introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

/-!
Self-contained extraction for Section 4.1 integers, sufficient for the target theorem.
-/

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

/-- Definition 4.1.2 (Multiplication of integers) -/
instance Int.instOfNat {n:ℕ} : OfNat Int n where
  ofNat := n —— 0

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

/-- Definition 4.1.4 (Negation of integers) / Exercise 4.1.2 -/
instance Int.instNeg : Neg Int where
  neg := Quotient.lift (fun ⟨ a, b ⟩ ↦ b —— a) (by sorry)

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Lemma 4.1.11(d) (Negation reverses order) / Exercise 4.1.7 -/
theorem Int.neg_ge_neg {a b:Int} (h: b ≤ a) : -a ≤ -b := by
end Section_4_1
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Section_4_1

/-- Lemma 4.1.11(d) (Negation reverses order) / Exercise 4.1.7,
now stated for the standard integers `ℤ` with the usual order. -/
theorem Int.neg_ge_neg {a b : ℤ} (h : b ≤ a) : -a ≤ -b := by
end Section_4_1

end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

/-- Definition 4.1.2 (Definition of addition) -/
theorem Int.add_eq (a b c d:ℕ) : a —— b + c —— d = (a+c)——(b+d) := Quotient.lift₂_mk _ _ _ _

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

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

/-- Definition 4.1.4 (Negation of integers) / Exercise 4.1.2 -/
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 := rfl

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Definition 4.1.10 (Ordering of the integers) -/
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 rfl

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

/-- Lemma 4.1.11(d) (Negation reverses order) / Exercise 4.1.7 -/
theorem Int.neg_gt_neg {a b:Int} (h: b < a) : -a < -b := by
end Section_4_1
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

/-- Definition 4.1.2 (Definition of addition) -/
theorem Int.add_eq (a b c d:ℕ) : a —— b + c —— d = (a+c)——(b+d) := Quotient.lift₂_mk _ _ _ _

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

/-- Exercise 4.1.8 -/
theorem Int.no_induction : ∃ P: Int → Prop, (P 0 ∧ ∀ n, P n → P (n+1)) ∧ ¬ ∀ n, P n := by
end Section_4_1
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Int.no_induction :
  ∃ P : Int → Prop,
    (P 0 ∧ ∀ n : Int, P n → P (n + 1)) ∧ ¬ ∀ n : Int, P n := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

/-!
# Analysis I, Section 4.1: The integers

(Extracted minimal context for the target theorem.)
-/

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Decidability of equality -/
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

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq_diff (n:Int) : ∃ a b, n = a —— b := by apply n.ind _; intro ⟨ a, b ⟩; use a, b

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

/-- Definition 4.1.2 (Definition of addition) -/
theorem Int.add_eq (a b c d:ℕ) : a —— b + c —— d = (a+c)——(b+d) := Quotient.lift₂_mk _ _ _ _

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

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

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Definition 4.1.10 (Ordering of the integers) -/
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 rfl

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

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7 -/
theorem Int.not_gt_and_eq (a b:Int) : ¬ (a > b ∧ a = b):= by
end Section_4_1
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7 -/
theorem Int.not_gt_and_eq (a b : Int) : ¬ (a > b ∧ a = b) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

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

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLT : LT Int where
  lt n m := n ≤ m ∧ n ≠ m

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7 -/
theorem Int.not_gt_and_lt (a b:Int) : ¬ (a > b ∧ a < b):= by
end Section_4_1
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

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

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

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Definition 4.1.10 (Ordering of the integers) -/
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 rfl


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

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7 -/
theorem Int.not_lt_and_eq (a b:Int) : ¬ (a < b ∧ a = b):= by
end Section_4_1
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

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

/-- Provide a coercion from ℕ to Section_4_1.Int to allow writing `x = m` for `m : ℕ`. -/
instance Aux_coeNatToSectionInt : Coe Nat Section_4_1.Int where
  coe n := (n : Section_4_1.Int)

/-- Lemma 4.1.3 (Multiplication well-defined) -/
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
  simp only [eq] at *
  calc
    _ = c*(a+b') + d*(a'+b) := by ring
    _ = c*(a'+b) + d*(a+b') := by rw [h]
    _ = _ := by ring

/-- Lemma 4.1.3 (Multiplication well-defined) -/
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
  simp only [eq] at *
  calc
    _ = a*(c+d') + b*(c'+d) := by ring
    _ = a*(c'+d) + b*(c+d') := by rw [h]
    _ = _ := by ring

/-- Lemma 4.1.3 (Multiplication well-defined) -/
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
  rw [mul_congr_left a b a' b' c d h1, mul_congr_right a' b' c d c' d' h2]

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)
    )

/-- Exercise 4.1.9 -/
theorem Int.sq_nonneg' (n:Int) : ∃ (m:Nat), n*n = m := by
end Section_4_1
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Section_4_1

/-- Exercise 4.1.9 (Squares of integers are nonnegative) -/
theorem Int.sq_nonneg' (n : ℤ) : ∃ m : ℕ, n * n = m := by
end Section_4_1

end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

namespace Section_4_1

structure PreInt where
  minuend : ℕ
  subtrahend : ℕ

/-- Definition 4.1.1 -/
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
      -- This proof is written to follow the structure of the original text.
      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 rfl

abbrev Int := Quotient PreInt.instSetoid

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

infix:100 " —— " => Int.formalDiff

/-- Definition 4.1.1 (Integers) -/
theorem Int.eq (a b c d:ℕ): a —— b = c —— d ↔ a + d = c + b :=
  ⟨ Quotient.exact, by intro h; exact Quotient.sound h ⟩

/-- Lemma 4.1.3 (Addition well-defined) -/
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)

/-- Definition 4.1.2 (Definition of addition) -/
theorem Int.add_eq (a b c d:ℕ) : a —— b + c —— d = (a+c)——(b+d) := Quotient.lift₂_mk _ _ _ _

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

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

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLE : LE Int where
  le n m := ∃ a:ℕ, m = n + a

/-- Definition 4.1.10 (Ordering of the integers) -/
instance Int.instLT : LT Int where
  lt n m := n ≤ m ∧ n ≠ m

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7 -/
theorem Int.trichotomous' (a b:Int) : a > b ∨ a < b ∨ a = b := by
end Section_4_1
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 4.1.11(f) (Order trichotomy) / Exercise 4.1.7,
stated for the standard integers in Mathlib. -/
theorem Int.trichotomous' (a b : Int) :
    a > b ∨ a < b ∨ a = b := by
end TaoBench
import Mathlib.Data.Int.Basic

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
    }

end Section_4_2
import Mathlib

namespace Section_4_2

-- <CONTEXT>
structure PreRat where
  numerator : ℤ
  denominator : ℤ
  nonzero : denominator ≠ 0
-- </CONTEXT>
-- <TARGET_THEOREM>
instance PreRat.instSetoid : Setoid PreRat := by
-- </TARGET_THEOREM>

end Section_4_2
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
instance PreRat.instSetoid :
    Equivalence
      (fun a b : {p : ℤ × ℤ // p.2 ≠ 0} =>
        a.1.1 * b.1.2 = b.1.1 * a.1.2) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms

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]

/-- Provide a zero element for Rat. -/
instance Aux_zero_Rat : Zero Rat where
  zero := (0:ℤ) // 1

/-- 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 (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
)

end Section_4_2