Refactor the DIY equational reasoning library to be a bit more like
the generic pre-order reasoning library:
Change the `...` notation into a constructor for a new `Step` datatype.
This seems to help idris disambiguate between the two kinds of
reasoning when they're used in the same file (e.g., frex).
Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk>
Division Theorem. For every natural number `x` and positive natural
number `n`, there is a unique decomposition:
`x = q*n + r`
with `q`,`r` natural and `r` < `n`.
`q` is the quotient when dividing `x` by `n`
`r` is the remainder when dividing `x` by `n`.
This commit adds a proof for this fact, in case
we want to reason about modular arithmetic (for example, when dealing
with binary representations). A future, more systematic, development could
perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics:
https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idrhttps://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idrhttps://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr
In the process, it bulks up the stdlib with:
+ a generic PreorderReasoning module for arbitrary preorders,
analogous for the equational reasoning module
+ some missing facts about Nat operations.
+ Refactor some Nat order properties using a 'reflect' function
Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk>
Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
* [contrib] Add misc libraries to contrib
Expose some `private` function in libs/base that I needed, and seem like
their visibility was forgotten
I'd appreciate a code review, especially to tell me I'm
re-implementing something that's already elsewhere in the library
Mostly extending existing functionality:
* `Data/Void.idr`: add some utility functions for manipulating absurdity.
* `Decidable/Decidable/Extra.idr`: add support for double negation elimination in decidable relations
* `Data/Fun/Extra.idr`:
+ add `application` (total and partil) for n-ary functions
+ add (slightly) dependent versions of these operations
* `Decidable/Order/Strict.idr`: a strict preorder is what you get when
you remove the diagonal from a pre-order. For example, `<` is the
associated preorder for `<=` over `Nat`.
Analogous to `Decidable.Order`. The proof search mechanism struggled
a bit, so I had to hack it --- sorry.
Eventually we should move `Data.Fun.Extra.Pointwise` to `Data.Vect.Quantifiers` in base
but we don't have any interesting uses for it at the moment so it's not
urgent.
Co-authored by @gallais
For lack of a better place, I've put it in `Syntax.PreorderReasoning`
These equations are natural in equational reasoning, but less so when
rewriting, so that's why it's there
Apparently the concrete syntax is controversial ("{apples}"
vs. "$oranges"), so I'm just adding a simple DYI version until we
agree on a concrete syntax
It's worth delaying in case doing some more work (and some more
interface resolution) can make the type more concrete. This makes
test linear011 work more smoothly, and will help with this sort of
program in general.
A better way, later, would be to try elaborating and delay only if the
type is not yet known. We have the facilities, but it's too fiddly for
me to want to implement it right now...
We need to check below top level too, since there could be holes that
we're happy to resolve by searching. The linearity test added
illustrates a place where this is needed.
Doing this is awkward, for a number of reasons, including 'pure' not
being linear in its argument - there's no guarantee it'll be used
linearly after all - and lack of multiplicity polymorphism. Fortunately
the user doesn't have to know about the ugliness underneath!
We can look at ways to make it less horrible later :). For now, this is
starting to look like something that allows us to write linear protocols
without too much extra machinery on top of IO.
Following a fairly detailed discussion on slack, the feeling is
generally that it's better to have a single interface. While precision
is nice, it doesn't appear to buy us anything here. If that turns out to
be wrong, or limiting somehow, we can revisit it later. Also:
- it's easier for backend authors if the type of IO operations is
slightly less restrictive. For example, if it's in HasIO, that limits
alternative implementations, which might be awkward for some
alternative back ends.
- it's one less extra detail to learn. This is minor, but there needs to
be a clear advantage if there's more detail to learn.
- It is difficult to think of an underlying type that can't have a Monad
instance (I have personally never encountered one - if they turns out
to exist, again, we can revisit!)
