The Fortress Language Specification - CiteSeerX

Proof. For the purpose of contradiction suppose there are two declarations f (P) and f (Q) in σ ′ . Since both f (P) and
f (Q) are applicable with coercion to f (A) and |Σ| = 0 there must exist a coercion from some type P ′ to P and a
coercion from some type Q ′ to Q such that A ≼ P ′ ∩ Q ′ . Therefore it is not the case that P Q. By the overloading
restrictions, P ≠ Q and either P ⊳ Q or Q ⊳ P or for all P ′ ∈ S and Q ′ ∈ T either P ′ ♦ Q ′ , or there is a declaration
f (P ′ ∩Q ′ ) visible at Z. If P ⊳ Q or Q ⊳ P then we contradict our assumption. Otherwise, if there exists a declaration
f (P ′ ∩ Q ′ ) visible at Z then this declaration is applicable to f (A) without coercion. This contradicts |Σ| = 0. If such
a declaration does not exist then it must be the case that P ′ ♦ Q ′ . Then both f (P) and f (Q) can not be applicable to
the call f (A) which is a contradiction.
Theorem 1. If |Σ| = 0 and |Σ ′ | ≠ 0 then |σ ′ | = 1.
Proof. Follows from Lemmas 2 and 3.
B.2 Proof of Overloading Resolution for Functions
This section proves that the restrictions placed on overloaded function declarations are sufficient to guarantee no
undefined nor ambiguous call at run time (the case for methods is analogous).
Consider a static function call f (A) at some program point Z and its corresponding dynamic function call f (X). Let
∆ be the set of parameter types of function declarations of f that are visible at Z and applicable to the dynamic call
f (X). Let Σ be the set of parameter types of function declarations of f that are visible at Z and applicable to the static
call f (A). Moreover, let σ be the subset of Σ for which no type in Σ is more specific and let δ be the subset of ∆ for
which no type in ∆ is more specific:
σ = {S ∈ Σ | ¬∃S ′ ∈ Σ : S ′ ≺ S }
δ = {D ∈ ∆ | ¬∃D ′ ∈ ∆ : D ′ ≺ D }.
Below we prove:
|Σ| ≠ 0 implies |σ| = 1, and
|Σ| ≠ 0 implies |δ| = 1.
Informally, if any declaration is applicable to a static call then there exists a single most specific declaration that is
applicable to the static call and a single most specific declaration that is applicable to the corresponding dynamic call.
Lemma 4. Σ ⊆ ∆.
Proof. Notice that X ≼ A by type soundness. If f (P) is applicable to the call f (A) then A ≼ P. Notice that X ≼ A
implies X ≼ P. Therefore f (P) is applicable to the call f (X).
Lemma 5. If |∆| ≥ 1 then |δ| ≥ 1. Also, if |Σ| ≥ 1 then |σ| ≥ 1.
Proof. Follows from Lemma 1 where S is the set of all types, A is ∆ and Σ respectively, and the relation ≺ is acyclic
and irreflexive.
Lemma 6. If |Σ| ≥ 1 then |δ| ≥ 1.
Proof. Follows from Lemmas 4 and 5.
Lemma 7. |σ| ≤ 1. Also, |δ| ≤ 1.
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