import Mathlib

namespace Section_4_1

-- <CONTEXT>
structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

instance PreInt.instSetoid : Setoid PreInt where
  r a b := a.minuend + b.subtrahend = b.minuend + a.subtrahend
  iseqv := {
    refl := by sorry
    symm := by sorry
    trans := by
      intro ⟨ a,b ⟩ ⟨ c,d ⟩ ⟨ e,f ⟩ h1 h2; simp_all
      have h3 := congrArg₂ (· + ·) h1 h2; simp at h3
      have : (a + f) + (c + d) = (e + b) + (c + d) := calc
        (a + f) + (c + d) = a + d + (c + f) := by abel
        _ = c + b + (e + d) := h3
        _ = (e + b) + (c + d) := by abel
      exact Nat.add_right_cancel this
    }

@[simp]
theorem PreInt.eq (a b c d:ℕ) : (⟨ a,b ⟩: PreInt) ≈ ⟨ c,d ⟩ ↔ a + d = c + b := by
  sorry

abbrev Int := Quotient PreInt.instSetoid

abbrev Int.formalDiff (a b:ℕ)  : Int := Quotient.mk PreInt.instSetoid ⟨ a,b ⟩

infix:100 " —— " => Int.formalDiff

theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b := by
  sorry

instance Int.decidableEq : DecidableEq Int := by
  intro a b
  have : ∀ (n:PreInt) (m: PreInt),
      Decidable (Quotient.mk PreInt.instSetoid n = Quotient.mk PreInt.instSetoid m) := by
    intro ⟨ a,b ⟩ ⟨ c,d ⟩
    rw [eq]
    exact decEq _ _
  exact Quotient.recOnSubsingleton₂ a b this

theorem Int.eq_diff (n:Int) : ∃ a b, n = a —— b := by
  sorry

instance Int.instAdd : Add Int where
  add := Quotient.lift₂ (fun ⟨ a, b ⟩ ⟨ c, d ⟩ ↦ (a+c) —— (b+d) ) (by
    intro ⟨ a, b ⟩ ⟨ c, d ⟩ ⟨ a', b' ⟩ ⟨ c', d' ⟩ h1 h2
    simp [eq] at *
    omega)

theorem Int.add_eq (a b c d:ℕ) : a —— b + c —— d = (a+c)——(b+d) := by
  sorry

theorem Int.mul_congr_left (a b a' b' c d : ℕ) (h: a —— b = a' —— b') :
    (a*c+b*d) —— (a*d+b*c) = (a'*c+b'*d) —— (a'*d+b'*c) := by
  sorry

theorem Int.mul_congr_right (a b c d c' d' : ℕ) (h: c —— d = c' —— d') :
    (a*c+b*d) —— (a*d+b*c) = (a*c'+b*d') —— (a*d'+b*c') := by
  sorry

theorem Int.mul_congr {a b c d a' b' c' d' : ℕ} (h1: a —— b = a' —— b') (h2: c —— d = c' —— d') :
  (a*c+b*d) —— (a*d+b*c) = (a'*c'+b'*d') —— (a'*d'+b'*c') := by
  sorry

instance Int.instMul : Mul Int where
  mul := Quotient.lift₂ (fun ⟨ a, b ⟩ ⟨ c, d ⟩ ↦ (a * c + b * d) —— (a * d + b * c)) (by
    intro ⟨ a, b ⟩ ⟨ c, d ⟩ ⟨ a', b' ⟩ ⟨ c', d' ⟩ h1 h2
    exact mul_congr (Quotient.eq.mpr h1) (Quotient.eq.mpr h2)
    )

theorem Int.mul_eq (a b c d:ℕ) : a —— b * c —— d = (a*c+b*d) —— (a*d+b*c) := by
  sorry

instance Int.instOfNat {n:ℕ} : OfNat Int n where
  ofNat := n —— 0

instance Int.instNatCast : NatCast Int where
  natCast n := n —— 0

theorem Int.ofNat_eq (n:ℕ) : ofNat(n) = n —— 0 := by
  sorry

theorem Int.natCast_eq (n:ℕ) : (n:Int) = n —— 0 := by
  sorry

@[simp]
theorem Int.natCast_ofNat (n:ℕ) : ((ofNat(n):ℕ): Int) = ofNat(n) := by
  sorry

@[simp]
theorem Int.ofNat_inj (n m:ℕ) : (ofNat(n) : Int) = (ofNat(m) : Int) ↔ ofNat(n) = ofNat(m) := by
  sorry

@[simp]
theorem Int.natCast_inj (n m:ℕ) : (n : Int) = (m : Int) ↔ n = m := by
  sorry

lemma Int.cast_eq_0_iff_eq_0 (n : ℕ) : (n : Int) = 0 ↔ n = 0 := by
  sorry

instance Int.instNeg : Neg Int where
  neg := Quotient.lift (fun ⟨ a, b ⟩ ↦ b —— a) (by sorry)

theorem Int.neg_eq (a b:ℕ) : -(a —— b) = b —— a := by
  sorry

abbrev Int.IsPos (x:Int) : Prop := ∃ (n:ℕ), n > 0 ∧ x = n
abbrev Int.IsNeg (x:Int) : Prop := ∃ (n:ℕ), n > 0 ∧ x = -n

theorem Int.trichotomous (x:Int) : x = 0 ∨ x.IsPos ∨ x.IsNeg := by
  sorry

theorem Int.not_pos_zero (x:Int) : x = 0 ∧ x.IsPos → False := by
  sorry

theorem Int.not_neg_zero (x:Int) : x = 0 ∧ x.IsNeg → False := by
  sorry

theorem Int.not_pos_neg (x:Int) : x.IsPos ∧ x.IsNeg → False := by
  sorry

instance Int.instAddGroup : AddGroup Int :=
  AddGroup.ofLeftAxioms (by sorry) (by sorry) (by sorry)

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

instance Int.instCommMonoid : CommMonoid Int where
  mul_comm := by sorry
  mul_assoc := by
    intro x y z
    obtain ⟨ a, b, rfl ⟩ := eq_diff x
    obtain ⟨ c, d, rfl ⟩ := eq_diff y
    obtain ⟨ e, f, rfl ⟩ := eq_diff z
    simp_rw [mul_eq]; congr 1 <;> ring
  one_mul := by sorry
  mul_one := by sorry

instance Int.instCommRing : CommRing Int where
  left_distrib := by sorry
  right_distrib := by sorry
  zero_mul := by sorry
  mul_zero := by sorry

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

theorem Int.sub_eq_formal_sub (a b:ℕ) : (a:Int) - (b:Int) = a —— b := by
  sorry

instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

instance Int.instLT : LT Int where
  lt n m := n ≤ m ∧ n ≠ m

theorem Int.le_iff (a b:Int) : a ≤ b ↔ ∃ t:ℕ, b = a + t := by
  sorry

theorem Int.lt_iff (a b:Int): a < b ↔ (∃ t:ℕ, b = a + t) ∧ a ≠ b := by
  sorry

instance Int.decidableRel : DecidableRel (· ≤ · : Int → Int → Prop) := by
  intro n m
  have : ∀ (n:PreInt) (m: PreInt),
      Decidable (Quotient.mk PreInt.instSetoid n ≤ Quotient.mk PreInt.instSetoid m) := by
    intro ⟨ a,b ⟩ ⟨ c,d ⟩
    change Decidable (a —— b ≤ c —— d)
    cases (a + d).decLe (b + c) with
      | isTrue h =>
        apply isTrue
        sorry
      | isFalse h =>
        apply isFalse
        sorry
  exact Quotient.recOnSubsingleton₂ n m this

lemma Int.is_additive_identity_iff_eq_0 (b : Int) : (∀ a, a = a + b) ↔ b = 0 := by
  sorry

instance Int.instLinearOrder : LinearOrder Int where
  le_refl := sorry
  le_trans := sorry
  lt_iff_le_not_ge := sorry
  le_antisymm := sorry
  le_total := sorry
  toDecidableLE := decidableRel
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Int.no_induction : ∃ P: Int → Prop, (P 0 ∧ ∀ n, P n → P (n+1)) ∧ ¬ ∀ n, P n := by
-- </TARGET_THEOREM>

end Section_4_1