import Mathlib
import Analysis.Section_5_2
import Analysis.Tags.Section_5_2

namespace Chapter5

-- <CONTEXT>
def _root_.Rat.Close (ε : ℚ) (x y : ℚ) := |x - y| ≤ ε

@[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 grind

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 grind

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

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

@[aesop safe apply]
lemma Sequence.vanish_coe (a:ℕ→ℚ) : ∀ n, n < (0:ℤ) → (a:Sequence).seq n = 0 := by
  sorry

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

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

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

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

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

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

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

theorem CauchySequence.ext' {a b: CauchySequence} (h: a.seq = b.seq) : a = b := by
  sorry

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

@[simp]
theorem CauchySequence.coe_eq {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) :
    (mk' ha).toSequence = (a:Sequence) := by
  sorry

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

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

@[simp]
theorem CauchySequence.coe_coe {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) : mk' ha = a := by
  sorry

instance CauchySequence.instSetoid : Setoid CauchySequence where
  r := fun a b ↦ Sequence.Equiv a b
  iseqv := {
     refl := sorry
     symm := sorry
     trans := sorry
  }

theorem CauchySequence.equiv_iff (a b: CauchySequence) : a ≈ b ↔ Sequence.Equiv a b := by
  sorry

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

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 LIM_def {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) :
    LIM a = Quotient.mk _ (CauchySequence.mk' ha) := by
  sorry

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

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 x y ↦
    Quotient.liftOn₂ x y (fun a b ↦ LIM (a + b)) (by
      intro a b a' b' _ _
      change LIM ((a:ℕ → ℚ) + (b:ℕ → ℚ)) = LIM ((a':ℕ → ℚ) + (b':ℕ → ℚ))
      rw [LIM_eq_LIM]
      . solve_by_elim [Sequence.add_equiv]
      all_goals apply Sequence.IsCauchy.add <;> rw [CauchySequence.coe_to_sequence] <;> convert @CauchySequence.cauchy ?_
      )

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

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

noncomputable instance Real.mul_inst : Mul Real where
  mul := fun x y ↦
    Quotient.liftOn₂ x y (fun a b ↦ LIM (a * b)) (by
      sorry)

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

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

theorem Real.ratCast_def (q:ℚ) : (q:Real) = LIM (fun _ ↦ q) := by
  sorry

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

instance Real.instNatCast : NatCast Real where
  natCast n := ((n:ℚ):Real)

@[simp]
theorem Real.LIM.zero : LIM (fun _ ↦ (0:ℚ)) = 0 := by
  sorry

instance Real.instIntCast : IntCast Real where
  intCast n := ((n:ℚ):Real)

theorem Real.ratCast_add (a b:ℚ) : (a:Real) + (b:Real) = (a+b:ℚ) := by
  sorry

theorem Real.ratCast_mul (a b:ℚ) : (a:Real) * (b:Real) = (a*b:ℚ) := by
  sorry

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

theorem Real.neg_ratCast (a:ℚ) : -(a:Real) = (-a:ℚ) := by
  sorry

theorem Real.neg_LIM (a:ℕ → ℚ) (ha: (a:Sequence).IsCauchy) : -LIM a = LIM (-a) := by
  sorry

theorem Sequence.IsCauchy.neg (a:ℕ → ℚ) (ha: (a:Sequence).IsCauchy) :
    ((-a:ℕ → ℚ):Sequence).IsCauchy := by
  sorry

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

theorem Real.sub_eq_add_neg (x y:Real) : x - y = x + (-y) := by
  sorry

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

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

theorem Real.ratCast_sub (a b:ℚ) : (a:Real) - (b:Real) = (a-b:ℚ) := by
  sorry

noncomputable instance Real.instAddCommGroup : AddCommGroup Real where
  add_comm := by sorry

noncomputable instance Real.instCommMonoid : CommMonoid Real where
  mul_comm := by sorry
  mul_assoc := by sorry
  one_mul := by sorry
  mul_one := by sorry

noncomputable instance Real.instCommRing : CommRing Real where
  left_distrib := by sorry
  right_distrib := by sorry
  zero_mul := by sorry
  mul_zero := by sorry
  mul_assoc := by sorry
  natCast_succ := by sorry
  intCast_negSucc := by sorry

abbrev Real.ratCast_hom : ℚ →+* Real where
  toFun := RatCast.ratCast
  map_zero' := by sorry
  map_one' := by sorry
  map_add' := by sorry
  map_mul' := by sorry

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

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

theorem Real.boundedAwayZero_of_nonzero {x:Real} (hx: x ≠ 0) :
    ∃ a:ℕ → ℚ, (a:Sequence).IsCauchy ∧ BoundedAwayZero a ∧ x = LIM a := by
  sorry

theorem Real.nonzero_of_boundedAwayZero {a:ℕ → ℚ} (ha: BoundedAwayZero a) (n: ℕ) : a n ≠ 0 := by
  sorry

theorem Real.inv_isCauchy_of_boundedAwayZero {a:ℕ → ℚ} (ha: BoundedAwayZero a)
  (ha_cauchy: (a:Sequence).IsCauchy) :
    ((a⁻¹:ℕ → ℚ):Sequence).IsCauchy := by
  sorry

theorem Real.inv_of_equiv {a b:ℕ → ℚ} (ha: BoundedAwayZero a)
  (ha_cauchy: (a:Sequence).IsCauchy) (hb: BoundedAwayZero b)
  (hb_cauchy: (b:Sequence).IsCauchy) (hlim: LIM a = LIM b) :
    LIM a⁻¹ = LIM b⁻¹ := by
  sorry

open Classical in
noncomputable instance Real.instInv : Inv Real where
  inv x := if h: x ≠ 0 then LIM (boundedAwayZero_of_nonzero h).choose⁻¹ else 0

theorem Real.inv_def {a:ℕ → ℚ} (h: BoundedAwayZero a) (hc: (a:Sequence).IsCauchy) :
    (LIM a)⁻¹ = LIM a⁻¹ := by
  sorry

@[simp]
theorem Real.inv_zero : (0:Real)⁻¹ = 0 := by
  sorry

theorem Real.self_mul_inv {x:Real} (hx: x ≠ 0) : x * x⁻¹ = 1 := by
  sorry

theorem Real.inv_mul_self {x:Real} (hx: x ≠ 0) : x⁻¹ * x = 1 := by
  sorry

lemma BoundedAwayZero.const {q : ℚ} (hq : q ≠ 0) : BoundedAwayZero fun _ ↦ q := by
  sorry

theorem Real.inv_ratCast (q:ℚ) : (q:Real)⁻¹ = (q⁻¹:ℚ) := by
  sorry

noncomputable instance Real.instDivInvMonoid : DivInvMonoid Real where

theorem Real.div_eq (x y:Real) : x/y = x * y⁻¹ := by
  sorry

noncomputable instance Real.instField : Field Real where
  exists_pair_ne := by sorry
  mul_inv_cancel := by sorry
  inv_zero := by sorry
  ratCast_def := by sorry
  qsmul := _
  nnqsmul := _

theorem Real.mul_right_cancel₀ {x y z:Real} (hz: z ≠ 0) (h: x * z = y * z) : x = y := by
  sorry

theorem Real.mul_right_nocancel : ¬ ∀ (x y z:Real), (hz: z = 0) → (x * z = y * z) → x = y := by
  sorry

theorem Sequence.IsCauchy.harmonic' : ((fun n ↦ 1/((n:ℚ)+1): ℕ → ℚ):Sequence).IsCauchy := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.LIM.harmonic : LIM (fun n ↦ 1/((n:ℚ)+1)) = 0 := by
-- </TARGET_THEOREM>

end Chapter5