add

formal source

 143instance : Neg LogicRat := ⟨neg⟩
 144
 145/-- Addition: `(a/b) + (c/d) = (a*d + c*b) / (b*d)`. -/
 146def add : LogicRat → LogicRat → LogicRat :=
 147  Quotient.lift₂
 148    (fun (p q : PreRat) =>
 149       mk (p.num * q.den + q.num * p.den) (p.den * q.den)
 150         (fun h => p.den_nonzero ((mul_eq_zero.mp h).resolve_right q.den_nonzero)))
 151    (by
 152      rintro ⟨a, b, hb⟩ ⟨c, d, hd⟩ ⟨a', b', hb'⟩ ⟨c', d', hd'⟩ hab hcd
 153      show mk (a * d + c * b) (b * d) _
 154            = mk (a' * d' + c' * b') (b' * d') _
 155      apply sound
 156      show (a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)
 157      rw [eq_iff_toInt_eq]
 158      simp only [toInt_add, toInt_mul]
 159      have hab' : toInt a * toInt b' = toInt a' * toInt b := by
 160        have := congrArg toInt hab
 161        rwa [toInt_mul, toInt_mul] at this
 162      have hcd' : toInt c * toInt d' = toInt c' * toInt d := by
 163        have := congrArg toInt hcd
 164        rwa [toInt_mul, toInt_mul] at this
 165      linear_combination (toInt d * toInt d') * hab' + (toInt b * toInt b') * hcd')
 166
 167instance : Add LogicRat := ⟨add⟩
 168
 169/-- Multiplication: `(a/b) * (c/d) = (a*c) / (b*d)`. -/
 170def mul : LogicRat → LogicRat → LogicRat :=
 171  Quotient.lift₂
 172    (fun (p q : PreRat) =>
 173       mk (p.num * q.num) (p.den * q.den)
 174         (fun h => p.den_nonzero ((mul_eq_zero.mp h).resolve_right q.den_nonzero)))
 175    (by
 176      rintro ⟨a, b, hb⟩ ⟨c, d, hd⟩ ⟨a', b', hb'⟩ ⟨c', d', hd'⟩ hab hcd