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

def Rat.Close (ε : ℚ) (x y : ℚ) : Prop := |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
    intro n hn
    have hneg : ¬ n ≥ 0 := by exact Int.not_le.mpr hn
    have : (if n ≥ 0 then a n.toNat else 0) = 0 := by simp [hneg]
    simpa using this

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

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

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

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
    have : ¬ n ≥ n₀ := by exact Int.not_le.mpr hn
    simp [this]

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 → Rat.Close ε (a n) (b n)

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

namespace Chapter5

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

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

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

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

@[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 sorry
  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
  sorry

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

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

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' _ _
      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' _ _
      sorry)

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.lt_trans {x y z:Real} (hxy: x < y) (hyz: y < z) : x < z := by
-- </TARGET_THEOREM>

end Chapter5