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>
Nipping this historical artifact in the bud before it roots. It's often
useful to be able to `map` directly to the result of a StateT computation
and due to how Functor works this is made harder when the tuple is
(a,state) vs (state,a)
* [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
Until now namespaces were stored as (reversed) lists of strings.
It led to:
* confusing code where we work on the underlying representation of
namespaces rather than say what we mean (using `isSuffixOf` to mean
`isParentOf`)
* potentially introducing errors by not respecting the invariant cf.
bug report #616 (but also name generation in the scheme backend
although that did not lead to bugs as it was self-consistent AFAICT)
* ad-hoc code to circumvent overlapping interface implementation when
showing / pretty-printing namespaces
This PR introduces a `Namespace` newtype containing a list of strings.
Nested namespaces are still stored in reverse order but the exposed
interface aims to support programming by saying what we mean
(`isParentOf`, `isApproximationOf`, `X <.> Y` computes to `X.Y`, etc.)
irrespective of the underlying representation.
Until now namespaces were stored as (reversed) lists of strings.
It led to:
* confusing code where we work on the representation rather than say
what we mean (e.g. using `isSuffixOf` to mean `isParentOf`)
* potentially introducing errors by not respecting the invariant cf.
bug report #616 (but also name generation in the scheme backend
although that did not lead to bugs as it was self-consistent AFAICT)
* ad-hoc code to circumvent overlapping interface implementations when
showing / pretty-printing namespaces
This introduces a Namespace newtype containing non-empty lists of
strings. Nested namespaces are still stored in reverse order but the
exposed interface aims to support programming by saying what we mean
(`isParentOf`, `isApproximationOf`, `X <.> Y` computes to `X.Y`, etc.)
irrespective of the underlying representation.
Main change
===========
The main change is to the type of function dealing with an untouched
segment of the local scope. e.g.
```
weak : {outer, vars : _} -> (ns : List Name) ->
tm (outer ++ inner) -> tm (outer ++ ns ++ inner)
```
Instead we now write
```
weak : SizeOf ns -> tm (outer ++ inner) -> tm (outer ++ ns ++ inner)
```
meaning that we do not need the values of `outer`, `inner` and `ns`
at runtime. Instead we only demand a `SizeOf ns` which is a `Nat`
together with an (erased) proof that `ns` is of that length.
Other modifications
===================
Quadratic behaviour
-------------------
A side effect of this refactor is the removal of two sources of
quadratic behaviour. They typically arise in a situation where
work is done on a scope of the form
```
outer ++ done ++ ns ++ inner
```
When `ns` is non-empty, some work is performed and then the variable
is moved to the pile of things we are `done` with. This leads to
recursive calls of the form `f done` -> `f (done ++ [v])` leading
to a cost quadratic in the size of `ns`.
Now that we only care about `SizeOf done`, the recursive call is
(once all the runtime irrelevant content is erased) for the form
`f n` -> `f (S n)`!
More runtime irrelevance
------------------------
In some places we used to rely on a list of names `vars` being
available. However once we only care about the length of `vars`,
the fact it is not available is not a limitation.
For instance a `SizeOf vars` can be reconstructed from an environment
assigning values to `vars` even if `vars` is irrelevant. Indeed the
size of the environment is the same as that of `vars`.
For Void and Either
This is because I ended up using them elsewhere, so why not include them in the stdlib.
Also expose left/rightInjective functions, as are used in the DecEq proofs.
In a 'Bind', normalise the result of the first action, rather than
quoting the HNF. This improves performance since the HNF could be quite
big when quoted back.
Ideally, we wouldn't have to quote and unquote here, and we can probably
achieve this by tinkering with the evaluator.
This has an unfortunate effect on the reflection002 test, in that the
"typed template Idris" example now evaluates too much. But, I think the
overall performance is too important for the primary motivation
behind elaborator reflection. I will return to this!
This didn't cause a problem before as it was likely just ignored by the C
function. According to Edwin the extra argument is a leftover from when this
was a pure scheme call.
The ports are rather straight forward and I have purposefully written
the documentation to be beginner friendly.
Note, I have diverged from Idris1 over the naming of the projection
functions to make them consistent with `Pair` and `DPair`.
Conditional variables with timeout in Chez didn't work, so changed to a
consistent meaning of the timeout (microseconds). Also fix linearity of
unsafePerformIO.
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!)
Backed by Data.IOArray. Also moved the array external primitives to a
separate module Data.IOArray.Prims, since the next step is to add a
linear bounded array type where the bounds checks are done at compile
time, so we'll want to read and write without bounds likes.
Having unsolved holes in a 'core' library unneccessarily pollutes the list of holes shown to the user.
Thus, having unfilled holes in a 'core' library is not right.
These constructs can be re-added once the holes have been filled in.
I'm playing with some linear structures and finding these useful a lot,
so good to have a consistent syntax for it. '#' is chosen because it's
short, looks a bit like a cross if you look at it from the right angle
(!) and so as not to clash with '@@' in preorder reasoning syntax.
Meaning that the FFI is aware of it, so you can send arbitrary byte data
to foreign calls. Fixes#209
This means that we no longer need the hacky way of reading and writing
binary data via scheme, so can have a more general interface for reading
and writing buffer data in files.
It will also enable more interesting high level interfaces to binary
data, with C calls being used where necessary.
Note that the Buffer primitive are unsafe! They always have been, of
course... so perhaps (later) they should have 'unsafe' as part of their
name and better high level safe interfaces on top.
This requires updating the scheme to support Buffer as an FFI primitive,
but shouldn't affect Idris2-boot which loads buffers its own way.
The old way only worked by chance, because the argumemt order happens to
be the same in all cases. I noticed due to some experiments elsewhere
with different ways of elaborating case, which broke that assumption.
The meaning of the list of Vars is actually the opposite of what it was
taken to be... fortunately, the performance works out roughly the same.
Also this way is (arguably) simpler, which is usually a good sign.
Allows quoting a term back to a TTImp. Test reflection007 shows one
possible use for this, building a reflected, type safe, representation
of an expression.
So the type of Elab now gives the expected type that's being elaborated
to, meaning that we can run 'check' in the middle of scripts and use the
result.
Get the names of local variables. and add the ability to look up their
types.
When we get a reflected TTImp, either checking the Goal or looking up a
type, it's not impossible that there'll be some repeated binder names,
so also make sure binders are unique relative to the current context.
Ideally we'd also rename things in the environment to guarantee that all
names are unique, but we don't yet.
(This would be much easier if reflected terms were typed such that they
were well scoped, but that would also make reflection harder to use.)
