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Rodolphe Lepigre
Iris
Commits
125aecf0
Commit
125aecf0
authored
May 30, 2019
by
Robbert Krebbers
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Write documentation for the functor combinators in ProofGuide.
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ProofGuide.md
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@@ 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 userchosen locally contractive bifunctor from COFEs to unital
Camaras (a stepindexed 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 pointfree
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
stepindexed 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.mpisws.org/iris/appendix3.1.pdf
)
; here we
describe the Coq aspects of this approach.
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