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

def Rat.Close (ε : ℚ) (x y : ℚ) := |x - y| ≤ ε

namespace Section_4_3

abbrev dist (x y : ℚ) := |x - y|

theorem dist_eq (x y : ℚ) : dist x y = |x - y| := by
  sorry

theorem close_iff (ε x y : ℚ) : ε.Close x y ↔ |x - y| ≤ ε := by
  sorry

theorem close_refl (x : ℚ) : (0 : ℚ).Close x x := by
  sorry

theorem add_close {ε δ x y z w : ℚ} (hxy : ε.Close x y) (hzw : δ.Close z w) :
    (ε + δ).Close (x + z) (y + w) := by
  sorry

end Section_4_3

namespace Chapter5

@[ext]
structure Sequence where
  n₀ : ℤ
  seq : ℤ → ℚ
  vanish : ∀ n < n₀, seq n = 0

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

@[coe]
def Sequence.ofNatFun (a : ℕ → ℚ) : Sequence where
  n₀ := 0
  seq n := if n ≥ 0 then a n.toNat else 0
  vanish := by
    sorry

instance : Coe (ℕ → ℚ) Sequence where
  coe := Sequence.ofNatFun

abbrev Sequence.mk' (n₀ : ℤ) (a : { n // n ≥ n₀ } → ℚ) : Sequence where
  n₀ := n₀
  seq n := if h : n ≥ n₀ then a ⟨n, h⟩ else 0
  vanish := by
    sorry

@[simp]
lemma Sequence.n0_coe (a : ℕ → ℚ) : (a : Sequence).n₀ = 0 := by
  sorry

abbrev Sequence.from (a : Sequence) (n₁ : ℤ) : Sequence :=
  mk' (max a.n₀ n₁) (fun n ↦ a (n : ℤ))

end Chapter5

abbrev Rat.CloseSeq (ε : ℚ) (a b : Chapter5.Sequence) : Prop :=
  ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → ε.Close (a n) (b n)

abbrev Rat.EventuallyClose (ε : ℚ) (a b : Chapter5.Sequence) : Prop :=
  ∃ N, ε.CloseSeq (a.from N) (b.from N)

abbrev Rat.Steady (ε : ℚ) (a : Chapter5.Sequence) : Prop :=
  ∀ n ≥ a.n₀, ∀ m ≥ a.n₀, ε.Close (a n) (a m)

abbrev Rat.EventuallySteady (ε : ℚ) (a : Chapter5.Sequence) : Prop := ∃ N ≥ a.n₀, ε.Steady (a.from N)

namespace Chapter5

abbrev Sequence.IsCauchy (a : Sequence) : Prop := ∀ ε > (0 : ℚ), ε.EventuallySteady a

@[ext]
class CauchySequence extends Sequence where
  zero : n₀ = 0
  cauchy : toSequence.IsCauchy

instance CauchySequence.instCoeFun : CoeFun CauchySequence (fun _ ↦ ℕ → ℚ) where
  coe a n := a.toSequence (n : ℤ)

abbrev CauchySequence.mk' {a : ℕ → ℚ} (ha : (a : Sequence).IsCauchy) : CauchySequence where
  n₀ := 0
  seq := (a : Sequence).seq
  vanish := by
    sorry
  zero := rfl
  cauchy := ha

@[simp]
theorem CauchySequence.coe_to_sequence (a : CauchySequence) :
    ((a : ℕ → ℚ) : Sequence) = a.toSequence := by
  sorry

theorem Sequence.IsCauchy.const (a : ℚ) : ((fun _ : ℕ ↦ a) : Sequence).IsCauchy := by
  sorry

abbrev Sequence.Equiv (a b : ℕ → ℚ) : Prop :=
  ∀ ε > (0 : ℚ), ε.EventuallyClose (a : Sequence) (b : Sequence)

theorem Sequence.equiv_trans {a b c : ℕ → ℚ} (hab : Equiv a b) (hbc : Equiv b c) :
    Equiv a c := by
  sorry

instance CauchySequence.instSetoid : Setoid CauchySequence where
  r := fun a b ↦ Sequence.Equiv a b
  iseqv := {
    refl := by sorry
    symm := by sorry
    trans := by
      intro a b c hab hbc
      exact Sequence.equiv_trans hab hbc }

instance CauchySequence.instZero : Zero CauchySequence where
  zero := CauchySequence.mk' (a := fun _ : ℕ ↦ 0) (Sequence.IsCauchy.const (0 : ℚ))

abbrev Real := Quotient CauchySequence.instSetoid

open Classical in
noncomputable abbrev LIM (a : ℕ → ℚ) : Real :=
  Quotient.mk _ (if h : (a : Sequence).IsCauchy then CauchySequence.mk' h else (0 : CauchySequence))

theorem Real.LIM_eq_LIM {a b : ℕ → ℚ} (ha : (a : Sequence).IsCauchy) (hb : (b : Sequence).IsCauchy) :
    LIM a = LIM b ↔ Sequence.Equiv a b := by
  sorry

theorem Sequence.IsCauchy.add {a b : ℕ → ℚ} (ha : (a : Sequence).IsCauchy) (hb : (b : Sequence).IsCauchy) :
    (a + b : Sequence).IsCauchy := by
  sorry

theorem Sequence.add_equiv_left {a a' : ℕ → ℚ} (b : ℕ → ℚ) (haa' : Equiv a a') :
    Equiv (a + b) (a' + b) := by
  sorry

theorem Sequence.add_equiv_right {b b' : ℕ → ℚ} (a : ℕ → ℚ) (hbb' : Equiv b b') :
    Equiv (a + b) (a + b') := by
  sorry

theorem Sequence.add_equiv {a b a' b' : ℕ → ℚ} (haa' : Equiv a a')
    (hbb' : Equiv b b') :
    Equiv (a + b) (a' + b') := by
  sorry

noncomputable instance Real.add_inst : Add Real where
  add := fun _ _ ↦ Quotient.mk _ (0 : CauchySequence)

theorem Sequence.IsCauchy.mul {a b : ℕ → ℚ} (ha : (a : Sequence).IsCauchy) (hb : (b : Sequence).IsCauchy) :
    (a * b : Sequence).IsCauchy := by
  sorry

theorem Sequence.mul_equiv_left {a a' : ℕ → ℚ} (b : ℕ → ℚ) (hb : (b : Sequence).IsCauchy) (haa' : Equiv a a') :
    Equiv (a * b) (a' * b) := by
  sorry

theorem Sequence.mul_equiv_right {b b' : ℕ → ℚ} (a : ℕ → ℚ) (ha : (a : Sequence).IsCauchy) (hbb' : Equiv b b') :
    Equiv (a * b) (a * b') := by
  sorry

theorem Sequence.mul_equiv
    {a b a' b' : ℕ → ℚ}
    (ha : (a : Sequence).IsCauchy)
    (hb' : (b' : Sequence).IsCauchy)
    (haa' : Equiv a a')
    (hbb' : Equiv b b') : Equiv (a * b) (a' * b') := by
  sorry

noncomputable instance Real.mul_inst : Mul Real where
  mul := fun _ _ ↦ Quotient.mk _ (0 : CauchySequence)

instance Real.instRatCast : RatCast Real where
  ratCast := fun q ↦
    Quotient.mk _ (CauchySequence.mk' (a := fun _ ↦ q) (Sequence.IsCauchy.const q))

instance Real.instOfNat {n : ℕ} : OfNat Real n where
  ofNat := ((n : ℚ) : Real)

noncomputable instance Real.instNeg : Neg Real where
  neg x := ((-1 : ℚ) : Real) * x

noncomputable instance Real.addGroup_inst : AddGroup Real :=
  AddGroup.ofLeftAxioms (by sorry) (by sorry) (by sorry)

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

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.not_gt_and_lt (x y : Real) : ¬ (x > y ∧ x < y) := by
-- </TARGET_THEOREM>

end Chapter5