The Fortress Language Specification - CiteSeerX

Chapter 25
Numbers
25.1 Rational Numbers
The trait Q ( QQ ) encompasses all finite rational numbers, the result of dividing any integer by any nonzero integer.
The trait Q ∗ ( QQ_star ) is Q with two extra elements, +∞ and −∞ . The trait Q # ( QQ_splat ) is Q ∗ with
one additional element, the indefinite rational (written 0/0 ), which is used as the result of dividing zero by zero or of
adding −∞ to +∞ .
Often it is desirable to indicate that a variable ranges over only a subset of the rationals, such as only positive values
or only nonnegative values or only nonzero values. Unfortunately, traditional notations such as Q + are not used
consistently in the literature; one author may use Q + to mean the set of strictly positive rationals and another may
use it to mean the set of nonnegative rationals. Fortress therefore uses a notation that is novel but unambiguous:
Q ( QQ ) is the set of rationals (it is a subtype of R and Q ∗ ).
Q < ( QQ_LT ) is the set of strictly negative rationals (it is a subtype of R < , Q ∗ < , Q , Q ≤ , and Q≠ ).
Q ≤ ( QQ_LE ) is the set of nonpositive rationals, that is, Q < ∪ {0} (it is a subtype of R ≤ , Q ∗ ≤ , and Q ).
Q ≥ ( QQ_GE ) is the set of nonnegative rationals, that is, Q > ∪ {0} (it is a subtype of R ≥ , Q ∗ ≥ , and Q ).
Q > ( QQ_GT ) is the set of strictly positive rationals (it is a subtype of R > , Q ∗ > , Q , Q ≥ , and Q≠ ).
Q≠ ( QQ_NE ) is the set of strictly nonzero rationals (that is, Q < ∪ Q > ) (it is a subtype of R≠ , Q ∗
≠
, and Q ).
Q ∗ ( QQ_star ) is Q with extra elements +∞ and −∞ (it is a subtype of R ∗ and Q # ).
Q ∗ < ( QQ_star_LT ) is Q < with extra element −∞ (it is a subtype of R ∗ < , Q # < , Q ∗ , Q ∗ ≤ , and Q∗ ≠ ).
Q ∗ ≤ ( QQ_star_LE ) is Q ≤ with extra element −∞ (it is a subtype of R ∗ ≤ , Q# ≤ , and Q∗ ).
Q ∗ ≥ ( QQ_star_GE ) is Q ≥ with extra element +∞ (it is a subtype of R ∗ ≥ , Q# ≥ , and Q∗ ).
Q ∗ > ( QQ_star_GT ) is Q > with extra element +∞ (it is a subtype of R ∗ > , Q # > , Q ∗ , Q ∗ ≥ , and Q∗ ≠ ).
Q ∗
≠ ( QQ_star_NE ) is Q ≠ with extra elements +∞ and −∞ (it is a subtype of R ∗
≠ , Q# ≠ , and Q∗ ).
Q # ( QQ_splat ) is Q ∗ with extra element 0/0 (it is a subtype of R # ).
Q # < ( QQ_splat_LT ) is Q ∗ < with extra element 0/0 (it is a subtype of R # < , Q # , Q # ≤
, and Q#
≠ ).
Q # ≤ ( QQ_splat_LE ) is Q∗ ≤ with extra element 0/0 (it is a subtype of R# ≤ and Q# ).
Q # ≥ ( QQ_splat_GE ) is Q∗ ≥ with extra element 0/0 (it is a subtype of R# ≥ and Q# ).
Q # > ( QQ_splat_GT ) is Q ∗ > with extra element 0/0 (it is a subtype of R # > , Q # , Q # ≥
, and Q#
≠ ).
Q #
≠
( QQ_splat_NE ) is Q∗ ≠
with extra element 0/0 (it is a subtype of R#
≠ and Q# ).
The Fortress type system tracks these types closely through various arithmetic operations; for example, adding two
values of type Q > produces a result of type Q > , and adding a value of type Q ∗ > and a value of type Q ≥ produces
a value of type Q ∗ ≥ . 193