Early in the development of Hoare logic, Owicki and Gries introduced auxiliary variables as a way of encoding information about the history of a program's execution that is useful for verifying its correctness. Over a decade later, Abadi and Lamport observed that it is sometimes also necessary to know in advance what a program will do in the future. To address this need, they proposed prophecy variables, originally as a proof technique for refinement mappings between state machines. However, despite the fact that prophecy variables are a clearly useful reasoning mechanism, there is (surprisingly) almost no work that attempts to integrate them into Hoare logic. In this paper, we present the first account of prophecy variables in a Hoare-style program logic that is flexible enough to verify logical atomicity (a relative of linearizability) for classic examples from the concurrency literature like RDCSS and the Herlihy-Wing queue. Our account is formalized in the Iris framework for separation logic in Coq. It makes essential use of ownership to encode the exclusive right to resolve a prophecy, which in turn lets us enforce soundness of prophecies with a very simple set of proof rules.
When using a proof assistant to reason in an embedded logic -- like separation logic -- one cannot benefit from the proof contexts and basic tactics of the proof assistant. This results in proofs that are at a too low level of abstraction because they are cluttered with bookkeeping code related to manipulating the object logic. In this paper, we introduce a so-called proof mode that extends the Coq proof assistant with (spatial and non-spatial) named proof contexts for the object logic. We show that thanks to these contexts we can implement high-level tactics for introduction and elimination of the connectives of the object logic, and thereby make reasoning in the embedded logic as seamless as reasoning in the meta logic of the proof assistant. We apply our method to Iris: a state of the art higher-order impredicative concurrent separation logic. We show that our method is very general, and is not just limited to program verification. We demonstrate its generality by formalizing correctness proofs of fine-grained concurrent algorithms, derived constructs of the Iris logic, and a unary and binary logical relation for a language with concurrency, higher-order store, polymorphism, and recursive types. This is the first formalization of a binary logical relation for such an expressive language. We also show how to use the logical relation to prove contextual refinement of fine-grained concurrent algorithms.
We present a logical relations model of a higher-order functional programming language with impredicative polymorphism, recursive types, and a Haskell-style ST monad type with runST. We use our logical relations model to show that runST provides proper encapsulation of state, by showing that effectful computations encapsulated by runST are heap independent. Furthermore, we show that contextual refinements and equivalences that are expected to hold for pure computations do indeed hold in the presence of runST. This is the first time such relational results have been proven for a language with monadic encapsulation of state. We have formalized all the technical development and results in Coq.
A number of tools have been developed for carrying out separation-logic proofs mechanically using an interactive proof assistant. One of the most advanced such tools is the Iris Proof Mode (IPM) for Coq, which offers a rich set of tactics for making separation-logic proofs look and feel like ordinary Coq proofs. However, IPM is tied to a particular separation logic (namely, Iris), thus limiting its applicability. In this paper, we propose MoSeL, a general and extensible Coq framework that brings the benefits of IPM to a much larger class of separation logics. Unlike IPM, MoSeL is applicable to both affine and linear separation logics (and combinations thereof), and provides generic tactics that can be easily extended to account for the bespoke connectives of the logics with which it is instantiated. To demonstrate the effectiveness of MoSeL, we have instantiated it to provide effective tactical support for interactive and semi-automated proofs in six very different separation logics.
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