The Fortress Language Specification - CiteSeerX

property ∀(a:T) ¬(a ∼ a)
end
An irreflexive predicate always returns false when its operands are the same. (Note that it is possible for a predicate
to be neither reflexive nor irreflexive.)
trait SymmetricT extends SymmetricT, ∼,opr ∼
extends { BinaryPredicateT, ∼ }
property ∀(a:T, b: T) (a ∼ b) ↔ (b ∼ a)
end
A symmetric predicate doesn’t care in which order its arguments are presented; the result is the same either way.
trait TransitiveT extends TransitiveT, ∼,opr ∼
extends { BinaryPredicateT, ∼ }
property ∀(a:T, b: T, c:T) ((a ∼ b) ∧ (b ∼ c)) → (a ∼ c)
end
A transitive predicate has the property that if a is related to b and b is related to c , then a is related to c .
trait EquivalenceRelationT extends EquivalenceRelationT, ∼,opr ∼
extends { ReflexiveT, ∼, SymmetricT, ∼, TransitiveT, ∼ }
end
An equivalence relation is any predicate that is reflexive, symmetric, and transitive. You can think of an equivalence
relation as describing a way to separate a set of items into categories, such that each item belongs to exactly one
category; the predicate is true of two items if and only if they are in the same category.
trait IdentityEqualityT extends IdentityEqualityT
extends { EquivalenceRelationT, = }
opr =(self,other: T): Boolean = (self = other)
end
This trait provides a concrete implementation of the operator = (defining it to behave the same as the operator = )
and states that = is an equivalence relation over instances of the type T .
trait UnaryPredicateSubstitutionLawsT extends UnaryPredicateSubstitutionLawsT, ∼, ≃,
opr ∼,opr ≃
extends { UnaryPredicateT, ∼, BinaryPredicateT, ≃ }
property ∀(a:T, a ′ : T) (a ≃ a ′ ) →:((∼ a) ↔ (∼ a ′ ))
end
This handy trait states that the unary predicate ∼ is consistent under substitutions described by the relation ≃ (which
is typically, but not always, an equivalence relation); that is, the result produced by ∼ is unchanged if its argument is
replaced by some other value that is equivalent.
trait BinaryPredicateSubstitutionLawsT extends BinaryPredicateSubstitutionLawsT, ∼, ≃,
opr ∼,opr ≃
extends { BinaryPredicateT, ∼, BinaryPredicateT, ≃ }
property ∀(a:T, a ′ : T) (a ≃ a ′ ) →: ∀(b: T) (a ∼ b) ↔ (a ′ ∼ b)
property ∀(b: T, b ′ : T) (b ≃ b ′ ) →: ∀(a: T) (a ∼ b) ↔ (a ∼ b ′ )
end
This equally handy trait states that the binary predicate ∼ is consistent under substitutions described by the relation
≃ (which is typically, but not always, an equivalence relation); that is, the result produced by ∼ is unchanged if either
247