The Monotone construction

This project formalizes the monotone construction for reasoning about monotonicity with respect to arbitrary preorder relations in separation logic. There are three folders in this project: iris-src, categorical, and examples.

iris-src

This folder includes the formalization of the monotone construction as an Iris resource algebra. This resource algebra is defined in the file iris-src/monotone.v

compile instructions

Use make in iris-src to compile the development. This development has the following dependencies:

1. Coq 8.12 / 8.13
2. The std++ project (Iris depends on this) (commit hash f63bb74b)
3. The Iris program logic (commit hash d08de6e8)

categorical

This folder formalizes the theory behind the monotone construction and establishes the canonicity of this construction. In particular, it establishes that the monotone construction arises as a free functor. The files in this development are as follows:

• categorical/PreOrder.v formalizes the category of preorder relations and monotone functions.
• categorical/PartialOrder.v formalizes the category of partial order relations and monotone functions. It also establishes that this category is a reflective subcategory of the category of preorders.
• categorical/PCM.v formalizes the category of partial commutative monoids. This file also formalizes the extension functor that maps a PCM to its extension order in the category of preorders.
• categorical/lattice.v formalizes the category of join-semilattices with a bottom element and shows that this category is a full subcategory of the category of partial commutative monoids.
• categorical/monotone.v formalizes the monotone construction as a construction that given a partial order relation constructs a join-semilattice with a bottom element.
• categorical/adjunction.v establishes that the monotone functor from the category of partial orders to the category of join-semilattices with a bottom element is the left adjoint to the forgetful functor.

compile instructions

Use make in categorical to compile the development. This development has the following dependencies:

1. Coq 8.12
2. The category theory library: https://github.com/amintimany/Categories commit hash: 96ce5ad61a1078d8c8ac73befa33c147650caf4d

examples

The sub-folder examples includes three examples of using the monotone resource algebra and one simple example of how one reasons about monotonicity in separation logic using basic iris resource algebras.

• examples/mon_ref.v constructs monotone references that are associated with a preorder relation. A monotone reference can only be updated if they respect the associated preorder relation. Any Iris points-to proposition can be turned into a monotone reference and back into a an ordinary points-to proposition (possibly multiple times).

• examples/exclude_path.v shows how the monotone resource algebra instantiated with a very simple preorder can be used to prove that certain paths in the program execution are unreachable.

• examples/causal-closure.v is an example inspired by verification of causally consistent databases. It shows how the monotone resource algebra can be used to enforce a relation between ghost resources that allows us to prove that if an event is observed then any other event that it depends on is also observed.

• examples/mono_counter.v verifies correctness of a monotone counter with basic iris resource algebras, i.e., without the monotone resource algebra. This formalization serves as a simple example of how one reasons about monotonicity in separation logic.

compile instructions

Use make in examples to compile the development. This development has the same dependencies as the iris-src folder.