The Fortress Language Specification - CiteSeerX

Appendix B
Overloaded Functional Declarations
As mentioned in Chapter 33, this appendix proves that the restrictions discussed in Chapter 33 guarantee no undefined
nor ambiguous call at run time.
B.1 Proof of Coercion Resolution for Functions
This section proves that the restrictions discussed in the previous sections guarantee the static resolution of coercion
(described in Section 17.5) is well defined for functions (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
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).
Let Σ ′ be the set of parameter types of function declarations of f that are visible at Z and applicable with coercion to
the static call f (A). Moreover, let σ ′ be the subset of Σ ′ for which no type in Σ ′ is more specific:
σ ′ = {S ∈ Σ ′ | ¬∃S ′ ∈ Σ ′ : S ′ ⊳ S }.
We prove the following:
|Σ| = 0 and |Σ ′ | ≠ 0 imply |σ ′ | = 1.
Informally, if no declaration is applicable to a static call but there is a declaration that is applicable with coercion then
there exists a single most specific declaration that is applicable with coercion to the static call.
Lemma 1. Given an acyclic, irreflexive binary relation R on a set S, and a finite nonempty subset A of S, the set
{a ∈ A|¬∃a ′ ∈ A : (a ′ , a) ∈ R} is nonempty.
Proof. Consider the relation R on S as a directed acyclic graph. Let A represent a subgraph. Then the Lemma amounts
to proving that there exists a node in the graph represented by A with no edges pointing to it. This follows from the
fact that A is finite and the graph is acyclic.
Lemma 2. If |Σ ′ | ≥ 1 then |σ ′ | ≥ 1.
Proof. Follows from Lemma 1 where S is the set of all types, A is Σ ′ , and the relation ⊳ is acyclic and irreflexive.
Lemma 3. If |Σ| = 0 then |σ ′ | ≤ 1.
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