Method Subtyping Sample Clauses

Method Subtyping. Method types are of the following form: <T¯ α¯ > req F; ens G; U m(V0 ı0;V¯ ı¯) In method types, we make the self-parameter explicit, separated from the other formal parameters by a semicolon. In the scheme above, ı0 is the self-parameter. Before presenting the method subtyping rule in full generality, we present its in- stance for method types without logic parameters: U,V0,V¯ j <: U j,V0j,V¯ Γ, ı0 : V0, ı¯ : V¯ j; ı0; true € Fj -* (F *(fa U result)(G -* Gj)) Γ € req F; ens G; U m(V0 ı0;V¯ ı¯) <: req Fj; xxx Xx; U x x(V0j ı0;V¯ j ı¯) This rule has the following two derived rules (where types are elided): € Fj -* F € Gj -* G € req F; ens G <: req Fj; xxx Xx € req F; ens G <: req F * H; ens G * H The first of these derived rules is standard behavioural subtyping, the second one ab- stracts separation logic’s frame rule. In order to see that these two rules follow from the above rule, note that the following two formulas are tautologies (as can be easily proven by natural deduction): (Fj -* F)* H -* Fj -* F * H F * H -* F *(fa U x)(G -* G * H) The general method subtyping rule also accounts for logic parameters:8 m run T¯ j,U,V0,V¯ j <: T¯ ,U j,V0j,V¯ Γ, ı0 : V0; ı0; true € (fa T¯ j α¯ )(fa V¯ j ı¯)(Fj -* (ex W¯ α¯ j)(F *(fa U result)(G -* Gj))) Γ € <T¯ α¯ ,W¯ α¯ j> req F; ens G; U m(V0 ı0;V¯ ı¯) <: <T¯ j α¯ > req Fj; xxx Xx; U x x(V0j ı0;V¯ j ı¯) Note that the subtype may have more logic parameters than the supertype. For instance, we obtain the following derived rule: € <T α> req F; ens G <: req (ex T α)(F); ens (ex T α)(G) This derived rule is an abstraction of separation logic’s auxiliary variable rule. It follows from the method subtyping rule by the following tautology: (ex T α)(F) -* (ex T α)(F *(fa U x)(G -* (ex T α)(G)))
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Related to Method Subtyping

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