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

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

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
    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

@[simp]
lemma Sequence.eval_coe_at_int (n:ℤ) (a: ℕ → ℚ) : (a:Sequence) n = if n ≥ 0 then a n.toNat else 0 := by
  sorry

end Chapter5

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

namespace Chapter5

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

end Chapter5

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

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)

namespace Chapter5

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

-- <CONTEXT>
@[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
-- </CONTEXT>
-- <TARGET_THEOREM>
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
-- </TARGET_THEOREM>

end Chapter5