The Fortress Language Specification - CiteSeerX

trait PartialOrderOperatorsT extends PartialOrderOperatorsT, ≺, , , ≻,CMP,
opr ≺,opr ,opr ,opr ≻,opr CMP
extends { StrictPartialOrderT, ≺, PartialOrderT, ,
PartialOrderT, , StrictPartialOrderT, ≻ }
opr CMP(self,other:T): Comparison
opr ≺(self,other: T): Boolean = (aCMP b) = LessThan
opr (self,other: T): Boolean = (aCMP b) = LessThan ∨ (aCMP b) = EqualTo
opr =(self,other: T): Boolean = (aCMP b) = EqualTo
opr (self,other: T): Boolean = (aCMP b) = GreaterThan ∨ (aCMP b) = EqualTo
opr ≻(self,other: T): Boolean = (aCMP b) = GreaterThan
property ∀(a:T, b: T) ((aCMP b) = LessThan) ↔ ((bCMP a) = GreaterThan)
property ∀(a:T, b: T) ((aCMP b) = EqualTo) ↔ ((bCMP a) = EqualTo)
property ∀(a:T, b: T) ((aCMP b) = Unordered) ↔ ((bCMP a) = Unordered)
property ∀(a:T, b: T) (a ≺ b) ↔ ((aCMP b) = LessThan)
property ∀(a:T, b: T) (a b) ↔ ((aCMP b) = LessThan ∨ (aCMP b) = EqualTo)
property ∀(a:T, b: T) (a = b) ↔ ((aCMP b) = EqualTo)
property ∀(a:T, b: T) (a b) ↔ ((aCMP b) = GreaterThan ∨ (aCMP b) = EqualTo)
property ∀(a:T, b: T) (a ≻ b) ↔ ((aCMP b) = GreaterThan)
end
For practical programming, we assume that partial order operators come in groups of five: a “less than” predicate, a
“less than or equal to” predicate, a “greater than or equal to” predicate, a “greater than” predicate, and a “comparison”
operator that returns one of the four values LessThan, EqualTo, GreaterThan, and Unordered. The trait
PartialOrderOperators declares such a set of five operators and describes the necessary algebraic constraints among
them and the equality predicate = . It also provides default definitions for the predicates (including = ) in terms of
the comparison operator.
trait TotalOrderOperatorsT extends TotalOrderOperatorsT, ≺, , , ≻,CMP,
opr ≺,opr ,opr ,opr ≻,opr CMP
extends { PartialOrderOperatorsT, ≺, , , ≻,CMP,
StrictTotalOrderT, ≺, TotalOrderT, ,
TotalOrderT, , StrictTotalOrderT, ≻ }
opr CMP(self,other:T): TotalComparison
end
Total order operators likewise come in groups of five: a “less than” predicate, a “less than or equal to” predicate, a
“greater than or equal to” predicate, a “greater than” predicate, and a “comparison” operator that returns one of the
three values LessThan, EqualTo, and GreaterThan. The trait TotalOrderOperators declares such a set of five
operators and describes the necessary algebraic constraints among them and the equality predicate = . For a total
order, the comparison operator CMP never returns Unordered.
trait PartialOrderBasedOnLET extends PartialOrderBasedOnLET, ≺, , , ≻,CMP,
opr ≺,opr ,opr ,opr ≻,opr CMP
extends { PartialOrderOperatorsT, ≺, , , ≻,CMP }
opr CMP(self,other:T): Comparison =
if a b then
if b a then EqualTo else LessThan end
else
if b a then GreaterThan else Unordered end
end
opr ≺(self,other: T): Boolean = (self other) ∧ ¬(other self)
opr =(self,other: T): Boolean = (self other) ∧ (other self)
opr (self,other: T): Boolean = (other self)
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