import Mathlib
import Analysis.Section_5_3
import Analysis.Tags.Section_5_3

namespace Chapter5

-- <CONTEXT>
abbrev Real := ℝ

structure Sequence where
  seq : ℕ → ℚ

instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℕ → ℚ) where
  coe a := a.seq

instance Sequence.instCoe : Coe (ℕ → ℚ) Sequence where
  coe a := ⟨a⟩

namespace Sequence

abbrev IsCauchy (a:Sequence) : Prop := True

end Sequence

noncomputable abbrev LIM (a:ℕ → ℚ) : Real := 0

abbrev BoundedAwayZero (a:ℕ → ℚ) : Prop :=
  ∃ (c:ℚ), c > 0 ∧ ∀ n, |a n| ≥ c

abbrev BoundedAwayPos (a:ℕ → ℚ) : Prop :=
  ∃ (c:ℚ), c > 0 ∧ ∀ n, a n ≥ c

abbrev BoundedAwayNeg (a:ℕ → ℚ) : Prop :=
  ∃ (c:ℚ), c > 0 ∧ ∀ n, a n ≤ -c

theorem boundedAwayPos_def (a:ℕ → ℚ) : BoundedAwayPos a ↔ ∃ (c:ℚ), c > 0 ∧ ∀ n, a n ≥ c := by
  sorry

theorem boundedAwayNeg_def (a:ℕ → ℚ) : BoundedAwayNeg a ↔ ∃ (c:ℚ), c > 0 ∧ ∀ n, a n ≤ -c := by
  sorry

theorem BoundedAwayZero.boundedAwayPos {a:ℕ → ℚ} (ha: BoundedAwayPos a) : BoundedAwayZero a := by
  sorry

theorem BoundedAwayZero.boundedAwayNeg {a:ℕ → ℚ} (ha: BoundedAwayNeg a) : BoundedAwayZero a := by
  sorry

theorem not_boundedAwayPos_boundedAwayNeg {a:ℕ → ℚ} : ¬ (BoundedAwayPos a ∧ BoundedAwayNeg a) := by
  sorry

abbrev Real.IsPos (x:Real) : Prop :=
  ∃ a:ℕ → ℚ, BoundedAwayPos a ∧ (a:Sequence).IsCauchy ∧ x = LIM a

abbrev Real.IsNeg (x:Real) : Prop :=
  ∃ a:ℕ → ℚ, BoundedAwayNeg a ∧ (a:Sequence).IsCauchy ∧ x = LIM a

theorem Real.isPos_def (x:Real) :
    IsPos x ↔ ∃ a:ℕ → ℚ, BoundedAwayPos a ∧ (a:Sequence).IsCauchy ∧ x = LIM a := by
  sorry

theorem Real.isNeg_def (x:Real) :
    IsNeg x ↔ ∃ a:ℕ → ℚ, BoundedAwayNeg a ∧ (a:Sequence).IsCauchy ∧ x = LIM a := by
  sorry

theorem Real.nonzero_of_pos {x:Real} (hx: x.IsPos) : x ≠ 0 := by
  sorry

theorem Real.nonzero_of_neg {x:Real} (hx: x.IsNeg) : x ≠ 0 := by
  sorry

theorem Real.pos_of_coe (q:ℚ) : (q:Real).IsPos ↔ q > 0 := by
  sorry

theorem Real.neg_of_coe (q:ℚ) : (q:Real).IsNeg ↔ q < 0 := by
  sorry

open Classical in
noncomputable abbrev Real.abs (x:Real) : Real := if x.IsPos then x else (if x.IsNeg then -x else 0)

@[simp]
theorem Real.abs_of_pos (x:Real) (hx: x.IsPos) : abs x = x := by
  sorry

@[simp]
theorem Real.abs_of_neg (x:Real) (hx: x.IsNeg) : abs x = -x := by
  sorry

@[simp]
theorem Real.abs_of_zero : abs 0 = 0 := by
  sorry

instance Real.instLT : LT Real where
  lt x y := (x-y).IsNeg

instance Real.instLE : LE Real where
  le x y := (x < y) ∨ (x = y)

theorem Real.lt_iff (x y:Real) : x < y ↔ (x-y).IsNeg := by
  sorry

theorem Real.le_iff (x y:Real) : x ≤ y ↔ (x < y) ∨ (x = y) := by
  sorry

theorem Real.gt_iff (x y:Real) : x > y ↔ (x-y).IsPos := by
  sorry

theorem Real.ge_iff (x y:Real) : x ≥ y ↔ (x > y) ∨ (x = y) := by
  sorry

theorem Real.lt_of_coe (q q':ℚ): q < q' ↔ (q:Real) < (q':Real) := by
  sorry

theorem Real.gt_of_coe (q q':ℚ): q > q' ↔ (q:Real) > (q':Real) := by
  sorry

theorem Real.isPos_iff (x:Real) : x.IsPos ↔ x > 0 := by
  sorry

theorem Real.isNeg_iff (x:Real) : x.IsNeg ↔ x < 0 := by
  sorry

theorem Real.mul_pos_neg {x y:Real} (hx: x.IsPos) (hy: y.IsNeg) : (x * y).IsNeg := by
  sorry

open Classical in
noncomputable instance Real.instLinearOrder : LinearOrder Real where
  le_refl := sorry
  le_trans := sorry
  lt_iff_le_not_ge := sorry
  le_antisymm := sorry
  le_total := sorry
  toDecidableLE := Classical.decRel _
  min_def := by sorry
  max_def := by sorry
  compare_eq_compareOfLessAndEq := by sorry

theorem Real.abs_eq_abs (x:Real) : |x| = abs x := by
  sorry

theorem Real.inv_of_pos {x:Real} (hx: x.IsPos) : x⁻¹.IsPos := by
  sorry

theorem Real.div_of_pos {x y:Real} (hx: x.IsPos) (hy: y.IsPos) : (x/y).IsPos := by
  sorry

theorem Real.inv_of_gt {x y:Real} (hx: x.IsPos) (hy: y.IsPos) (hxy: x > y) : x⁻¹ < y⁻¹ := by
  sorry

instance Real.instIsStrictOrderedRing : IsStrictOrderedRing Real where
  add_le_add_left := by sorry
  add_le_add_right := by sorry
  mul_lt_mul_of_pos_left := by sorry
  mul_lt_mul_of_pos_right := by sorry
  le_of_add_le_add_left := by sorry
  zero_le_one := by sorry

theorem Real.LIM_of_nonneg {a: ℕ → ℚ} (ha: ∀ n, a n ≥ 0) (hcauchy: (a:Sequence).IsCauchy) :
    LIM a ≥ 0 := by
  sorry

theorem Real.LIM_mono {a b:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy)
  (hmono: ∀ n, a n ≤ b n) :
    LIM a ≤ LIM b := by
  sorry

theorem Real.LIM_mono_fail :
    ∃ (a b:ℕ → ℚ), (a:Sequence).IsCauchy
    ∧ (b:Sequence).IsCauchy
    ∧ (∀ n, a n > b n)
    ∧ ¬LIM a > LIM b := by
  sorry

theorem Real.exists_rat_le_and_nat_gt {x:Real} (hx: x.IsPos) :
    (∃ q:ℚ, q > 0 ∧ (q:Real) ≤ x) ∧ ∃ N:ℕ, x < (N:Real) := by
  sorry

theorem Real.le_mul {ε:Real} (hε: ε.IsPos) (x:Real) : ∃ M:ℕ, M > 0 ∧ M * ε > x := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.le_add_eps_iff (x y:Real) : (∀ ε > 0, x ≤ y+ε) ↔ x ≤ y := by
-- </TARGET_THEOREM>

end Chapter5