Theory Quadratic_Discriminant

(*  Title:       HOL/Library/Quadratic_Discriminant.thy
    Author:      Tim Makarios <tjm1983 at gmail.com>, 2012

Originally from the AFP entry Tarskis_Geometry
*)

section "Roots of real quadratics"

theory Quadratic_Discriminant
imports Complex_Main
begin

definition discrim :: "real ⇒ real ⇒ real ⇒ real"
  where "discrim a b c ≡ b⇧2 - 4 * a * c"

lemma complete_square:
  "a ≠ 0 ⟹ a * x⇧2 + b * x + c = 0 ⟷ (2 * a * x + b)⇧2 = discrim a b c"
by (simp add: discrim_def) algebra

lemma discriminant_negative:
  fixes a b c x :: real
  assumes "a ≠ 0"
    and "discrim a b c < 0"
  shows "a * x⇧2 + b * x + c ≠ 0"
proof -
  have "(2 * a * x + b)⇧2 ≥ 0"
    by simp
  with ‹discrim a b c < 0› have "(2 * a * x + b)⇧2 ≠ discrim a b c"
    by arith
  with complete_square and ‹a ≠ 0› show "a * x⇧2 + b * x + c ≠ 0"
    by simp
qed

lemma plus_or_minus_sqrt:
  fixes x y :: real
  assumes "y ≥ 0"
  shows "x⇧2 = y ⟷ x = sqrt y ∨ x = - sqrt y"
proof
  assume "x⇧2 = y"
  then have "sqrt (x⇧2) = sqrt y"
    by simp
  then have "sqrt y = ¦x¦"
    by simp
  then show "x = sqrt y ∨ x = - sqrt y"
    by auto
next
  assume "x = sqrt y ∨ x = - sqrt y"
  then have "x⇧2 = (sqrt y)⇧2 ∨ x⇧2 = (- sqrt y)⇧2"
    by auto
  with ‹y ≥ 0› show "x⇧2 = y"
    by simp
qed

lemma divide_non_zero:
  fixes x y z :: real
  assumes "x ≠ 0"
  shows "x * y = z ⟷ y = z / x"
proof
  show "y = z / x" if "x * y = z"
    using ‹x ≠ 0› that by (simp add: field_simps)
  show "x * y = z" if "y = z / x"
    using ‹x ≠ 0› that by simp
qed

lemma discriminant_nonneg:
  fixes a b c x :: real
  assumes "a ≠ 0"
    and "discrim a b c ≥ 0"
  shows "a * x⇧2 + b * x + c = 0 ⟷
    x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
    x = (-b - sqrt (discrim a b c)) / (2 * a)"
proof -
  from complete_square and plus_or_minus_sqrt and assms
  have "a * x⇧2 + b * x + c = 0 ⟷
    (2 * a) * x + b = sqrt (discrim a b c) ∨
    (2 * a) * x + b = - sqrt (discrim a b c)"
    by simp
  also have "… ⟷ (2 * a) * x = (-b + sqrt (discrim a b c)) ∨
    (2 * a) * x = (-b - sqrt (discrim a b c))"
    by auto
  also from ‹a ≠ 0› and divide_non_zero [of "2 * a" x]
  have "… ⟷ x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
    x = (-b - sqrt (discrim a b c)) / (2 * a)"
    by simp
  finally show "a * x⇧2 + b * x + c = 0 ⟷
    x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
    x = (-b - sqrt (discrim a b c)) / (2 * a)" .
qed

lemma discriminant_zero:
  fixes a b c x :: real
  assumes "a ≠ 0"
    and "discrim a b c = 0"
  shows "a * x⇧2 + b * x + c = 0 ⟷ x = -b / (2 * a)"
  by (simp add: discriminant_nonneg assms)

theorem discriminant_iff:
  fixes a b c x :: real
  assumes "a ≠ 0"
  shows "a * x⇧2 + b * x + c = 0 ⟷
    discrim a b c ≥ 0 ∧
    (x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
     x = (-b - sqrt (discrim a b c)) / (2 * a))"
proof
  assume "a * x⇧2 + b * x + c = 0"
  with discriminant_negative and ‹a ≠ 0› have "¬(discrim a b c < 0)"
    by auto
  then have "discrim a b c ≥ 0"
    by simp
  with discriminant_nonneg and ‹a * x⇧2 + b * x + c = 0› and ‹a ≠ 0›
  have "x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
      x = (-b - sqrt (discrim a b c)) / (2 * a)"
    by simp
  with ‹discrim a b c ≥ 0›
  show "discrim a b c ≥ 0 ∧
    (x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
     x = (-b - sqrt (discrim a b c)) / (2 * a))" ..
next
  assume "discrim a b c ≥ 0 ∧
    (x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
     x = (-b - sqrt (discrim a b c)) / (2 * a))"
  then have "discrim a b c ≥ 0" and
    "x = (-b + sqrt (discrim a b c)) / (2 * a) ∨
     x = (-b - sqrt (discrim a b c)) / (2 * a)"
    by simp_all
  with discriminant_nonneg and ‹a ≠ 0› show "a * x⇧2 + b * x + c = 0"
    by simp
qed

lemma discriminant_nonneg_ex:
  fixes a b c :: real
  assumes "a ≠ 0"
    and "discrim a b c ≥ 0"
  shows "∃ x. a * x⇧2 + b * x + c = 0"
  by (auto simp: discriminant_nonneg assms)

lemma discriminant_pos_ex:
  fixes a b c :: real
  assumes "a ≠ 0"
    and "discrim a b c > 0"
  shows "∃x y. x ≠ y ∧ a * x⇧2 + b * x + c = 0 ∧ a * y⇧2 + b * y + c = 0"
proof -
  let ?x = "(-b + sqrt (discrim a b c)) / (2 * a)"
  let ?y = "(-b - sqrt (discrim a b c)) / (2 * a)"
  from ‹discrim a b c > 0› have "sqrt (discrim a b c) ≠ 0"
    by simp
  then have "sqrt (discrim a b c) ≠ - sqrt (discrim a b c)"
    by arith
  with ‹a ≠ 0› have "?x ≠ ?y"
    by simp
  moreover from assms have "a * ?x⇧2 + b * ?x + c = 0" and "a * ?y⇧2 + b * ?y + c = 0"
    using discriminant_nonneg [of a b c ?x]
      and discriminant_nonneg [of a b c ?y]
    by simp_all
  ultimately show ?thesis
    by blast
qed

lemma discriminant_pos_distinct:
  fixes a b c x :: real
  assumes "a ≠ 0"
    and "discrim a b c > 0"
  shows "∃ y. x ≠ y ∧ a * y⇧2 + b * y + c = 0"
proof -
  from discriminant_pos_ex and ‹a ≠ 0› and ‹discrim a b c > 0›
  obtain w and z where "w ≠ z"
    and "a * w⇧2 + b * w + c = 0" and "a * z⇧2 + b * z + c = 0"
    by blast
  show "∃y. x ≠ y ∧ a * y⇧2 + b * y + c = 0"
  proof (cases "x = w")
    case True
    with ‹w ≠ z› have "x ≠ z"
      by simp
    with ‹a * z⇧2 + b * z + c = 0› show ?thesis
      by auto
  next
    case False
    with ‹a * w⇧2 + b * w + c = 0› show ?thesis
      by auto
  qed
qed

lemma Rats_solution_QE:
  assumes "a ∈ ℚ" "b ∈ ℚ" "a ≠ 0"
  and "a*x^2 + b*x + c = 0"
  and "sqrt (discrim a b c) ∈ ℚ"
  shows "x ∈ ℚ" 
using assms(1,2,5) discriminant_iff[THEN iffD1, OF assms(3,4)] by auto

lemma Rats_solution_QE_converse:
  assumes "a ∈ ℚ" "b ∈ ℚ"
  and "a*x^2 + b*x + c = 0"
  and "x ∈ ℚ"
  shows "sqrt (discrim a b c) ∈ ℚ"
proof -
  from assms(3) have "discrim a b c = (2*a*x+b)^2" unfolding discrim_def by algebra
  hence "sqrt (discrim a b c) = ¦2*a*x+b¦" by (simp)
  thus ?thesis using ‹a ∈ ℚ› ‹b ∈ ℚ› ‹x ∈ ℚ› by (simp)
qed

end