Theory Cubic_Quartic

theory Cubic_Quartic
imports Complex_Main
```(*  Title:      HOL/ex/Cubic_Quartic.thy
Author:     Amine Chaieb
*)

section ‹The Cubic and Quartic Root Formulas›

theory Cubic_Quartic
imports Complex_Main
begin

section ‹The Cubic Formula›

definition "ccbrt z = (SOME (w::complex). w^3 = z)"

lemma ccbrt: "(ccbrt z) ^ 3 = z"
proof -
from rcis_Ex obtain r a where ra: "z = rcis r a"
by blast
let ?r' = "if r < 0 then - root 3 (-r) else root 3 r"
let ?a' = "a/3"
have "rcis ?r' ?a' ^ 3 = rcis r a"
by (cases "r < 0") (simp_all add: DeMoivre2)
then have *: "∃w. w^3 = z"
unfolding ra by blast
from someI_ex [OF *] show ?thesis
unfolding ccbrt_def by blast
qed

text ‹The reduction to a simpler form:›

lemma cubic_reduction:
fixes a :: complex
assumes
"a ≠ 0 ∧ x = y - b / (3 * a) ∧  p = (3* a * c - b^2) / (9 * a^2) ∧
q = (9 * a * b * c - 2 * b^3 - 27 * a^2 * d) / (54 * a^3)"
shows "a * x^3 + b * x^2 + c * x + d = 0 ⟷ y^3 + 3 * p * y - 2 * q = 0"
proof -
from assms have "3 * a ≠ 0" "9 * a^2 ≠ 0" "54 * a^3 ≠ 0" by auto
then have *:
"x = y - b / (3 * a) ⟷ (3*a) * x = (3*a) * y - b"
"p = (3* a * c - b^2) / (9 * a^2) ⟷ (9 * a^2) * p = (3* a * c - b^2)"
"q = (9 * a * b * c - 2 * b^3 - 27 * a^2 * d) / (54 * a^3) ⟷
(54 * a^3) * q = (9 * a * b * c - 2 * b^3 - 27 * a^2 * d)"
from assms [unfolded *] show ?thesis
by algebra
qed

text ‹The solutions of the special form:›

lemma cubic_basic:
fixes s :: complex
assumes
"s^2 = q^2 + p^3 ∧
s1^3 = (if p = 0 then 2 * q else q + s) ∧
s2 = -s1 * (1 + i * t) / 2 ∧
s3 = -s1 * (1 - i * t) / 2 ∧
i^2 + 1 = 0 ∧
t^2 = 3"
shows
"if p = 0
then y^3 + 3 * p * y - 2 * q = 0 ⟷ y = s1 ∨ y = s2 ∨ y = s3
else s1 ≠ 0 ∧
(y^3 + 3 * p * y - 2 * q = 0 ⟷ (y = s1 - p / s1 ∨ y = s2 - p / s2 ∨ y = s3 - p / s3))"
proof (cases "p = 0")
case True
with assms show ?thesis
next
case False
with assms have *: "s1 ≠ 0" by (simp add: field_simps) algebra
with assms False have "s2 ≠ 0" "s3 ≠ 0"
with * have **:
"y = s1 - p / s1 ⟷ s1 * y = s1^2 - p"
"y = s2 - p / s2 ⟷ s2 * y = s2^2 - p"
"y = s3 - p / s3 ⟷ s3 * y = s3^2 - p"
from assms False show ?thesis
unfolding ** by (simp add: field_simps) algebra
qed

text ‹Explicit formula for the roots:›

lemma cubic:
assumes a0: "a ≠ 0"
shows
"let
p = (3 * a * c - b^2) / (9 * a^2);
q = (9 * a * b * c - 2 * b^3 - 27 * a^2 * d) / (54 * a^3);
s = csqrt(q^2 + p^3);
s1 = (if p = 0 then ccbrt(2 * q) else ccbrt(q + s));
s2 = -s1 * (1 + 𝗂 * csqrt 3) / 2;
s3 = -s1 * (1 - 𝗂 * csqrt 3) / 2
in
if p = 0 then
a * x^3 + b * x^2 + c * x + d = 0 ⟷
x = s1 - b / (3 * a) ∨
x = s2 - b / (3 * a) ∨
x = s3 - b / (3 * a)
else
s1 ≠ 0 ∧
(a * x^3 + b * x^2 + c * x + d = 0 ⟷
x = s1 - p / s1 - b / (3 * a) ∨
x = s2 - p / s2 - b / (3 * a) ∨
x = s3 - p / s3 - b / (3 * a))"
proof -
let ?p = "(3 * a * c - b^2) / (9 * a^2)"
let ?q = "(9 * a * b * c - 2 * b^3 - 27 * a^2 * d) / (54 * a^3)"
let ?s = "csqrt (?q^2 + ?p^3)"
let ?s1 = "if ?p = 0 then ccbrt(2 * ?q) else ccbrt(?q + ?s)"
let ?s2 = "- ?s1 * (1 + 𝗂 * csqrt 3) / 2"
let ?s3 = "- ?s1 * (1 - 𝗂 * csqrt 3) / 2"
let ?y = "x + b / (3 * a)"
from a0 have zero: "9 * a^2 ≠ 0" "a^3 * 54 ≠ 0" "(a * 3) ≠ 0"
by auto
have eq: "a * x^3 + b * x^2 + c * x + d = 0 ⟷ ?y^3 + 3 * ?p * ?y - 2 * ?q = 0"
by (rule cubic_reduction) (auto simp add: field_simps zero a0)
have "csqrt 3^2 = 3" by (rule power2_csqrt)
then have th0:
"?s^2 = ?q^2 + ?p ^ 3 ∧ ?s1^ 3 = (if ?p = 0 then 2 * ?q else ?q + ?s) ∧
?s2 = - ?s1 * (1 + 𝗂 * csqrt 3) / 2 ∧
?s3 = - ?s1 * (1 - 𝗂 * csqrt 3) / 2 ∧
𝗂^2 + 1 = 0 ∧ csqrt 3^2 = 3"
using zero by (simp add: field_simps ccbrt)
from cubic_basic[OF th0, of ?y]
show ?thesis
apply (simp only: Let_def eq)
using zero apply (simp add: field_simps ccbrt)
using zero
apply (cases "a * (c * 3) = b^2")
done
qed

section ‹The Quartic Formula›

lemma quartic:
"(y::real)^3 - b * y^2 + (a * c - 4 * d) * y - a^2 * d + 4 * b * d - c^2 = 0 ∧
R^2 = a^2 / 4 - b + y ∧
s^2 = y^2 - 4 * d ∧
(D^2 = (if R = 0 then 3 * a^2 / 4 - 2 * b + 2 * s
else 3 * a^2 / 4 - R^2 - 2 * b + (4 * a * b - 8 * c - a^3) / (4 * R))) ∧
(E^2 = (if R = 0 then 3 * a^2 / 4 - 2 * b - 2 * s
else 3 * a^2 / 4 - R^2 - 2 * b - (4 * a * b - 8 * c - a^3) / (4 * R)))
⟹ x^4 + a * x^3 + b * x^2 + c * x + d = 0 ⟷
x = -a / 4 + R / 2 + D / 2 ∨
x = -a / 4 + R / 2 - D / 2 ∨
x = -a / 4 - R / 2 + E / 2 ∨
x = -a / 4 - R / 2 - E / 2"
apply (cases "R = 0")
apply (simp_all add: field_simps divide_minus_left[symmetric] del: divide_minus_left)
apply algebra
apply algebra
done

end
```