = map_poly the_int p"
have of_int_the_int: "of_int (the_int x) = x" if "x ∈ ℤ" for x
unfolding the_int_def by (rule sym, rule theI') (insert that, auto simp: Ints_def)
have the_int_0_iff: "the_int x = 0 ⟷ x = 0" if "x ∈ ℤ"
using of_int_the_int[OF that] by auto
have "map_poly of_int p' = map_poly (of_int ∘ the_int) p"
by (simp add: p'_def map_poly_map_poly)
also from 1 of_int_the_int have "… = p"
by (subst poly_eq_iff) (auto simp: coeff_map_poly)
finally have p_p': "map_poly of_int p' = p" .
show ?case
proof (intro exI conjI notI)
from 1 show "ipoly p' x = 0" by (simp add: p_p')
next
assume "p' = 0"
hence "p = 0" by (simp add: p_p' [symmetric])
with ‹p ≠ 0› show False by contradiction
qed
next
case (2 p)
thus ?case by (intro exI[of _ "map_poly of_int p"], auto)
qed

text ‹Definition of being algebraic with explicit witness polynomial.›

definition represents :: "int poly ⇒ 'a :: field_char_0 ⇒ bool" (infix "represents" 51)
where "p represents x = (ipoly p x = 0 ∧ p ≠ 0)"

lemma representsI[intro]: "ipoly p x = 0 ⟹ p ≠ 0 ⟹ p represents x"
unfolding represents_def by auto

lemma representsD:
assumes "p represents x" shows "p ≠ 0" and "ipoly p x = 0" using assms unfolding represents_def by auto

lemma representsE:
assumes "p represents x" and "p ≠ 0 ⟹ ipoly p x = 0 ⟹ thesis"
shows thesis using assms unfolding represents_def by auto

lemma represents_imp_degree:
fixes x :: "'a :: field_char_0"
assumes "p represents x" shows "degree p ≠ 0"
proof-
from assms have "p ≠ 0" and px: "ipoly p x = 0" by (auto dest:representsD)
then have "(of_int_poly p :: 'a poly) ≠ 0" by auto
then have "degree (of_int_poly p :: 'a poly) ≠ 0" by (fold poly_zero[OF px])
then show ?thesis by auto
qed

lemma representsE_full[elim]:
assumes rep: "p represents x"
and main: "p ≠ 0 ⟹ ipoly p x = 0 ⟹ degree p ≠ 0 ⟹ thesis"
shows thesis
by (rule main, insert represents_imp_degree[OF rep] rep, auto elim: representsE)

lemma represents_of_rat[simp]: "p represents (of_rat x) = p represents x" by (auto elim!:representsE)
lemma represents_of_real[simp]: "p represents (of_real x) = p represents x" by (auto elim!:representsE)

lemma algebraic_iff_represents: "algebraic x ⟷ (∃ p. p represents x)"
unfolding algebraic_altdef_ipoly represents_def ..

lemma represents_irr_non_0:
assumes irr: "irreducible p" and ap: "p represents x" and x0: "x ≠ 0"
shows "poly p 0 ≠ 0"
proof
have nu: "¬ [:0,1::int:] dvd 1" by (auto simp: poly_dvd_1)
assume "poly p 0 = 0"
hence dvd: "[: 0, 1 :] dvd p" by (unfold dvd_iff_poly_eq_0, simp)
then obtain q where pq: "p = [:0,1:] * q" by (elim dvdE)
from irreducibleD[OF irr this] nu have "q dvd 1" by auto
from this obtain r where "q = [:r:]" "r dvd 1" by (auto simp add: poly_dvd_1 dest: degree0_coeffs)
with pq have "p = [:0,r:]" by auto
with ap have "x = 0" by (auto simp: of_int_hom.map_poly_pCons_hom)
with x0 show False by auto
qed

text ‹The polynomial encoding a rational number.›

definition poly_rat :: "rat ⇒ int poly" where
"poly_rat x = (case quotient_of x of (n,d) ⇒ [:-n,d:])"

definition abs_int_poly:: "int poly ⇒ int poly" where
"abs_int_poly p ≡ if lead_coeff p < 0 then -p else p"

lemma pos_poly_abs_poly[simp]:
shows "lead_coeff (abs_int_poly p) > 0 ⟷ p ≠ 0"
proof-
have "p ≠ 0 ⟷ lead_coeff p * sgn (lead_coeff p) > 0" by (fold abs_sgn, auto)
then show ?thesis by (auto simp: abs_int_poly_def mult.commute)
qed

lemma abs_int_poly_0[simp]: "abs_int_poly 0 = 0"
by (auto simp: abs_int_poly_def)

lemma abs_int_poly_eq_0_iff[simp]: "abs_int_poly p = 0 ⟷ p = 0"
by (auto simp: abs_int_poly_def sgn_eq_0_iff)

lemma degree_abs_int_poly[simp]: "degree (abs_int_poly p) = degree p"
by (auto simp: abs_int_poly_def sgn_eq_0_iff)

lemma abs_int_poly_dvd[simp]: "abs_int_poly p dvd q ⟷ p dvd q"
by (unfold abs_int_poly_def, auto)

(*TODO: move & generalize *)
lemma (in idom) irreducible_uminus[simp]: "irreducible (-x) ⟷ irreducible x"
proof-
have "-x = -1 * x" by simp
also have "irreducible ... ⟷ irreducible x" by (rule irreducible_mult_unit_left, auto)
finally show ?thesis.
qed

lemma irreducible_abs_int_poly[simp]:
"irreducible (abs_int_poly p) ⟷ irreducible p"
by (unfold abs_int_poly_def, auto)

lemma coeff_abs_int_poly[simp]:
"coeff (abs_int_poly p) n = (if lead_coeff p < 0 then - coeff p n else coeff p n)"
by (simp add: abs_int_poly_def)

"lead_coeff (abs_int_poly p) = abs (lead_coeff p)"
by auto

lemma ipoly_abs_int_poly_eq_zero_iff[simp]:
"ipoly (abs_int_poly p) (x :: 'a :: comm_ring_1) = 0 ⟷ ipoly p x = 0"
by (auto simp: abs_int_poly_def sgn_eq_0_iff of_int_poly_hom.hom_uminus)

lemma abs_int_poly_represents[simp]:
"abs_int_poly p represents x ⟷ p represents x" by (auto elim!:representsE)

(* TODO: Move *)
lemma content_pCons[simp]: "content (pCons a p) = gcd a (content p)"
by (unfold content_def coeffs_pCons_eq_cCons cCons_def, auto)

lemma content_uminus[simp]:
fixes p :: "'a :: ring_gcd poly" shows "content (-p) = content p"
by (induct p, auto)

lemma primitive_abs_int_poly[simp]:
"primitive (abs_int_poly p) ⟷ primitive p"
by (auto simp: abs_int_poly_def)

lemma abs_int_poly_inv[simp]: "smult (sgn (lead_coeff p)) (abs_int_poly p) = p"
by (cases "lead_coeff p > 0", auto simp: abs_int_poly_def)

definition cf_pos :: "int poly ⇒ bool" where
"cf_pos p = (content p = 1 ∧ lead_coeff p > 0)"

definition cf_pos_poly :: "int poly ⇒ int poly" where
"cf_pos_poly f = (let
c = content f;
d = (sgn (lead_coeff f) * c)
in sdiv_poly f d)"

lemma sgn_is_unit[intro!]:
fixes x :: "'a :: linordered_idom" (* find/make better class *)
assumes "x ≠ 0"
shows "sgn x dvd 1" using assms by(cases x "0::'a" rule:linorder_cases, auto)

lemma cf_pos_poly_0[simp]: "cf_pos_poly 0 = 0" by (unfold cf_pos_poly_def sdiv_poly_def, auto)

lemma cf_pos_poly_eq_0[simp]: "cf_pos_poly f = 0 ⟷ f = 0"
proof(cases "f = 0")
case True
thus ?thesis unfolding cf_pos_poly_def Let_def by (simp add: sdiv_poly_def)
next
case False
then have lc0: "lead_coeff f ≠ 0" by auto
then have s0: "sgn (lead_coeff f) ≠ 0" (is "?s ≠ 0") and "content f ≠ 0" (is "?c ≠ 0") by (auto simp: sgn_0_0)
then have sc0: "?s * ?c ≠ 0" by auto
{ fix i
from content_dvd_coeff sgn_is_unit[OF lc0]
have "?s * ?c dvd coeff f i" by (auto simp: unit_dvd_iff)
then have "coeff f i div (?s * ?c) = 0 ⟷ coeff f i = 0" by (auto simp:dvd_div_eq_0_iff)
} note * = this
show ?thesis unfolding cf_pos_poly_def Let_def sdiv_poly_def poly_eq_iff by (auto simp: coeff_map_poly *)
qed

lemma
shows cf_pos_poly_main: "smult (sgn (lead_coeff f) * content f) (cf_pos_poly f) = f" (is ?g1)
and content_cf_pos_poly[simp]: "content (cf_pos_poly f) = (if f = 0 then 0 else 1)" (is ?g2)
and lead_coeff_cf_pos_poly[simp]: "lead_coeff (cf_pos_poly f) > 0 ⟷ f ≠ 0" (is ?g3)
and cf_pos_poly_dvd[simp]: "cf_pos_poly f dvd f" (is ?g4)
proof(atomize(full), (cases "f = 0"; intro conjI))
case True
then show ?g1 ?g2 ?g3 ?g4 by simp_all
next
case f0: False
let ?s = "sgn (lead_coeff f)"
have s: "?s ∈ {-1,1}" using f0 unfolding sgn_if by auto
define g where "g ≡ smult ?s f"
define d where "d ≡ ?s * content f"
have "content g = content ([:?s:] * f)" unfolding g_def by simp
also have "… = content [:?s:] * content f" unfolding content_mult by simp
also have "content [:?s:] = 1" using s by (auto simp: content_def)
finally have cg: "content g = content f" by simp
from f0
have d: "cf_pos_poly f = sdiv_poly f d"  by (auto simp: cf_pos_poly_def Let_def d_def)
let ?g = "primitive_part g"
define ng where "ng = primitive_part g"
note d
also have "sdiv_poly f d = sdiv_poly g (content g)" unfolding cg unfolding g_def d_def
by (rule poly_eqI, unfold coeff_sdiv_poly coeff_smult, insert s, auto simp: div_minus_right)
finally have fg: "cf_pos_poly f = primitive_part g" unfolding primitive_part_alt_def .
have "lead_coeff f ≠ 0" using f0 by auto
hence lg: "lead_coeff g > 0" unfolding g_def lead_coeff_smult
by (meson linorder_neqE_linordered_idom sgn_greater sgn_less zero_less_mult_iff)
hence g0: "g ≠ 0" by auto
from f0 content_primitive_part[OF this]
show ?g2 unfolding fg by auto
from g0 have "content g ≠ 0" by simp
with arg_cong[OF content_times_primitive_part[of g], of lead_coeff, unfolded lead_coeff_smult]
lg content_ge_0_int[of g] have lg': "lead_coeff ng > 0" unfolding ng_def
by (metis dual_order.antisym dual_order.strict_implies_order zero_less_mult_iff)
with f0 show ?g3 unfolding fg ng_def by auto

have d0: "d ≠ 0" using s f0 by (force simp add: d_def)
have "smult d (cf_pos_poly f) = smult ?s (smult (content f) (sdiv_poly (smult ?s f) (content f)))"
unfolding fg primitive_part_alt_def cg by (simp add: g_def d_def)
also have "sdiv_poly (smult ?s f) (content f) = smult ?s (sdiv_poly f (content f))"
using s by (metis cg g_def primitive_part_alt_def primitive_part_smult_int sgn_sgn)
finally have "smult d (cf_pos_poly f) = smult (content f) (primitive_part f)"
unfolding primitive_part_alt_def using s by auto
also have "… = f" by (rule content_times_primitive_part)
finally have df: "smult d (cf_pos_poly f) = f" .
with d0 show ?g1 by (auto simp: d_def)
from df have *: "f = cf_pos_poly f * [:d:]" by simp
from dvdI[OF this] show ?g4.
qed

(* TODO: remove *)
lemma irreducible_connect_int:
fixes p :: "int poly"
assumes ir: "irreducible⇩d p" and c: "content p = 1"
shows "irreducible p"
using c primitive_iff_content_eq_1 ir irreducible_primitive_connect by blast

lemma
fixes x :: "'a :: {idom,ring_char_0}"
shows ipoly_cf_pos_poly_eq_0[simp]: "ipoly (cf_pos_poly p) x = 0 ⟷ ipoly p x = 0"
and degree_cf_pos_poly[simp]: "degree (cf_pos_poly p) = degree p"
and cf_pos_cf_pos_poly[intro]: "p ≠ 0 ⟹ cf_pos (cf_pos_poly p)"
proof-
show "degree (cf_pos_poly p) = degree p"
by (subst(3) cf_pos_poly_main[symmetric], auto simp:sgn_eq_0_iff)
{
assume p: "p ≠ 0"
show "cf_pos (cf_pos_poly p)" using cf_pos_poly_main p by (auto simp: cf_pos_def)
have "(ipoly (cf_pos_poly p) x = 0) = (ipoly p x = 0)"
apply (subst(3) cf_pos_poly_main[symmetric]) by (auto simp: sgn_eq_0_iff hom_distribs)
}
then show "(ipoly (cf_pos_poly p) x = 0) = (ipoly p x = 0)" by (cases "p = 0", auto)
qed

lemma cf_pos_poly_eq_1: "cf_pos_poly f = 1 ⟷ degree f = 0 ∧ f ≠ 0" (is "?l ⟷ ?r")
proof(intro iffI conjI)
assume ?r
then have df0: "degree f = 0" and f0: "f ≠ 0" by auto
from  degree0_coeffs[OF df0] obtain f0 where f: "f = [:f0:]" by auto
show "cf_pos_poly f = 1" using f0 unfolding f cf_pos_poly_def Let_def sdiv_poly_def
by (auto simp: content_def mult_sgn_abs)
next
assume l: ?l
then have "degree (cf_pos_poly f) = 0" by auto
then show "degree f = 0" by simp
from l have "cf_pos_poly f ≠ 0" by auto
then show "f ≠ 0" by simp
qed

lemma irr_cf_poly_rat[simp]: "irreducible (poly_rat x)"
"lead_coeff (poly_rat x) > 0" "primitive (poly_rat x)"
proof -
obtain n d where x: "quotient_of x = (n,d)" by force
hence id: "poly_rat x = [:-n,d:]" by (auto simp: poly_rat_def)
from quotient_of_denom_pos[OF x] have d: "d > 0" by auto
show "lead_coeff (poly_rat x) > 0" "primitive (poly_rat x)"
unfolding id cf_pos_def using d quotient_of_coprime[OF x] by (auto simp: content_def)
from this[unfolded cf_pos_def]
show irr: "irreducible (poly_rat x)" unfolding id using d by (auto intro!: linear_irreducible_int)
qed

lemma poly_rat[simp]: "ipoly (poly_rat x) (of_rat x :: 'a :: field_char_0) = 0" "ipoly (poly_rat x) x = 0"
"poly_rat x ≠ 0" "ipoly (poly_rat x) y = 0 ⟷ y = (of_rat x :: 'a)"
proof -
from irr_cf_poly_rat(1)[of x] show "poly_rat x ≠ 0"
unfolding Factorial_Ring.irreducible_def by auto
obtain n d where x: "quotient_of x = (n,d)" by force
hence id: "poly_rat x = [:-n,d:]" by (auto simp: poly_rat_def)
from quotient_of_denom_pos[OF x] have d: "d ≠ 0" by auto
have "y * of_int d = of_int n ⟹ y = of_int n / of_int d" using d
by (simp add: eq_divide_imp)
with d id show "ipoly (poly_rat x) (of_rat x) = 0" "ipoly (poly_rat x) x = 0"
"ipoly (poly_rat x) y = 0 ⟷ y = (of_rat x :: 'a)"
by (auto simp: of_rat_minus of_rat_divide simp: quotient_of_div[OF x])
qed

lemma poly_rat_represents_of_rat: "(poly_rat x) represents (of_rat x)" by auto

lemma ipoly_smult_0_iff: assumes c: "c ≠ 0"
shows "(ipoly (smult c p) x = (0 :: real)) = (ipoly p x = 0)"
using c by (simp add: hom_distribs)

(* TODO *)
lemma not_irreducibleD:
assumes "¬ irreducible x" and "x ≠ 0" and "¬ x dvd 1"
shows "∃y z. x = y * z ∧ ¬ y dvd 1 ∧ ¬ z dvd 1" using assms
apply (unfold Factorial_Ring.irreducible_def) by auto

lemma cf_pos_poly_represents[simp]: "(cf_pos_poly p) represents x ⟷ p represents x"
unfolding represents_def by auto

lemma coprime_prod: (* TODO: move *)
"a ≠ 0 ⟹ c ≠ 0 ⟹ coprime (a * b) (c * d) ⟹ coprime b (d::'a::{semiring_gcd})"
by auto

lemma smult_prod: (* TODO: move or find corresponding lemma *)
"smult a b = monom a 0 * b"
by (simp add: monom_0)

lemma degree_map_poly_2:
assumes "f (lead_coeff p) ≠ 0"
shows   "degree (map_poly f p) = degree p"
proof (cases "p=0")
case False thus ?thesis
unfolding degree_eq_length_coeffs Polynomial.coeffs_map_poly
using assms by (simp add:coeffs_def)
qed auto

lemma irreducible_cf_pos_poly:
assumes irr: "irreducible p" and deg: "degree p ≠ 0"
shows "irreducible (cf_pos_poly p)" (is "irreducible ?p")
proof (unfold irreducible_altdef, intro conjI allI impI)
from irr show "?p ≠ 0" by auto
from deg have "degree ?p ≠ 0" by simp
then show "¬ ?p dvd 1" unfolding poly_dvd_1 by auto
fix b assume "b dvd cf_pos_poly p"
also note cf_pos_poly_dvd
finally have "b dvd p".
with irr[unfolded irreducible_altdef] have "p dvd b ∨ b dvd 1" by auto
then show "?p dvd b ∨ b dvd 1" by (auto dest: dvd_trans[OF cf_pos_poly_dvd])
qed

locale dvd_preserving_hom = comm_semiring_1_hom +
assumes hom_eq_mult_hom_imp: "hom x = hom y * hz ⟹ ∃z. hz = hom z ∧ x = y * z"
begin

lemma hom_dvd_hom_iff[simp]: "hom x dvd hom y ⟷ x dvd y"
proof
assume "hom x dvd hom y"
then obtain hz where "hom y = hom x * hz" by (elim dvdE)
from hom_eq_mult_hom_imp[OF this] obtain z
where "hz = hom z" and mult: "y = x * z" by auto
then show "x dvd y" by auto
qed auto

sublocale unit_preserving_hom
proof unfold_locales
fix x assume "hom x dvd 1" then have "hom x dvd hom 1" by simp
then show "x dvd 1" by (unfold hom_dvd_hom_iff)
qed

sublocale zero_hom_0
proof (unfold_locales)
fix a :: 'a
assume "hom a = 0"
then have "hom 0 dvd hom a" by auto
then have "0 dvd a" by (unfold hom_dvd_hom_iff)
then show "a = 0" by auto
qed

end

lemma smult_inverse_monom:"p ≠ 0 ⟹ smult (inverse c) (p::rat poly) = 1 ⟷ p = [: c :]"
proof (cases "c=0")
case True thus "p ≠ 0 ⟹ ?thesis" by auto
next
case False thus ?thesis by (metis left_inverse right_inverse smult_1 smult_1_left smult_smult)
qed

lemma of_int_monom:"of_int_poly p = [:rat_of_int c:] ⟷ p = [: c :]" by (induct p, auto)

lemma degree_0_content:
fixes p :: "int poly"
assumes deg: "degree p = 0" shows "content p = abs (coeff p 0)"
proof-
from deg obtain a where p: "p = [:a:]" by (auto dest: degree0_coeffs)
show ?thesis by (auto simp: p)
qed

lemma prime_elem_imp_gcd_eq:
fixes x::"'a:: ring_gcd"
shows "prime_elem x ⟹ gcd x y = normalize x ∨ gcd x y = 1"
using prime_elem_imp_coprime [of x y]
by (auto simp add: gcd_proj1_iff intro: coprime_imp_gcd_eq_1)

lemma irreducible_pos_gcd:
fixes p :: "int poly"
assumes ir: "irreducible p" and pos: "lead_coeff p > 0" shows "gcd p q ∈ {1,p}"
proof-
from pos have "[:sgn (lead_coeff p):] = 1" by auto
with prime_elem_imp_gcd_eq[of p, unfolded prime_elem_iff_irreducible, OF ir, of q]
show ?thesis by (auto simp: normalize_poly_def)
qed

lemma irreducible_pos_gcd_twice:
fixes p q :: "int poly"
assumes p: "irreducible p" "lead_coeff p > 0"
and q: "irreducible q" "lead_coeff q > 0"
shows "gcd p q = 1 ∨ p = q"
proof (cases "gcd p q = 1")
case False note pq = this
have "p = gcd p q" using irreducible_pos_gcd [OF p, of q] pq
by auto
also have "… = q" using irreducible_pos_gcd [OF q, of p] pq
by (auto simp add: ac_simps)
finally show ?thesis by auto
qed simp

interpretation of_rat_hom: field_hom_0' of_rat..

lemma poly_zero_imp_not_unit:
assumes "poly p x = 0" shows "¬ p dvd 1"
proof (rule notI)
assume "p dvd 1"
from poly_hom.hom_dvd_1[OF this] have "poly p x dvd 1" by auto
with assms show False by auto
qed

lemma poly_prod_mset_zero_iff:
fixes x :: "'a :: idom"
shows "poly (prod_mset F) x = 0 ⟷ (∃f ∈# F. poly f x = 0)"
by (induct F, auto simp: poly_mult_zero_iff)

lemma algebraic_imp_represents_irreducible:
fixes x :: "'a :: field_char_0"
assumes "algebraic x"
shows "∃p. p represents x ∧ irreducible p"
proof -
from assms obtain p
where px0: "ipoly p x = 0" and p0: "p ≠ 0" unfolding algebraic_altdef_ipoly by auto
from poly_zero_imp_not_unit[OF px0]
have "¬ p dvd 1" by (auto dest: of_int_poly_hom.hom_dvd_1[where 'a = 'a])
from mset_factors_exist[OF p0 this]
obtain F where F: "mset_factors F p" by auto
then have "p = prod_mset F" by auto
also have "(of_int_poly ... :: 'a poly) = prod_mset (image_mset of_int_poly F)" by simp
finally have "poly ... x = 0" using px0 by auto
from this[unfolded poly_prod_mset_zero_iff]
obtain f where "f ∈# F" and fx0: "ipoly f x = 0" by auto
with F have "irreducible f" by auto
with fx0 show ?thesis by auto
qed

lemma algebraic_imp_represents_irreducible_cf_pos:
assumes "algebraic (x::'a::field_char_0)"
shows "∃p. p represents x ∧ irreducible p ∧ lead_coeff p > 0 ∧ primitive p"
proof -
from algebraic_imp_represents_irreducible[OF assms(1)]
obtain p where px: "p represents x" and irr: "irreducible p" by auto
let ?p = "cf_pos_poly p"
from px irr represents_imp_degree
have 1: "?p represents x" and 2: "irreducible ?p" and 3: "cf_pos ?p"
by (auto intro: irreducible_cf_pos_poly)
then show ?thesis by (auto intro: exI[of _ ?p] simp: cf_pos_def)
qed

lemma gcd_of_int_poly: "gcd (of_int_poly f) (of_int_poly g :: 'a :: {field_char_0,field_gcd} poly) =
smult (inverse (of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
proof -
let ?ia = "of_int_poly :: _ ⇒ 'a poly"
let ?ir = "of_int_poly :: _ ⇒ rat poly"
let ?ra = "map_poly of_rat :: _ ⇒ 'a poly"
have id: "?ia x = ?ra (?ir x)" for x by (subst map_poly_map_poly, auto)
show ?thesis
unfolding id
unfolding of_rat_hom.map_poly_gcd[symmetric]
unfolding gcd_rat_to_gcd_int by (auto simp: hom_distribs)
qed

lemma algebraic_imp_represents_unique:
fixes x :: "'a :: {field_char_0,field_gcd}"
assumes "algebraic x"
shows "∃! p. p represents x ∧ irreducible p ∧ lead_coeff p > 0" (is "Ex1 ?p")
proof -
from assms obtain p
where p: "?p p" and cfp: "cf_pos p"
by (auto simp: cf_pos_def dest: algebraic_imp_represents_irreducible_cf_pos)
show ?thesis
proof (rule ex1I)
show "?p p" by fact
fix q
assume q: "?p q"
then have "q represents x" by auto
from represents_imp_degree[OF this] q irreducible_content[of q]
have cfq: "cf_pos q" by (auto simp: cf_pos_def)
show "q = p"
proof (rule ccontr)
let ?ia = "map_poly of_int :: int poly ⇒ 'a poly"
assume "q ≠ p"
with irreducible_pos_gcd_twice[of p q] p q cfp cfq have gcd: "gcd p q = 1" by auto
from p q have rt: "ipoly p x = 0" "ipoly q x = 0" unfolding represents_def by auto
define c :: 'a where "c = inverse (of_int (lead_coeff (gcd p q)))"
have rt: "poly (?ia p) x = 0" "poly (?ia q) x = 0" using rt by auto
hence "[:-x,1:] dvd ?ia p" "[:-x,1:] dvd ?ia q"
unfolding poly_eq_0_iff_dvd by auto
hence "[:-x,1:] dvd gcd (?ia p) (?ia q)" by (rule gcd_greatest)
also have "… = smult c (?ia (gcd p q))" unfolding gcd_of_int_poly c_def ..
also have "?ia (gcd p q) = 1" by (simp add: gcd)
also have "smult c 1 = [: c :]" by simp
finally show False using c_def gcd by (simp add: dvd_iff_poly_eq_0)
qed
qed
qed

lemma ipoly_poly_compose:
fixes x :: "'a :: idom"
shows "ipoly (p ∘⇩p q) x = ipoly p (ipoly q x)"
proof (induct p)
case (pCons a p)
have "ipoly ((pCons a p) ∘⇩p q) x = of_int a + ipoly (q * p ∘⇩p q) x" by (simp add: hom_distribs)
also have "ipoly (q * p ∘⇩p q) x = ipoly q x * ipoly (p ∘⇩p q) x" by (simp add: hom_distribs)
also have "ipoly (p ∘⇩p q) x = ipoly p (ipoly q x)" unfolding pCons(2) ..
also have "of_int a + ipoly q x * … = ipoly (pCons a p) (ipoly q x)"
unfolding map_poly_pCons[OF pCons(1)] by simp
finally show ?case .
qed simp

lemma algebraic_0[simp]: "algebraic 0"
unfolding algebraic_altdef_ipoly
by (intro exI[of _ "[:0,1:]"], auto)

lemma algebraic_1[simp]: "algebraic 1"
unfolding algebraic_altdef_ipoly
by (intro exI[of _ "[:-1,1:]"], auto)

text ‹Polynomial for unary minus.›

definition poly_uminus :: "'a :: ring_1 poly ⇒ 'a poly" where [code del]:
"poly_uminus p ≡ ∑i≤degree p. monom ((-1)^i * coeff p i) i"

lemma poly_uminus_pCons_pCons[simp]:
"poly_uminus (pCons a (pCons b p)) = pCons a (pCons (-b) (poly_uminus p))" (is "?l = ?r")
proof(cases "p = 0")
case False
then have deg: "degree (pCons a (pCons b p)) = Suc (Suc (degree p))" by simp
show ?thesis
by (unfold poly_uminus_def deg sum.atMost_Suc_shift monom_Suc monom_0 sum_pCons_0_commute, simp)
next
case True
then show ?thesis by (auto simp add: poly_uminus_def monom_0 monom_Suc)
qed

fun poly_uminus_inner :: "'a :: ring_1 list ⇒ 'a poly"
where "poly_uminus_inner [] = 0"
|   "poly_uminus_inner [a] = [:a:]"
|   "poly_uminus_inner (a#b#cs) = pCons a (pCons (-b) (poly_uminus_inner cs))"

lemma poly_uminus_code[code,simp]: "poly_uminus p = poly_uminus_inner (coeffs p)"
proof-
have "poly_uminus (Poly as) = poly_uminus_inner as" for as :: "'a list"
proof (induct "length as" arbitrary:as rule: less_induct)
case less
show ?case
proof(cases as)
case Nil
then show ?thesis by (simp add: poly_uminus_def)
next
case [simp]: (Cons a bs)
show ?thesis
proof (cases bs)
case Nil
then show ?thesis by (simp add: poly_uminus_def monom_0)
next
case [simp]: (Cons b cs)
show ?thesis by (simp add: less)
qed
qed
qed
from this[of "coeffs p"]
show ?thesis by simp
qed

lemma poly_uminus_inner_0[simp]: "poly_uminus_inner as = 0 ⟷ Poly as = 0"
by (induct as rule: poly_uminus_inner.induct, auto)

lemma degree_poly_uminus_inner[simp]: "degree (poly_uminus_inner as) = degree (Poly as)"
by (induct as rule: poly_uminus_inner.induct, auto)

lemma ipoly_uminus_inner[simp]:
"ipoly (poly_uminus_inner as) (x::'a::comm_ring_1) = ipoly (Poly as) (-x)"
by (induct as rule: poly_uminus_inner.induct, auto simp: hom_distribs ring_distribs)

lemma represents_uminus: assumes alg: "p represents x"
shows "(poly_uminus p) represents (-x)"
proof -
from representsD[OF alg] have "p ≠ 0" and rp: "ipoly p x = 0" by auto
hence 0: "poly_uminus p ≠ 0" by simp
show ?thesis
by (rule representsI[OF _ 0], insert rp, auto)
qed

lemma content_poly_uminus_inner[simp]:
fixes as :: "'a :: ring_gcd list"
shows "content (poly_uminus_inner as) = content (Poly as)"
by (induct as rule: poly_uminus_inner.induct, auto)

text ‹Multiplicative inverse is represented by @{const reflect_poly}.›

lemma inverse_pow_minus: assumes "x ≠ (0 :: 'a :: field)"
and "i ≤ n"
shows "inverse x ^ n * x ^ i = inverse x ^ (n - i)"
using assms by (simp add: field_class.field_divide_inverse power_diff power_inverse)

lemma (in inj_idom_hom) reflect_poly_hom:
"reflect_poly (map_poly hom p) = map_poly hom (reflect_poly p)"
proof -
obtain xs where xs: "rev (coeffs p) = xs" by auto
show ?thesis unfolding reflect_poly_def coeffs_map_poly_hom rev_map
xs by (induct xs, auto simp: hom_distribs)
qed

lemma ipoly_reflect_poly: assumes x: "(x :: 'a :: field_char_0) ≠ 0"
shows "ipoly (reflect_poly p) x = x ^ (degree p) * ipoly p (inverse x)" (is "?l = ?r")
proof -
let ?or = "of_int :: int ⇒ 'a"
have hom: "inj_idom_hom ?or" ..
show ?thesis
using poly_reflect_poly_nz[OF x, of "map_poly ?or p"] by (simp add: inj_idom_hom.reflect_poly_hom[OF hom])
qed

lemma represents_inverse: assumes x: "x ≠ 0"
and alg: "p represents x"
shows "(reflect_poly p) represents (inverse x)"
proof (intro representsI)
from representsD[OF alg] have "p ≠ 0" and rp: "ipoly p x = 0" by auto
then show "reflect_poly p ≠ 0" by (metis reflect_poly_0 reflect_poly_at_0_eq_0_iff)
show "ipoly (reflect_poly p) (inverse x) = 0" by (subst ipoly_reflect_poly, insert x, auto simp:rp)
qed

lemma inverse_roots: assumes x: "(x :: 'a :: field_char_0) ≠ 0"
shows "ipoly (reflect_poly p) x = 0 ⟷ ipoly p (inverse x) = 0"
using x by (auto simp: ipoly_reflect_poly)

context
fixes n :: nat
begin
text ‹Polynomial for n-th root.›

definition poly_nth_root :: "'a :: idom poly ⇒ 'a poly" where
"poly_nth_root p = p ∘⇩p monom 1 n"

lemma ipoly_nth_root:
fixes x :: "'a :: idom"
shows "ipoly (poly_nth_root p) x = ipoly p (x ^ n)"
unfolding poly_nth_root_def ipoly_poly_compose by (simp add: map_poly_monom poly_monom)

context
assumes n: "n ≠ 0"
begin
lemma poly_nth_root_0[simp]: "poly_nth_root p = 0 ⟷ p = 0"
unfolding poly_nth_root_def
by (rule pcompose_eq_0, insert n, auto simp: degree_monom_eq)

lemma represents_nth_root:
assumes y: "y^n = x" and alg: "p represents x"
shows "(poly_nth_root p) represents y"
proof -
from representsD[OF alg] have "p ≠ 0" and rp: "ipoly p x = 0" by auto
hence 0: "poly_nth_root p ≠ 0" by simp
show ?thesis
by (rule representsI[OF _ 0], unfold ipoly_nth_root y rp, simp)
qed

lemma represents_nth_root_odd_real:
assumes alg: "p represents x" and odd: "odd n"
shows "(poly_nth_root p) represents (root n x)"
by (rule represents_nth_root[OF odd_real_root_pow[OF odd] alg])

lemma represents_nth_root_pos_real:
assumes alg: "p represents x" and pos: "x > 0"
shows "(poly_nth_root p) represents (root n x)"
proof -
from n have id: "Suc (n - 1) = n" by auto
show ?thesis
proof (rule represents_nth_root[OF _ alg])
show "root n x ^ n = x" using id pos by auto
qed
qed

lemma represents_nth_root_neg_real:
assumes alg: "p represents x" and neg: "x < 0"
shows "(poly_uminus (poly_nth_root (poly_uminus p))) represents (root n x)"
proof -
have rt: "root n x = - root n (-x)" unfolding real_root_minus by simp
show ?thesis unfolding rt
by (rule represents_uminus[OF represents_nth_root_pos_real[OF represents_uminus[OF alg]]], insert neg, auto)
qed
end
end

lemma represents_csqrt:
assumes alg: "p represents x" shows "(poly_nth_root 2 p) represents (csqrt x)"
by (rule represents_nth_root[OF _ _ alg], auto)

lemma represents_sqrt:
assumes alg: "p represents x" and pos: "x ≥ 0"
shows "(poly_nth_root 2 p) represents (sqrt x)"
by (rule represents_nth_root[OF _ _ alg], insert pos, auto)

lemma represents_degree:
assumes "p represents x" shows "degree p ≠ 0"
proof
assume "degree p = 0"
from degree0_coeffs[OF this] obtain c where p: "p = [:c:]" by auto
from assms[unfolded represents_def p]
show False by auto
qed

text ‹Polynomial for multiplying a rational number with an algebraic number.›

definition poly_mult_rat_main where
"poly_mult_rat_main n d (f :: 'a :: idom poly) = (let fs = coeffs f; k = length fs in
poly_of_list (map (λ (fi, i). fi * d ^ i * n ^ (k - Suc i)) (zip fs [0 ..< k])))"

definition poly_mult_rat :: "rat ⇒ int poly ⇒ int poly" where
"poly_mult_rat r p ≡ case quotient_of r of (n,d) ⇒ poly_mult_rat_main n d p"

lemma coeff_poly_mult_rat_main: "coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i"
proof -
have id: "coeff (poly_mult_rat_main n d f) i = (coeff f i * d ^ i) * n ^ (length (coeffs f) - Suc i)"
unfolding poly_mult_rat_main_def Let_def poly_of_list_def coeff_Poly
unfolding nth_default_coeffs_eq[symmetric]
unfolding nth_default_def by auto
show ?thesis unfolding id by (simp add: degree_eq_length_coeffs)
qed

lemma degree_poly_mult_rat_main: "n ≠ 0 ⟹ degree (poly_mult_rat_main n d f) = (if d = 0 then 0 else degree f)"
proof (cases "d = 0")
case True
thus ?thesis unfolding degree_def unfolding coeff_poly_mult_rat_main by simp
next
case False
hence id: "(d = 0) = False" by simp
show "n ≠ 0 ⟹ ?thesis" unfolding degree_def coeff_poly_mult_rat_main id
by (simp add: id)
qed

lemma ipoly_mult_rat_main:
fixes x :: "'a :: {field,ring_char_0}"
assumes "d ≠ 0" and "n ≠ 0"
shows "ipoly (poly_mult_rat_main n d p) x = of_int n ^ degree p * ipoly p (x * of_int d / of_int n)"
proof -
from assms have d: "(if d = 0 then t else f) = f" for t f :: 'b by simp
show ?thesis
unfolding poly_altdef of_int_hom.coeff_map_poly_hom mult.assoc[symmetric] of_int_mult[symmetric]
sum_distrib_left
unfolding of_int_hom.degree_map_poly_hom degree_poly_mult_rat_main[OF assms(2)] d
proof (rule sum.cong[OF refl])
fix i
assume "i ∈ {..degree p}"
hence i: "i ≤ degree p" by auto
hence id: "of_int n ^ (degree p - i) = (of_int n ^ degree p / of_int n ^ i :: 'a)"
by (simp add: assms(2) power_diff)
thus "of_int (coeff (poly_mult_rat_main n d p) i) * x ^ i = of_int n ^ degree p * of_int (coeff p i) * (x * of_int d / of_int n) ^ i"
unfolding coeff_poly_mult_rat_main
by (simp add: field_simps)
qed
qed

lemma degree_poly_mult_rat[simp]: assumes "r ≠ 0" shows "degree (poly_mult_rat r p) = degree p"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" by auto
with assms r have n0: "n ≠ 0" by simp
from quot have id: "poly_mult_rat r p = poly_mult_rat_main n d p"  unfolding poly_mult_rat_def by simp
show ?thesis unfolding id degree_poly_mult_rat_main[OF n0] using d by simp
qed

lemma ipoly_mult_rat:
assumes r0: "r ≠ 0"
shows "ipoly (poly_mult_rat r p) x = of_int (fst (quotient_of r)) ^ degree p * ipoly p (x * inverse (of_rat r))"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" by auto
from r r0 have n: "n ≠ 0" by simp
from r d n have inv: "of_int d / of_int n = inverse r" by simp
from quot have id: "poly_mult_rat r p = poly_mult_rat_main n d p"  unfolding poly_mult_rat_def by simp
show ?thesis unfolding id ipoly_mult_rat_main[OF d n] quot fst_conv of_rat_inverse[symmetric] inv[symmetric]
by (simp add: of_rat_divide)
qed

lemma poly_mult_rat_main_0[simp]:
assumes "n ≠ 0" "d ≠ 0" shows "poly_mult_rat_main n d p = 0 ⟷ p = 0"
proof
assume "p = 0" thus "poly_mult_rat_main n d p = 0"
by (simp add: poly_mult_rat_main_def)
next
assume 0: "poly_mult_rat_main n d p = 0"
{
fix i
from 0 have "coeff (poly_mult_rat_main n d p) i = 0" by simp
hence "coeff p i = 0" unfolding coeff_poly_mult_rat_main using assms by simp
}
thus "p = 0" by (intro poly_eqI, auto)
qed

lemma poly_mult_rat_0[simp]: assumes r0: "r ≠ 0" shows "poly_mult_rat r p = 0 ⟷ p = 0"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" by auto
from r r0 have n: "n ≠ 0" by simp
from quot have id: "poly_mult_rat r p = poly_mult_rat_main n d p"  unfolding poly_mult_rat_def by simp
show ?thesis unfolding id using n d by simp
qed

lemma represents_mult_rat:
assumes r: "r ≠ 0" and "p represents x" shows "(poly_mult_rat r p) represents (of_rat r * x)"
using assms
unfolding represents_def ipoly_mult_rat[OF r] by (simp add: field_simps)

text ‹Polynomial for adding a rational number on an algebraic number.
Again, we do not have to factor afterwards.›

definition poly_add_rat :: "rat ⇒ int poly ⇒ int poly" where
"poly_add_rat r p ≡ case quotient_of r of (n,d) ⇒
(poly_mult_rat_main d 1 p ∘⇩p [:-n,d:])"

lemma poly_add_rat_code[code]: "poly_add_rat r p ≡ case quotient_of r of (n,d) ⇒
let p' = (let fs = coeffs p; k = length fs in poly_of_list (map (λ(fi, i). fi * d ^ (k - Suc i)) (zip fs [0..<k])));
p'' = p' ∘⇩p [:-n,d:]
in p''"
unfolding poly_add_rat_def poly_mult_rat_main_def Let_def by simp

lemma degree_poly_add_rat[simp]: "degree (poly_add_rat r p) = degree p"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" "d > 0" by auto
show ?thesis unfolding poly_add_rat_def quot split
by (simp add: degree_poly_mult_rat_main d)
qed

lemma ipoly_add_rat: "ipoly (poly_add_rat r p) x = (of_int (snd (quotient_of r)) ^ degree p) * ipoly p (x - of_rat r)"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" "d > 0" by auto
have id: "ipoly [:- n, 1:] (x / of_int d :: 'a) = - of_int n + x / of_int d" by simp
show ?thesis unfolding poly_add_rat_def quot split
by (simp add: ipoly_mult_rat_main ipoly_poly_compose d r degree_poly_mult_rat_main field_simps id of_rat_divide)
qed

lemma poly_add_rat_0[simp]: "poly_add_rat r p = 0 ⟷ p = 0"
proof -
obtain n d where quot: "quotient_of r = (n,d)" by force
from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
from quotient_of_denom_pos[OF quot] have d: "d ≠ 0" "d > 0" by auto
show ?thesis unfolding poly_add_rat_def quot split
by (simp add: d pcompose_eq_0)
qed

lemma add_rat_roots: "ipoly (poly_add_rat r p) x = 0 ⟷ ipoly p (x - of_rat r) = 0"
unfolding ipoly_add_rat using quotient_of_nonzero by auto

assumes "p represents x" shows "(poly_add_rat r p) represents (of_rat r + x)"
using assms unfolding represents_def ipoly_add_rat by simp

(* TODO: move? *)
lemmas pos_mult[simplified,simp] = mult_less_cancel_left_pos[of _ 0] mult_less_cancel_left_pos[of _ _ 0]

"ipoly (poly_add_rat r p) (x::'a::linordered_field) < 0 ⟷ ipoly p (x - of_rat r) < 0"
"ipoly (poly_add_rat r p) (x::'a::linordered_field) > 0 ⟷ ipoly p (x - of_rat r) > 0"
using quotient_of_nonzero unfolding ipoly_add_rat by auto

"sgn (ipoly (poly_add_rat r p) (x::'a::linordered_field)) = sgn (ipoly p (x - of_rat r))" (is "sgn ?l = sgn ?r")
using ipoly_add_rat_pos_neg[of r p x]
by (cases ?r "0::'a" rule: linorder_cases,auto simp:  sgn_1_pos sgn_1_neg sgn_eq_0_iff)

lemma deg_nonzero_represents:
assumes deg: "degree p ≠ 0" shows "∃ x :: complex. p represents x"
proof -
let ?p = "of_int_poly p :: complex poly"
from fundamental_theorem_algebra_factorized[of ?p]
obtain as c where id: "smult c (∏a←as. [:- a, 1:]) = ?p"
and len: "length as = degree ?p" by blast
have "degree ?p = degree p" by simp
with deg len obtain b bs where as: "as = b # bs" by (cases as, auto)
have "p represents b" unfolding represents_def id[symmetric] as using deg by auto
thus ?thesis by blast
qed

end
```