import Mathlib.Tactic

namespace Section_4_3
end Section_4_3

/-- Close relation on rationals, from Section 4.3 -/
def Rat.Close (ε : ℚ) (x y : ℚ) := |x - y| ≤ ε

namespace Chapter5

-- Definition 5.1.1 (Sequence)
@[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
    intro n hn; simp [hn]

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
    intro n hn; simp [hn]

@[simp]
lemma Sequence.eval_coe (n:ℕ) (a: ℕ → ℚ) : (a:Sequence) n = a n := by
  have : (n:ℤ) ≥ (0:ℤ) := by exact_mod_cast (Nat.zero_le n)
  simp [Sequence.ofNatFun, this]

@[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
  rfl

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

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

end Chapter5

-- ε-steadiness and eventual ε-steadiness
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

end Chapter5

-- Section 5.2: ε-close and equivalent sequences (at top level)
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)

lemma Sequence.equiv_iff (a b: ℕ → ℚ) :
    Equiv a b ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, |a n - b n| ≤ ε := by
  sorry

lemma Sequence.isCauchy_of_equiv {a b: ℕ → ℚ} (hab: Equiv a b) :
    (a:Sequence).IsCauchy ↔ (b:Sequence).IsCauchy := by
  sorry

-- Section 5.3: CauchySequence and Real
@[ext]
class CauchySequence extends Sequence where
  zero : n₀ = 0
  cauchy : toSequence.IsCauchy

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

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

@[simp]
lemma CauchySequence.coe_to_sequence (a: CauchySequence) :
    ((a:ℕ → ℚ):Sequence) = a.toSequence := by
  apply Sequence.ext (by simp [Sequence.n0_coe, a.zero])
  ext n; by_cases h:n ≥ 0 <;> simp_all
  rw [a.vanish]; rwa [a.zero]

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

-- Constant sequences are Cauchy (from Section 5.3)
lemma Sequence.IsCauchy.const (a:ℚ) : ((fun _ : ℕ ↦ a) : Sequence).IsCauchy := by
  sorry

-- We need Zero CauchySequence for the LIM default value
instance CauchySequence.instZero : Zero CauchySequence where
  zero := CauchySequence.mk' (a := fun _ : ℕ ↦ 0) (Sequence.IsCauchy.const 0)

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

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))

lemma LIM_def {a:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) :
    LIM a = Quotient.mk _ (CauchySequence.mk' ha) := by
  rw [LIM, dif_pos ha]

-- Equality of limits
lemma 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
  constructor
  · intro h; replace h := Quotient.exact h
    rwa [dif_pos ha, dif_pos hb] at h
  intro h; apply Quotient.sound
  rwa [dif_pos ha, dif_pos hb]

-- Addition on reals
lemma Sequence.IsCauchy.add {a b:ℕ → ℚ}  (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
    (a + b:Sequence).IsCauchy := by
  sorry

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

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

lemma 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 ?_)

-- Multiplication on reals
lemma Sequence.IsCauchy.mul {a b:ℕ → ℚ}  (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
    (a * b:Sequence).IsCauchy := by
  sorry

lemma 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

lemma 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

lemma 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 x y ↦
    Quotient.liftOn₂ x y (fun a b ↦ LIM (a * b)) (by
      intro a b a' b' haa' hbb'
      change LIM ((a:ℕ → ℚ) * (b:ℕ → ℚ)) = LIM ((a':ℕ → ℚ) * (b':ℕ → ℚ))
      rw [LIM_eq_LIM]
      · exact
          Sequence.mul_equiv (by rw [CauchySequence.coe_to_sequence]; exact a.cauchy)
            (by rw [CauchySequence.coe_to_sequence]; exact b'.cauchy) haa' hbb'
      all_goals
        apply Sequence.IsCauchy.mul <;> rw [CauchySequence.coe_to_sequence] <;> convert @CauchySequence.cauchy ?_)

-- Constant sequences -> reals
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)

-- Negation via multiplication by -1 (as in the source)
noncomputable instance Real.instNeg : Neg Real where neg x := ((-1:ℚ):Real) * x

-- Additive group structure (as in the source, with sorries in proofs)
noncomputable instance Real.addGroup_inst : AddGroup Real :=
  AddGroup.ofLeftAxioms (by sorry) (by sorry) (by sorry)

-- From Section 5.3: BoundedAwayZero
abbrev BoundedAwayZero (a:ℕ → ℚ) : Prop := ∃ c:ℚ, c > 0 ∧ ∀ n, |a n| ≥ c

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

-- Section 5.4: Positivity and order

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

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

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)

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

-- Exercise 5.4.9
 theorem Real.inv_min {x y :Real} (hx:x.IsPos) (hy:y.IsPos) : (min x y)⁻¹ = max x⁻¹ y⁻¹ := by
end Chapter5