import Mathlib

namespace Section_4_2

-- <CONTEXT>
structure PreRat where
  numerator : ℤ
  denominator : ℤ
  nonzero : denominator ≠ 0

instance PreRat.instSetoid : Setoid PreRat where
  r a b := a.numerator * b.denominator = b.numerator * a.denominator
  iseqv := {
    refl := by sorry
    symm := by sorry
    trans := by sorry
    }

@[simp]
theorem PreRat.eq (a b c d:ℤ) (hb: b ≠ 0) (hd: d ≠ 0) :
    (⟨ a,b,hb ⟩: PreRat) ≈ ⟨ c,d,hd ⟩ ↔ a * d = c * b := by
  sorry

abbrev Rat := Quotient PreRat.instSetoid

abbrev Rat.formalDiv (a b:ℤ) : Rat :=
  Quotient.mk PreRat.instSetoid (if h:b ≠ 0 then ⟨ a,b,h ⟩ else ⟨ 0, 1, by decide ⟩)

infix:100 " // " => Rat.formalDiv

theorem Rat.eq (a c:ℤ) {b d:ℤ} (hb: b ≠ 0) (hd: d ≠ 0): a // b = c // d ↔ a * d = c * b := by
  sorry

theorem Rat.eq_diff (n:Rat) : ∃ a b, b ≠ 0 ∧ n = a // b := by
  sorry

instance Rat.decidableEq : DecidableEq Rat := by
  sorry

instance Rat.add_inst : Add Rat where
  add := Quotient.lift₂ (fun ⟨ a, b, h1 ⟩ ⟨ c, d, h2 ⟩ ↦ (a*d+b*c) // (b*d)) (by
    intro ⟨ a, b, h1 ⟩ ⟨ c, d, h2 ⟩ ⟨ a', b', h1' ⟩ ⟨ c', d', h2' ⟩ h3 h4
    simp_all [Quotient.eq, PreRat.instSetoid]
    linear_combination d * d' * h3 + b * b' * h4
  )

theorem Rat.add_eq (a c:ℤ) {b d:ℤ} (hb: b ≠ 0) (hd: d ≠ 0) :
    (a // b) + (c // d) = (a*d + b*c) // (b*d) := by
  sorry

instance Rat.mul_inst : Mul Rat where
  mul := Quotient.lift₂ (fun ⟨ a, b, h1 ⟩ ⟨ c, d, h2 ⟩ ↦ (a*c) // (b*d)) (by sorry)

theorem Rat.mul_eq (a c:ℤ) {b d:ℤ} (hb: b ≠ 0) (hd: d ≠ 0) :
    (a // b) * (c // d) = (a*c) // (b*d) := by
  sorry

instance Rat.neg_inst : Neg Rat where
  neg := Quotient.lift (fun ⟨ a, b, h1 ⟩ ↦ (-a) // b) (by sorry)

theorem Rat.neg_eq (a:ℤ) {b:ℤ} (hb: b ≠ 0) : - (a // b) = (-a) // b := by
  sorry

instance Rat.instIntCast : IntCast Rat where
  intCast a := a // 1

instance Rat.instNatCast : NatCast Rat where
  natCast n := (n:ℤ) // 1

instance Rat.instOfNat {n:ℕ} : OfNat Rat n where
  ofNat := (n:ℤ) // 1

theorem Rat.coe_Int_eq (a:ℤ) : (a:Rat) = a // 1 := by
  sorry

theorem Rat.coe_Nat_eq (n:ℕ) : (n:Rat) = n // 1 := by
  sorry

theorem Rat.of_Nat_eq (n:ℕ) : (ofNat(n):Rat) = (ofNat(n):Nat) // 1 := by
  sorry

theorem Rat.natCast_succ (n: ℕ) : ((n + 1: ℕ): Rat) = (n: Rat) + 1 := by
  sorry

lemma Rat.intCast_add (a b:ℤ) : (a:Rat) + (b:Rat) = (a+b:ℤ) := by
  sorry

lemma Rat.intCast_mul (a b:ℤ) : (a:Rat) * (b:Rat) = (a*b:ℤ) := by
  sorry

lemma Rat.intCast_neg (a:ℤ) : - (a:Rat) = (-a:ℤ) := by
  sorry

theorem Rat.coe_Int_inj : Function.Injective (fun n:ℤ ↦ (n:Rat)) := by
  sorry

instance Rat.instInv : Inv Rat where
  inv := Quotient.lift (fun ⟨ a, b, h1 ⟩ ↦ b // a) (by
    sorry
)

lemma Rat.inv_eq (a:ℤ) {b:ℤ} (hb: b ≠ 0) : (a // b)⁻¹ = b // a := by
  sorry

@[simp]
theorem Rat.inv_zero : (0:Rat)⁻¹ = 0 := by
  sorry

instance Rat.addGroup_inst : AddGroup Rat :=
AddGroup.ofLeftAxioms (by
  intro x y z
  obtain ⟨ a, b, hb, rfl ⟩ := eq_diff x
  obtain ⟨ c, d, hd, rfl ⟩ := eq_diff y
  obtain ⟨ e, f, hf, rfl ⟩ := eq_diff z
  have hbd : b*d ≠ 0 := Int.mul_ne_zero hb hd
  have hdf : d*f ≠ 0 := Int.mul_ne_zero hd hf
  have hbdf : b*d*f ≠ 0 := Int.mul_ne_zero hbd hf
  rw [add_eq _ _ hb hd, add_eq _ _ hbd hf, add_eq _ _ hd hf,
      add_eq _ _ hb hdf, ←mul_assoc b, eq _ _ hbdf hbdf]
  ring
)
 (by sorry) (by sorry)

instance Rat.instAddCommGroup : AddCommGroup Rat where
  add_comm := by sorry

instance Rat.instCommMonoid : CommMonoid Rat where
  mul_comm := by sorry
  mul_assoc := by sorry
  one_mul := by sorry
  mul_one := by sorry

instance Rat.instCommRing : CommRing Rat where
  left_distrib := by sorry
  right_distrib := by sorry
  zero_mul := by sorry
  mul_zero := by sorry
  mul_assoc := by sorry
  natCast_succ := natCast_succ

instance Rat.instRatCast : RatCast Rat where
  ratCast q := q.num // q.den

theorem Rat.ratCast_inj : Function.Injective (fun n:ℚ ↦ (n:Rat)) := by
  sorry

theorem Rat.coe_Rat_eq (a:ℤ) {b:ℤ} (hb: b ≠ 0) : (a/b:ℚ) = a // b := by
  sorry

instance Rat.instDivInvMonoid : DivInvMonoid Rat where

theorem Rat.div_eq (q r:Rat) : q/r = q * r⁻¹ := by
  sorry

instance Rat.instField : Field Rat where
  exists_pair_ne := by sorry
  mul_inv_cancel := by sorry
  inv_zero := rfl
  ratCast_def := by
    intro q
    set num := q.num
    set den := q.den
    have hden : (den:ℤ) ≠ 0 := by simp [den, q.den_nz]
    rw [← Rat.num_div_den q]
    convert coe_Rat_eq _ hden
    rw [coe_Int_eq, coe_Nat_eq, div_eq, inv_eq, mul_eq, eq] <;> simp [num, den, q.den_nz]
  qsmul := _
  nnqsmul := _

theorem Rat.sub_eq (a b:Rat) : a - b = a + (-b) := by
  sorry

def Rat.coe_int_hom : ℤ →+* Rat where
  toFun n := (n:Rat)
  map_zero' := rfl
  map_one' := rfl
  map_add' := by sorry
  map_mul' := by sorry

def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

instance Rat.instLT : LT Rat where
  lt x y := (x-y).isNeg

instance Rat.instLE : LE Rat where
  le x y := (x < y) ∨ (x = y)

theorem Rat.lt_iff (x y:Rat) : x < y ↔ (x-y).isNeg := by
  sorry
theorem Rat.le_iff (x y:Rat) : x ≤ y ↔ (x < y) ∨ (x = y) := by
  sorry

theorem Rat.gt_iff (x y:Rat) : x > y ↔ (x-y).isPos := by
  sorry
theorem Rat.ge_iff (x y:Rat) : x ≥ y ↔ (x > y) ∨ (x = y) := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Rat.not_lt_and_eq (x y:Rat) : ¬ (x < y ∧ x = y):= by
-- </TARGET_THEOREM>

end Section_4_2