introvoyz041/TaoAnalysisBench huggingface dataset:
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
-- </CONTEXT>
-- <TARGET_THEOREM>
instance Rat.addGroup_inst : AddGroup Rat := 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

/-
Tao’s construction builds the rationals as a quotient of integer pairs
and then proves that this type `Rat` is an additive group. In Mathlib,
the rationals are already available as `ℚ` with their usual operations.
So the corresponding Mathlib-only statement is simply that `ℚ` carries
an `AddGroup` structure.
-/
instance Rat.addGroup_inst : AddGroup ℚ := 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

/-- 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
  )

/-- 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)

/-- 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)

/-- Provide a subtraction instance from `Add` and `Neg` so that `x - y` parses. -/
instance Aux_Rat_instSub : Sub Rat where
  sub a b := a + (-b)

/-- Embedding the integers in the rationals -/
instance Rat.instIntCast : IntCast Rat where
  intCast a := a // 1

/--
  Whereas the book leaves the inverse of 0 undefined, it is more convenient in Lean to assign a
  "junk" value to this inverse; we arbitrarily choose this junk value to be 0.
-/
instance Rat.instInv : Inv Rat where
  inv := Quotient.lift (fun ⟨ a, b, h1 ⟩ ↦ b // a) (by
    sorry -- hint: split into the `a=0` and `a≠0` cases
)

/-- Auxiliary: provide a bare `Div` instance so that `/` can be used without a full
`DivInvMonoid`. This does not appear in the textbook and is only to make the snippet compile. -/
instance Aux_Rat_instDiv : Div Rat where
  div a b := a * b⁻¹

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Definition 4.2.6 (negativity) -/
def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

/-- Definition 4.2.8 (Ordering of the rationals) -/
instance Rat.instLT : LT Rat where
  lt x y := (x-y).isNeg

/-- Proposition 4.2.9(b) (order is anti-symmetric) / Exercise 4.2.5 -/
theorem Rat.antisymm (x y:Rat) : x < y ↔ y > x := by
end Section_4_2
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 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

/-- 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)

/-- 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

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

namespace TaoBench

/-- Proposition 4.2.9(e) (positive multiplication preserves order) / Exercise 4.2.5,
rephrased using Mathlib's standard order on `ℚ`. -/
theorem Rat.mul_lt_mul_right {x y z : ℚ} (hxy : x < y) (hz : 0 < z) :
    x * z < y * z := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 4.2.6 (Tao, Analysis I, Section 4.2).  
If `x < y` and `z < 0` in `ℚ`, then right-multiplication by `z` reverses the inequality. -/
theorem Rat.mul_lt_mul_right_of_neg (x y z : ℚ) (hxy : x < y) (hz : z < 0) :
    x * z > y * z := by
end TaoBench
import Mathlib.Tactic

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
    }

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

/-- 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)

/-- Embedding the integers in the rationals -/
instance Rat.instIntCast : IntCast Rat where
  intCast a := a // 1

-- Auxiliary: a minimal subtraction to interpret `x - y` in the order definition below.
instance Aux_instSubRat : Sub Rat where
  sub x y := x   -- dummy; only needed so `(x - y)` typechecks in `instLT`

-- Auxiliary: a minimal division to interpret `a / b` (with `a b : Rat`) in `Rat.isPos`.
instance Aux_instDivRat : Div Rat where
  div x y := x   -- dummy; only needed so `a/b` typechecks in `Rat.isPos`

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Definition 4.2.6 (negativity) -/
def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

/-- Definition 4.2.8 (Ordering of the rationals) -/
instance Rat.instLT : LT Rat where
  lt x y := (x-y).isNeg

/-- Proposition 4.2.9(a) (order trichotomy) / Exercise 4.2.5 -/
theorem Rat.not_gt_and_eq (x y:Rat) : ¬ (x > y ∧ x = y):= by
end Section_4_2
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]

/-- 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
  )

/-- 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)

/-- Auxiliary: define subtraction directly from addition and negation (for this snippet). -/
instance Aux_instHSubRat : HSub Rat Rat Rat where
  hSub x y := x + (-y)

/-- Embedding the integers in the rationals -/
instance Rat.instIntCast : IntCast Rat where
  intCast a := a // 1

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Definition 4.2.6 (negativity) -/
def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

/-- Definition 4.2.8 (Ordering of the rationals) -/
instance Rat.instLT : LT Rat where
  lt x y := (x-y).isNeg

/-- Definition 4.2.8 (Ordering of the rationals) -/
instance Rat.instLE : LE Rat where
  le x y := (x < y) ∨ (x = y)

/-- Definition of subtraction -/
@[simp] theorem Rat.sub_eq (a b:Rat) : a - b = a + (-b) := by rfl

/-- Proposition 4.2.9(a) (order trichotomy) / Exercise 4.2.5 -/
theorem Rat.not_lt_and_eq (x y:Rat) : ¬ (x < y ∧ x = y):= by
end Section_4_2
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]

/-- 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)

/-- Embedding the integers in the rationals -/
instance Rat.instIntCast : IntCast Rat where
  intCast a := a // 1

/-- Provide a minimal division structure on Rat to parse `a / b` in `isPos`. -/
instance Aux_instDivRat : Div Rat where
  div a b := a

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Definition 4.2.6 (negativity) -/
def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4 -/
theorem Rat.not_pos_and_neg (x:Rat) : ¬(x.isPos ∧ x.isNeg) := by
end Section_4_2
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4.
In Mathlib terms: a rational number cannot be both positive and negative. -/
theorem Rat.not_pos_and_neg (x : ℚ) : ¬ (0 < x ∧ x < 0) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4 (Mathlib version).

In Tao's development, `x.isNeg` means “there exists a positive rational `r`
(with positive integer numerator and denominator) such that `x = -r`”.
This is mathematically equivalent, in the usual ℚ, to the condition `x < 0`.

So the original statement `¬(x = 0 ∧ x.isNeg)` becomes the standard fact
that no rational can be both zero and strictly negative. -/
theorem Rat.not_zero_and_neg (x : Rat) : ¬ (x = 0 ∧ x < 0) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Algebra.Group.MinimalAxioms
import Mathlib.Data.Rat.Defs

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
    }

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

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

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

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4 -/
theorem Rat.not_zero_and_pos (x:Rat) : ¬(x = 0 ∧ x.isPos) := by
end Section_4_2
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

/-- 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)

/-- 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

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

/-- Definition 4.2.6 (positivity) -/
def Rat.isPos (q:Rat) : Prop := ∃ a b:ℤ, a > 0 ∧ b > 0 ∧ q = a/b

/-- Definition 4.2.6 (negativity) -/
def Rat.isNeg (q:Rat) : Prop := ∃ r:Rat, r.isPos ∧ q = -r

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4 -/
theorem Rat.trichotomous (x:Rat) : x = 0 ∨ x.isPos ∨ x.isNeg := by
end Section_4_2
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Section_4_2

/-- Lemma 4.2.7 (trichotomy of rationals) / Exercise 4.2.4 (Tao, Analysis I). -/
theorem Rat.trichotomous (x : ℚ) : x = 0 ∨ 0 < x ∨ x < 0 := by
end Section_4_2

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

namespace TaoBench

/-- Trichotomy of the order on the rational numbers (Tao's `Rat.trichotomous'`). -/
theorem Rat.trichotomous' (x y : ℚ) : x > y ∨ x < y ∨ x = y := by
end TaoBench
import Mathlib

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

namespace Section_4_3

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_add (x y:ℚ) : |x + y| ≤ |x| + |y| := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 4.3.3(b) / Exercise 4.3.1 -/
theorem abs_add (x y : ℚ) : |x + y| ≤ |x| + |y| := by
end TaoBench
import Mathlib

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

namespace Section_4_3

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_eq_zero_iff (x : ℚ) : |x| = 0 ↔ x = 0 := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Rat.Basic

namespace Section_4_3

/-- Proposition 4.3.3(c) / Exercise 4.3.1 -/
theorem abs_le_iff (x y:ℚ) : -y ≤ x ∧ x ≤ y ↔ |x| ≤ y := by
end Section_4_3
import Mathlib

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

namespace Section_4_3

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_le_iff (x y:ℚ) : -y ≤ x ∧ x ≤ y ↔ |x| ≤ y := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Section_4_3

/-- Proposition 4.3.3(c) / Exercise 4.3.1 -/
theorem abs_le_iff (x y : ℚ) : -y ≤ x ∧ x ≤ y ↔ |x| ≤ y := by
end Section_4_3

end TaoBench
import Mathlib

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

namespace Section_4_3

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_mul (x y:ℚ) : |x * y| = |x| * |y| := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 4.3.3(d) / Exercise 4.3.1 -/
theorem abs_mul (x y : ℚ) : |x * y| = |x| * |y| := by
end TaoBench
import Mathlib

namespace Section_4_3

-- <CONTEXT>
def _root_.Rat.Close (ε : ℚ) (x y:ℚ) := |x-y| ≤ ε

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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_neg (x:ℚ) : |-x| = |x| := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib

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

namespace Section_4_3

-- <CONTEXT>
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
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem abs_nonneg (x: ℚ) : |x| ≥ 0 := by
-- </TARGET_THEOREM>

end Section_4_3
import Mathlib.Tactic

/--
  This definition needs to be made outside of the Section 4.3 namespace for technical reasons.
-/
def Rat.Close (ε : ℚ) (x y:ℚ) := |x-y| ≤ ε

namespace Section_4_3

/-- Proposition 4.3.7(d) / Exercise 4.3.2 -/
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
end Section_4_3