By Appel A.W., et al.

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The proof is longer than the program! Checking such a proof by hand might miss some errors. Automating the application of separation logic in a proof assistant ensures that there are no gaps in the proof. Better yet, perhaps parts of the construction, not just the checking, can be automated. Let us examine some of the key points in the proof. Just before the σ while loop (line 3), we have {w = 0 ∧ v 0}, that is, the initialization of w and the program precondition that the sequence σ is represented by the list starting at pointer v.

The discrete SA is useful for constructing SAs over tuples where only some of the components have interesting joins; the other components can be turned into discrete SAs. Note the compositionality of this construction using the discrete and product operators together. 7. OPERATORS 47 ON SEPARATION ALGEBRAS The discrete-SA construction is incompatible with the Sing-alg axiom (single-unit SAs). With single-unit SAs, one would instead have to “manually” construct an appropriate tupling operator. This is an important reason not to assume the single-unit axiom for all separation algebras.

For suppose we have two 7. OPERATORS 46 ON SEPARATION ALGEBRAS single-unit SAs 〈A, JA〉, 〈B, JB 〉. 5) Here A + B is the disjoint union of A and B with inl and inr as the left and right injections. This structure cannot be a single-unit separation algebra, for if uA and uB are the units of A and B, then core(inl uA) and core(inr uB ) must be equal, by the_unit_core. However, this is a contradiction as the injection functions for disjoint union always produce unequal elements. ) INDEXED SUM OPERATOR.

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