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

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

namespace Section_4_3

abbrev abs (x:ℚ) : ℚ := if x > 0 then x else (if x < 0 then -x else 0)

theorem abs_of_pos {x: ℚ} (hx: 0 < x) : abs x = x := by
  sorry

theorem abs_of_neg {x: ℚ} (hx: x < 0) : abs x = -x := by
  sorry

theorem abs_of_zero : abs 0 = 0 := by
  sorry

theorem abs_eq_abs (x: ℚ) : abs x = |x| := by
  sorry

abbrev dist (x y : ℚ) := |x - y|

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

theorem abs_nonneg (x: ℚ) : |x| ≥ 0 := by
  sorry

theorem abs_eq_zero_iff (x: ℚ) : |x| = 0 ↔ x = 0 := by
  sorry

theorem abs_add (x y:ℚ) : |x + y| ≤ |x| + |y| := by
  sorry

theorem abs_le_iff (x y:ℚ) : -y ≤ x ∧ x ≤ y ↔ |x| ≤ y := by
  sorry

theorem le_abs (x:ℚ) : -|x| ≤ x ∧ x ≤ |x| := by
  sorry

theorem abs_mul (x y:ℚ) : |x * y| = |x| * |y| := by
  sorry

theorem abs_neg (x:ℚ) : |-x| = |x| := by
  sorry

theorem dist_nonneg (x y:ℚ) : dist x y ≥ 0 := by
  sorry

theorem dist_eq_zero_iff (x y:ℚ) : dist x y = 0 ↔ x = y := by
  sorry

theorem dist_symm (x y:ℚ) : dist x y = dist y x := by
  sorry

theorem dist_le (x y z:ℚ) : dist x z ≤ dist x y + dist y z := by
  sorry

theorem close_iff (ε x y:ℚ): ε.Close x y ↔ |x - y| ≤ ε := by
  sorry

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

theorem eq_if_close (x y:ℚ) : x = y ↔ ∀ ε:ℚ, ε > 0 → ε.Close x y := by
  sorry

theorem close_symm (ε x y:ℚ) : ε.Close x y ↔ ε.Close y x := by
  sorry

theorem close_trans {ε δ x y z:ℚ} (hxy: ε.Close x y) (hyz: δ.Close y z) :
    (ε + δ).Close x z := by
  sorry

theorem 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 sub_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 close_mono {ε ε' x y:ℚ} (hxy: ε.Close x y) (hε: ε' ≥  ε) :
    ε'.Close x y := by
  sorry

theorem close_between {ε x y z w:ℚ} (hxy: ε.Close x y) (hxz: ε.Close x z)
  (hbetween: (y ≤ w ∧ w ≤ z) ∨ (z ≤ w ∧ w ≤ y)) : ε.Close x w := by
  sorry

theorem close_mul_right {ε x y z:ℚ} (hxy: ε.Close x y) :
    (ε*|z|).Close (x * z) (y * z) := by
  sorry

theorem close_mul_mul {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε*|z|+δ*|x|+ε*δ).Close (x * z) (y * w) := by
  sorry

theorem close_mul_mul' {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε*|z|+δ*|y|).Close (x * z) (y * w) := by
  sorry

lemma pow_zero (x:ℚ) : x^0 = 1 := by
  sorry

lemma pow_succ (x:ℚ) (n:ℕ) : x^(n+1) = x^n * x := by
  sorry

theorem pow_add (x:ℚ) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem pow_mul (x:ℚ) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem mul_pow (x y:ℚ) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem pow_eq_zero (x:ℚ) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem pow_nonneg {x:ℚ} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem pow_pos {x:ℚ} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem pow_ge_pow (x y:ℚ) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem pow_gt_pow (x y:ℚ) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem pow_abs (x:ℚ) (n:ℕ) : |x|^n = |x^n| := by
  sorry

theorem zpow_neg (x:ℚ) (n:ℕ) : x^(-(n:ℤ)) = 1/(x^n) := by
  sorry

theorem pow_eq_zpow (x:ℚ) (n:ℕ): x^(n:ℤ) = x^n := by
  sorry

theorem zpow_add (x:ℚ) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem zpow_mul (x:ℚ) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem mul_zpow (x y:ℚ) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem zpow_pos {x:ℚ} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem zpow_ge_zpow {x y:ℚ} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem zpow_ge_zpow_ofneg {x y:ℚ} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem zpow_inj {x y:ℚ} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem zpow_abs (x:ℚ) (n:ℤ) : |x|^n = |x^n| := by
  sorry

theorem two_pow_geq (N:ℕ) : 2^N ≥ N := by
  sorry

end Section_4_3

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
    sorry

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
    sorry

lemma Sequence.eval_mk {n n₀:ℤ} (a: { n // n ≥ n₀ } → ℚ) (h: n ≥ n₀) :
    (Sequence.mk' n₀ a) n = a ⟨ n, h ⟩ := 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

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

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

lemma Sequence.from_eval (a:Sequence) {n₁ n:ℤ} (hn: n ≥ n₁) :
  (a.from n₁) n = a n := 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)

lemma Rat.steady_def (ε: ℚ) (a: Chapter5.Sequence) :
  ε.Steady a ↔ ∀ n ≥ a.n₀, ∀ m ≥ a.n₀, ε.Close (a n) (a m) := by
  sorry

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

lemma Rat.eventuallySteady_def (ε: ℚ) (a: Chapter5.Sequence) :
  ε.EventuallySteady a ↔ ∃ N ≥ a.n₀, ε.Steady (a.from N) := by
  sorry

namespace Chapter5

lemma Rat.Steady.coe (ε : ℚ) (a:ℕ → ℚ) :
    ε.Steady a ↔ ∀ n m : ℕ, ε.Close (a n) (a m) := by
  sorry

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

lemma Sequence.isCauchy_def (a:Sequence) :
  a.IsCauchy ↔ ∀ ε > (0:ℚ), ε.EventuallySteady a := by
  sorry

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

lemma Sequence.IsCauchy.mk {n₀:ℤ} (a: {n // n ≥ n₀} → ℚ) :
    (mk' n₀ a).IsCauchy ↔ ∀ ε > (0:ℚ), ∃ N ≥ n₀, ∀ j ≥ N, ∀ k ≥ N,
    Section_4_3.dist (mk' n₀ a j) (mk' n₀ a k) ≤ ε := by
  sorry

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

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

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

lemma 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
    sorry
  zero := by
    sorry
  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

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

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

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

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
  sorry

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
      sorry)

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

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
      sorry)

lemma 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))

lemma 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)

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)

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

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

lemma Real.LIM_sub {a b:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy) :
  LIM a - LIM b = LIM (a - b) := 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 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.lim_of_boundedAwayZero {a:ℕ → ℚ} (ha: BoundedAwayZero a)
  (ha_cauchy: (a:Sequence).IsCauchy) :
    LIM a ≠ 0 := 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

-- <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

theorem Real.mul_pos_neg {x y:Real} (hx: x.IsPos) (hy: y.IsNeg) : (x * y).IsNeg := by
  sorry

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

theorem Real.abs_eq_abs (x:Real) : |x| = abs x := by
  sorry

theorem Real.inv_of_pos {x:Real} (hx: x.IsPos) : x⁻¹.IsPos := by
  sorry

theorem Real.div_of_pos {x y:Real} (hx: x.IsPos) (hy: y.IsPos) : (x/y).IsPos := by
  sorry

theorem Real.inv_of_gt {x y:Real} (hx: x.IsPos) (hy: y.IsPos) (hxy: x > y) : x⁻¹ < y⁻¹ := by
  sorry

instance Real.instIsStrictOrderedRing : IsStrictOrderedRing Real where
  add_le_add_left := by sorry
  add_le_add_right := by sorry
  mul_lt_mul_of_pos_left := by sorry
  mul_lt_mul_of_pos_right := by sorry
  le_of_add_le_add_left := by sorry
  zero_le_one := by sorry

theorem Real.LIM_of_nonneg {a: ℕ → ℚ} (ha: ∀ n, a n ≥ 0) (hcauchy: (a:Sequence).IsCauchy) :
    LIM a ≥ 0 := by
  sorry

theorem Real.LIM_mono {a b:ℕ → ℚ} (ha: (a:Sequence).IsCauchy) (hb: (b:Sequence).IsCauchy)
  (hmono: ∀ n, a n ≤ b n) :
    LIM a ≤ LIM b := by
  sorry

theorem Real.LIM_mono_fail :
    ∃ (a b:ℕ → ℚ), (a:Sequence).IsCauchy
    ∧ (b:Sequence).IsCauchy
    ∧ (∀ n, a n > b n)
    ∧ ¬LIM a > LIM b := by
  sorry

theorem Real.exists_rat_le_and_nat_gt {x:Real} (hx: x.IsPos) :
    (∃ q:ℚ, q > 0 ∧ (q:Real) ≤ x) ∧ ∃ N:ℕ, x < (N:Real) := by
  sorry

theorem Real.le_mul {ε:Real} (hε: ε.IsPos) (x:Real) : ∃ M:ℕ, M > 0 ∧ M * ε > x := by
  sorry

theorem Real.max_eq (x y:Real) : max x y = if x ≥ y then x else y := by
  sorry

theorem Real.min_eq (x y:Real) : min x y = if x ≤ y then x else y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.min_add (x y z:Real) : min (x + z) (y + z) = min x y + z := by
-- </TARGET_THEOREM>

end Chapter5