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import Mathlib.Data.Real.Basic

namespace C03S05
section

variable {x y : ℝ}

namespace MyAbs

theorem le_abs_self (x : ℝ) : x ≤ abs x := by
  cases' le_or_gt 0 x with h h
  · rw [abs_of_nonneg h]
  rw [abs_of_neg h]
  linarith

theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by
  cases' le_or_gt 0 x with h h
  · rw [abs_of_nonneg h]
    linarith
  rw [abs_of_neg h]

theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by
  cases' le_or_gt 0 (x + y) with h h
  · rw [abs_of_nonneg h]
    linarith [le_abs_self x, le_abs_self y]
  rw [abs_of_neg h]
  linarith [neg_le_abs_self x, neg_le_abs_self y]

theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by
  cases' le_or_gt 0 y with h h
  · rw [abs_of_nonneg h]
    constructor
    · intro h'
      left
      exact h'
    intro h'
    cases' h' with h' h'
    · exact h'
    linarith
  rw [abs_of_neg h]
  constructor
  · intro h'
    right
    exact h'
  intro h'
  cases' h' with h' h'
  · linarith
  exact h'

theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by
  cases' le_or_gt 0 x with h h
  · rw [abs_of_nonneg h]
    constructor
    · intro h'
      constructor
      · linarith
      exact h'
    intro h'
    cases' h' with h1 h2
    exact h2
  rw [abs_of_neg h]
  constructor
  · intro h'
    constructor
    · linarith
    linarith
  intro h'
  linarith

end MyAbs

end

example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by
  rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y]

example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
  have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self]
  have h'' : (x + 1) * (x - 1) = 0 := by
    rw [← h']
    ring
  cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1
  · right
    exact eq_neg_iff_add_eq_zero.mpr h1
  left
  exact eq_of_sub_eq_zero h1

example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
  have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self]
  have h'' : (x + y) * (x - y) = 0 := by
    rw [← h']
    ring
  cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1
  · right
    exact eq_neg_iff_add_eq_zero.mpr h1
  left
  exact eq_of_sub_eq_zero h1

section
variable {R : Type _} [CommRing R] [IsDomain R]
variable (x y : R)

example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by
  have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self]
  have h'' : (x + 1) * (x - 1) = 0 := by
    rw [← h']
    ring
  cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1
  · right
    exact eq_neg_iff_add_eq_zero.mpr h1
  left
  exact eq_of_sub_eq_zero h1

example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by
  have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self]
  have h'' : (x + y) * (x - y) = 0 := by
    rw [← h']
    ring
  cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1
  · right
    exact eq_neg_iff_add_eq_zero.mpr h1
  left
  exact eq_of_sub_eq_zero h1

end

example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by
  constructor
  · intro h
    by_cases h' : P
    · right
      exact h h'
    left
    exact h'
  rintro (h | h)
  · intro h'
    exact absurd h' h
  intro
  exact h