formal/lean/mathlib/algebra/cubic_discriminant.lean

/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/

import field_theory.splitting_field

/-!
# Cubics and discriminants

This file defines cubic polynomials over a semiring and their discriminants over a splitting field.

## Main definitions

* `cubic`: the structure representing a cubic polynomial.
* `disc`: the discriminant of a cubic polynomial.

## Main statements

* `disc_ne_zero_iff_roots_nodup`: the cubic discriminant is not equal to zero if and only if
  the cubic has no duplicate roots.

## References

* https://en.wikipedia.org/wiki/Cubic_equation
* https://en.wikipedia.org/wiki/Discriminant

## Tags

cubic, discriminant, polynomial, root
-/

noncomputable theory

/-- The structure representing a cubic polynomial. -/
@[ext] structure cubic (R : Type*) := (a b c d : R)

namespace cubic

open cubic polynomial
open_locale polynomial

variables {R S F K : Type*}

instance [inhabited R] : inhabited (cubic R) := ⟨⟨default, default, default, default⟩⟩

instance [has_zero R] : has_zero (cubic R) := ⟨⟨0, 0, 0, 0⟩⟩

section basic

variables {P : cubic R} [semiring R]

/-- Convert a cubic polynomial to a polynomial. -/
def to_poly (P : cubic R) : R[X] := C P.a * X ^ 3 + C P.b * X ^ 2 + C P.c * X + C P.d

/-! ### Coefficients -/

section coeff

private lemma coeffs :
  (∀ n > 3, P.to_poly.coeff n = 0) ∧ P.to_poly.coeff 3 = P.a ∧ P.to_poly.coeff 2 = P.b
    ∧ P.to_poly.coeff 1 = P.c ∧ P.to_poly.coeff 0 = P.d :=
begin
  simp only [to_poly, coeff_add, coeff_C, coeff_C_mul_X, coeff_C_mul_X_pow],
  norm_num,
  intros n hn,
  repeat { rw [if_neg] },
  any_goals { linarith only [hn] },
  repeat { rw [zero_add] }
end

@[simp] lemma coeff_gt_three (n : ℕ) (hn : 3 < n) : P.to_poly.coeff n = 0 := coeffs.1 n hn

@[simp] lemma coeff_three : P.to_poly.coeff 3 = P.a := coeffs.2.1

@[simp] lemma coeff_two : P.to_poly.coeff 2 = P.b := coeffs.2.2.1

@[simp] lemma coeff_one : P.to_poly.coeff 1 = P.c := coeffs.2.2.2.1

@[simp] lemma coeff_zero : P.to_poly.coeff 0 = P.d := coeffs.2.2.2.2

lemma a_of_eq {Q : cubic R} (h : P.to_poly = Q.to_poly) : P.a = Q.a :=
by rw [← coeff_three, h, coeff_three]

lemma b_of_eq {Q : cubic R} (h : P.to_poly = Q.to_poly) : P.b = Q.b :=
by rw [← coeff_two, h, coeff_two]

lemma c_of_eq {Q : cubic R} (h : P.to_poly = Q.to_poly) : P.c = Q.c :=
by rw [← coeff_one, h, coeff_one]

lemma d_of_eq {Q : cubic R} (h : P.to_poly = Q.to_poly) : P.d = Q.d :=
by rw [← coeff_zero, h, coeff_zero]

@[simp] lemma to_poly_injective (P Q : cubic R) : P.to_poly = Q.to_poly ↔ P = Q :=
⟨λ h, cubic.ext _ _ (a_of_eq h) (b_of_eq h) (c_of_eq h) (d_of_eq h), congr_arg _⟩

@[simp] lemma of_a_eq_zero (ha : P.a = 0) : P.to_poly = C P.b * X ^ 2 + C P.c * X + C P.d :=
by rw [to_poly, C_eq_zero.mpr ha, zero_mul, zero_add]

@[simp] lemma of_a_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.to_poly = C P.c * X + C P.d :=
by rw [of_a_eq_zero ha, C_eq_zero.mpr hb, zero_mul, zero_add]

@[simp] lemma of_a_b_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.to_poly = C P.d :=
by rw [of_a_b_eq_zero ha hb, C_eq_zero.mpr hc, zero_mul, zero_add]

@[simp] lemma of_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) : P.to_poly = 0 :=
by rw [of_a_b_c_eq_zero ha hb hc, C_eq_zero.mpr hd]

@[simp] lemma zero : (0 : cubic R).to_poly = 0 := of_zero rfl rfl rfl rfl

@[simp] lemma eq_zero_iff : P.to_poly = 0 ↔ P = 0 := by rw [← zero, to_poly_injective]

lemma ne_zero (h0 : ¬P.a = 0 ∨ ¬P.b = 0 ∨ ¬P.c = 0 ∨ ¬P.d = 0) : P.to_poly ≠ 0 :=
by { contrapose! h0, rw [eq_zero_iff.mp h0], exact ⟨rfl, rfl, rfl, rfl⟩ }

lemma ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.to_poly ≠ 0 := (or_imp_distrib.mp ne_zero).1 ha

lemma ne_zero_of_b_ne_zero (hb : P.b ≠ 0) : P.to_poly ≠ 0 :=
(or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).1 hb

lemma ne_zero_of_c_ne_zero (hc : P.c ≠ 0) : P.to_poly ≠ 0 :=
(or_imp_distrib.mp (or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).2).1 hc

lemma ne_zero_of_d_ne_zero (hd : P.d ≠ 0) : P.to_poly ≠ 0 :=
(or_imp_distrib.mp (or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).2).2 hd

end coeff

/-! ### Degrees -/

section degree

/-- The equivalence between cubic polynomials and polynomials of degree at most three. -/
@[simps] def equiv : cubic R ≃ {p : R[X] // p.degree ≤ 3} :=
{ to_fun    := λ P, ⟨P.to_poly, degree_cubic_le⟩,
  inv_fun   := λ f, ⟨coeff f 3, coeff f 2, coeff f 1, coeff f 0⟩,
  left_inv  := λ P, by ext; simp only [subtype.coe_mk, coeffs],
  right_inv := λ f,
  begin
    ext (_ | _ | _ | _ | n); simp only [subtype.coe_mk, coeffs],
    have h3 : 3 < n + 4 := by linarith only,
    rw [coeff_gt_three _ h3,
        (degree_le_iff_coeff_zero (f : R[X]) 3).mp f.2 _ $ with_bot.coe_lt_coe.mpr h3]
  end }

lemma degree (ha : P.a ≠ 0) : P.to_poly.degree = 3 := degree_cubic ha

lemma degree_of_a_eq_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.to_poly.degree = 2 :=
by rw [of_a_eq_zero ha, degree_quadratic hb]

lemma degree_of_a_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) : P.to_poly.degree = 1 :=
by rw [of_a_b_eq_zero ha hb, degree_linear hc]

lemma degree_of_a_b_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d ≠ 0) :
  P.to_poly.degree = 0 :=
by rw [of_a_b_c_eq_zero ha hb hc, degree_C hd]

lemma degree_of_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) :
  P.to_poly.degree = ⊥ :=
by rw [of_zero ha hb hc hd, degree_zero]

lemma leading_coeff (ha : P.a ≠ 0) : P.to_poly.leading_coeff = P.a := leading_coeff_cubic ha

lemma leading_coeff_of_a_eq_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.to_poly.leading_coeff = P.b :=
by rw [of_a_eq_zero ha, leading_coeff_quadratic hb]

lemma leading_coeff_of_a_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
  P.to_poly.leading_coeff = P.c :=
by rw [of_a_b_eq_zero ha hb, leading_coeff_linear hc]

lemma leading_coeff_of_a_b_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
  P.to_poly.leading_coeff = P.d :=
by rw [of_a_b_c_eq_zero ha hb hc, leading_coeff_C]

end degree

/-! ### Map across a homomorphism -/

section map

variables [semiring S] {φ : R →+* S}

/-- Map a cubic polynomial across a semiring homomorphism. -/
def map (φ : R →+* S) (P : cubic R) : cubic S := ⟨φ P.a, φ P.b, φ P.c, φ P.d⟩

lemma map_to_poly : (map φ P).to_poly = polynomial.map φ P.to_poly :=
by simp only [map, to_poly, map_C, map_X, polynomial.map_add, polynomial.map_mul,
              polynomial.map_pow]

end map

end basic

section roots

open multiset

/-! ### Roots over an extension -/

section extension

variables {P : cubic R} [comm_ring R] [comm_ring S] {φ : R →+* S}

/-- The roots of a cubic polynomial. -/
def roots [is_domain R] (P : cubic R) : multiset R := P.to_poly.roots

lemma map_roots [is_domain S] : (map φ P).roots = (polynomial.map φ P.to_poly).roots :=
by rw [roots, map_to_poly]

theorem mem_roots_iff [is_domain R] (h0 : P.to_poly ≠ 0) (x : R) :
  x ∈ P.roots ↔ P.a * x ^ 3 + P.b * x ^ 2 + P.c * x + P.d = 0 :=
begin
  rw [roots, mem_roots h0, is_root, to_poly],
  simp only [eval_C, eval_X, eval_add, eval_mul, eval_pow]
end

theorem card_roots_le [is_domain R] [decidable_eq R] : P.roots.to_finset.card ≤ 3 :=
begin
  apply (to_finset_card_le P.to_poly.roots).trans,
  by_cases hP : P.to_poly = 0,
  { exact (card_roots' P.to_poly).trans (by { rw [hP, nat_degree_zero], exact zero_le 3 }) },
  { exact with_bot.coe_le_coe.1 ((card_roots hP).trans degree_cubic_le) }
end

end extension

variables {P : cubic F} [field F] [field K] {φ : F →+* K} {x y z : K}

/-! ### Roots over a splitting field -/

section split

theorem splits_iff_card_roots (ha : P.a ≠ 0) : splits φ P.to_poly ↔ (map φ P).roots.card = 3 :=
begin
  replace ha : (map φ P).a ≠ 0 := (ring_hom.map_ne_zero φ).mpr ha,
  nth_rewrite_lhs 0 [← ring_hom.id_comp φ],
  rw [roots, ← splits_map_iff, ← map_to_poly, splits_iff_card_roots,
      ← ((degree_eq_iff_nat_degree_eq $ ne_zero_of_a_ne_zero ha).mp $ degree ha : _ = 3)]
end

theorem splits_iff_roots_eq_three (ha : P.a ≠ 0) :
  splits φ P.to_poly ↔ ∃ x y z : K, (map φ P).roots = {x, y, z} :=
by rw [splits_iff_card_roots ha, card_eq_three]

theorem eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  (map φ P).to_poly = C (φ P.a) * (X - C x) * (X - C y) * (X - C z) :=
begin
  rw [map_to_poly, eq_prod_roots_of_splits $ (splits_iff_roots_eq_three ha).mpr $ exists.intro x $
        exists.intro y $ exists.intro z h3, leading_coeff ha, ← map_roots, h3],
  change C (φ P.a) * ((X - C x) ::ₘ (X - C y) ::ₘ {X - C z}).prod = _,
  rw [prod_cons, prod_cons, prod_singleton, mul_assoc, mul_assoc]
end

theorem eq_sum_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  map φ P = ⟨φ P.a, φ P.a * -(x + y + z), φ P.a * (x * y + x * z + y * z), φ P.a * -(x * y * z)⟩ :=
begin
  apply_fun to_poly,
  any_goals { exact λ P Q, (to_poly_injective P Q).mp },
  rw [eq_prod_three_roots ha h3, to_poly],
  simp only [C_neg, C_add, C_mul],
  ring1
end

theorem b_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  φ P.b = φ P.a * -(x + y + z) :=
by injection eq_sum_three_roots ha h3

theorem c_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  φ P.c = φ P.a * (x * y + x * z + y * z) :=
by injection eq_sum_three_roots ha h3

theorem d_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  φ P.d = φ P.a * -(x * y * z) :=
by injection eq_sum_three_roots ha h3

end split

/-! ### Discriminant over a splitting field -/

section discriminant

/-- The discriminant of a cubic polynomial. -/
def disc {R : Type*} [ring R] (P : cubic R) : R :=
P.b ^ 2 * P.c ^ 2 - 4 * P.a * P.c ^ 3 - 4 * P.b ^ 3 * P.d - 27 * P.a ^ 2 * P.d ^ 2
  + 18 * P.a * P.b * P.c * P.d

theorem disc_eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  φ P.disc = (φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 :=
begin
  simp only [disc, ring_hom.map_add, ring_hom.map_sub, ring_hom.map_mul, map_pow],
  simp only [ring_hom.map_one, map_bit0, map_bit1],
  rw [b_eq_three_roots ha h3, c_eq_three_roots ha h3, d_eq_three_roots ha h3],
  ring1
end

theorem disc_ne_zero_iff_roots_ne (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  P.disc ≠ 0 ↔ x ≠ y ∧ x ≠ z ∧ y ≠ z :=
begin
  rw [← ring_hom.map_ne_zero φ, disc_eq_prod_three_roots ha h3, pow_two],
  simp only [mul_ne_zero_iff, sub_ne_zero],
  rw [ring_hom.map_ne_zero],
  tautology
end

theorem disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
  P.disc ≠ 0 ↔ (map φ P).roots.nodup :=
begin
  rw [disc_ne_zero_iff_roots_ne ha h3, h3],
  change _ ↔ (x ::ₘ y ::ₘ {z}).nodup,
  rw [nodup_cons, nodup_cons, mem_cons, mem_singleton, mem_singleton],
  simp only [nodup_singleton],
  tautology
end

theorem card_roots_of_disc_ne_zero [decidable_eq K] (ha : P.a ≠ 0)
  (h3 : (map φ P).roots = {x, y, z}) (hd : P.disc ≠ 0) : (map φ P).roots.to_finset.card = 3 :=
begin
  rw [to_finset_card_of_nodup $ (disc_ne_zero_iff_roots_nodup ha h3).mp hd,
      ← splits_iff_card_roots ha, splits_iff_roots_eq_three ha],
  exact ⟨x, ⟨y, ⟨z, h3⟩⟩⟩
end

end discriminant

end roots

end cubic