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

namespace Chapter5

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

abbrev _root_.Section_4_3.dist (x y : ℚ) := |x - y|

theorem _root_.Section_4_3.dist_eq (x y: ℚ) : Section_4_3.dist x y = |x-y| := by
  sorry

theorem _root_.Section_4_3.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

theorem _root_.Section_4_3.close_refl (x:ℚ) : (0:ℚ).Close x x := by
  sorry

@[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

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

@[simp]
lemma Sequence.eval_coe (n:ℕ) (a: ℕ → ℚ) : (a:Sequence) n = a n := by
  sorry

@[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

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

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

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

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

lemma Sequence.IsCauchy.coe (a:ℕ → ℚ) :
    (a:Sequence).IsCauchy ↔ ∀ ε > (0:ℚ), ∃ N, ∀ j ≥ N, ∀ k ≥ N,
    Section_4_3.dist (a j) (a k) ≤ ε := by
  sorry

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

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

lemma Rat.closeSeq_def (ε: ℚ) (a b: Sequence) :
    ε.CloseSeq a b ↔ ∀ n, n ≥ a.n₀ → n ≥ b.n₀ → ε.Close (a n) (b n) := by
  sorry

lemma Rat.eventuallyClose_def (ε: ℚ) (a b: Sequence) :
    ε.EventuallyClose a b ↔ ∃ N, ε.CloseSeq (a.from N) (b.from N) := by
  sorry

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

lemma Sequence.equiv_def (a b: ℕ → ℚ) :
    Equiv a b ↔ ∀ (ε:ℚ), ε > 0 → ε.EventuallyClose (a:Sequence) (b:Sequence) := by
  sorry

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

@[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' {f:ℕ → ℚ} (hf: (f:Sequence).IsCauchy) : CauchySequence where
  n₀ := 0
  seq := (f:Sequence).seq
  vanish := by
    intro n hn
    have hneg : ¬ n ≥ 0 := by linarith
    simpa [Sequence.instCoeFun, Sequence.eval_coe_at_int, hneg]
  zero := rfl
  cauchy := hf

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

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

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

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 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 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
  {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 ?_
      )

instance Real.instRatCast : RatCast Real where
  ratCast := fun q ↦
    Quotient.mk _ (CauchySequence.mk' (f := 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)

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

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.antisymm (x y:Real) : x < y ↔ y > x := by
-- </TARGET_THEOREM>

end Chapter5