import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

/-!
# Analysis I, Section 4.2

This file is a translation of Section 4.2 of Analysis I to Lean 4.
All numbering refers to the original text.

I have attempted to make the translation as faithful a paraphrasing as possible of the original
text. When there is a choice between a more idiomatic Lean solution and a more faithful
translation, I have generally chosen the latter. In particular, there will be places where the
Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided
doing so.

Main constructions and results of this section:

- Definition of the "Section 4.2" rationals, `Section_4_2.Rat`, as formal quotients `a // b` of
  integers `a b:ℤ`, up to equivalence.  (This is a quotient of a scaffolding type
  `Section_4_2.PreRat`, which consists of formal quotients without any equivalence imposed.)

- Field operations and order on these rationals, as well as an embedding of ℕ and ℤ.

- Equivalence with the Mathlib rationals `_root_.Rat` (or `ℚ`), which we will use going forward.

Note: here (and in the sequel) we use Mathlib's natural numbers `ℕ` and integers `ℤ` rather than
the Chapter 2 natural numbers and Section 4.1 integers.

## Tips from past users

Users of the companion who have completed the exercises in this section are welcome to send their tips for future users in this section as PRs.

- (Add tip here)

-/

namespace Section_4_2

structure PreRat where
  numerator : ℤ
  denominator : ℤ
  nonzero : denominator ≠ 0

/-- Exercise 4.2.1 -/
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 rfl

abbrev Rat := Quotient PreRat.instSetoid

/-- We give division a "junk" value of 0//1 if the denominator is zero -/
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

/-- Definition 4.2.1 (Rationals) -/
theorem Rat.eq (a c:ℤ) {b d:ℤ} (hb: b ≠ 0) (hd: d ≠ 0): a // b = c // d ↔ a * d = c * b := by
  simp [formalDiv, hb, hd, Quotient.eq, PreRat.instSetoid]

/-- Definition 4.2.1 (Rationals) -/
theorem Rat.eq_diff (n:Rat) : ∃ a b, b ≠ 0 ∧ n = a // b := by
  apply Quotient.ind _ n; intro ⟨ a, b, h ⟩
  refine ⟨ a, b, h, ?_ ⟩
  simp [formalDiv, h]

/--
  Decidability of equality. Hint: modify the proof of `DecidableEq Int` from the previous
  section. However, because formal division handles the case of zero denominator separately, it
  may be more convenient to avoid that operation and work directly with the `Quotient` API.
-/
instance Rat.decidableEq : DecidableEq Rat := by
  sorry

/-- Lemma 4.2.3 (Addition well-defined) -/
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
  )

/-- Definition 4.2.2 (Addition of rationals) -/
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
  convert Quotient.lift₂_mk _ _ _ _ <;> simp [hb, hd]

/-- Lemma 4.2.3 (Multiplication well-defined) -/
instance Rat.mul_inst : Mul Rat where
  mul := Quotient.lift₂ (fun ⟨ a, b, h1 ⟩ ⟨ c, d, h2 ⟩ ↦ (a*c) // (b*d)) (by sorry)

/-- Definition 4.2.2 (Multiplication of rationals) -/
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
  convert Quotient.lift₂_mk _ _ _ _ <;> simp [hb, hd]

/-- Lemma 4.2.3 (Negation well-defined) -/
instance Rat.neg_inst : Neg Rat where
  neg := Quotient.lift (fun ⟨ a, b, h1 ⟩ ↦ (-a) // b) (by sorry)

/-- Definition 4.2.2 (Negation of rationals) -/
theorem Rat.neg_eq (a:ℤ) {b:ℤ} (hb: b ≠ 0) : - (a // b) = (-a) // b := by
  convert Quotient.lift_mk _ _ _ <;> simp [hb]

/-- Embedding the integers in the rationals -/
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 := rfl

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

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

/-- natCast distributes over successor -/
theorem Rat.natCast_succ (n: ℕ) : ((n + 1: ℕ): Rat) = (n: Rat) + 1 := by sorry

/-- intCast distributes over addition -/
lemma Rat.intCast_add (a b:ℤ) : (a:Rat) + (b:Rat) = (a+b:ℤ) := by sorry

/-- intCast distributes over multiplication -/
lemma Rat.intCast_mul (a b:ℤ) : (a:Rat) * (b:Rat) = (a*b:ℤ) := by sorry

/-- intCast commutes with negation -/
lemma Rat.intCast_neg (a:ℤ) : - (a:Rat) = (-a:ℤ) := rfl

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

/-- Proposition 4.2.4 (laws of algebra) / Exercise 4.2.3 -/
instance Rat.addGroup_inst : AddGroup Rat :=
AddGroup.ofLeftAxioms (by
  -- this proof is written to follow the structure of the original text.
  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     -- can also use `observe hbd : b*d ≠ 0` here
  have hdf : d*f ≠ 0 := Int.mul_ne_zero hd hf     -- can also use `observe hdf : d*f ≠ 0` here
  have hbdf : b*d*f ≠ 0 := Int.mul_ne_zero hbd hf -- can also use `observe hbdf : b*d*f ≠ 0` here
  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)

/-- Proposition 4.2.4 (laws of algebra) / Exercise 4.2.3 -/
instance Rat.instAddCommGroup : AddCommGroup Rat where
  add_comm := by sorry

/-- Proposition 4.2.4 (laws of algebra) / Exercise 4.2.3 -/
instance Rat.instCommMonoid : CommMonoid Rat where
  mul_comm := by sorry
  mul_assoc := by sorry
  one_mul := by sorry
  mul_one := by sorry

/-- Proposition 4.2.4 (laws of algebra) / Exercise 4.2.3 -/
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
  -- Usually CommRing will generate a natCast instance and a proof for this.
  -- However, we are using a custom natCast for which `natCast_succ` cannot
  -- be proven automatically by `rfl`. Luckily we have proven it already.
  natCast_succ := natCast_succ

end Section_4_2