### Write documentation for the functor combinators in ProofGuide.

parent 45d4e2a6
 ... ... @@ -6,16 +6,71 @@ This complements the tactic documentation for the [proof mode](ProofMode.md) and [HeapLang](HeapLang.md) as well as the documentation of syntactic conventions in the [style guide](StyleGuide.md). ## Combinators for functors In Iris, the type of propositions [iProp] is described by the solution to the recursive domain equation: ``` iProp ≅ uPred (F (iProp)) ``` Here, `F` is a user-chosen locally contractive bifunctor from COFEs to unital Camaras (a step-indexed generalization of unital resource algebras). To make it convenient to construct such functors out of smaller pieces, we provide a number of abstractions: - [`cFunctor`](theories/algebra/ofe.v): bifunctors from COFEs to OFEs. - [`rFunctor`](theories/algebra/cmra.v): bifunctors from COFEs to cameras. - [`urFunctor`](theories/algebra/cmra.v): bifunctors from COFEs to unital cameras. Besides, there are the classes `cFunctorContractive`, `rFunctorContractive`, and `urFunctorContractive` which describe the subset of the above functors that are contractive. To compose these functors, we provide a number of combinators, e.g.: - `constCF (A : ofeT) : cFunctor := λ (B,B⁻), A ` - `idCF : cFunctor := λ (B,B⁻), B` - `prodCF (F1 F2 : cFunctor) : cFunctor := λ (B,B⁻), F1 (B,B⁻) * F2 (B,B⁻)` - `ofe_morCF (F1 F2 : cFunctor) : cFunctor := λ (B,B⁻), F1 (B⁻,B) -n> F2 (B,B⁻)` - `laterCF (F : cFunctor) : cFunctor := λ (B,B⁻), later (F (B,B⁻))` - `agreeRF (F : cFunctor) : rFunctor := λ (B,B⁻), agree (F (B,B⁻))` - `gmapURF K (F : rFunctor) : urFunctor := λ (B,B⁻), gmap K (F (B,B⁻))` Using these combinators, one can easily construct bigger functors in point-free style, e.g: ``` F := gmapURF K (agreeRF (prodCF (constCF natC) (laterCF idCF))) ``` which effectively defines `F := λ (B,B⁻), gmap K (agree (nat * later B))`. Furthermore, for functors written using these combinators like the functor `F` above, Coq can automatically `urFunctorContractive F`. To make it a little bit more convenient to write down such functors, we make the constant functors (`constCF`, `constRF`, and `constURF`) a coercion, and provide the usual notation for products, etc. So the above functor can be written as follows (which is similar to the effective definition of `F` above): ``` F := gmapURF K (agreeRF (natC * ▶ ∙)) ``` ## Resource algebra management When using ghost state in Iris, you have to make sure that the resource algebras you need are actually available. Every Iris proof is carried out using a universally quantified list `Σ: gFunctors` defining which resource algebras are available. You can think of this as a list of resource algebras, though in reality it is a list of functors from OFEs to Cameras (where Cameras are a step-indexed generalization of resource algebras). This is the *global* list of resources that the entire proof can use. We keep it universally quantified to enable composition of proofs. The formal side of this is described in §7.4 of reality it is a list of locally contractive functors from COFEs to Cameras, which are typically defined using the combinators for functors described above. The `Σ` is the *global* list of resources that the entire proof can use. We keep the `Σ` universally quantified to enable composition of proofs. The formal side of this is described in §7.4 of [The Iris Documentation](http://plv.mpi-sws.org/iris/appendix-3.1.pdf); here we describe the Coq aspects of this approach. ... ...
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