* Add utility functions to treat All as a heterogeneous container
* Distinguish RefC Int and Bits types
* Change RefC Integers to be arbitrary precision
* Add RefC Bits maths operations
* Make RefC div and mod Euclidean
* Add RefC bit-ops tests
* Add RefC integer comparison tests
* Add RefC IntN support
We should really get these automatically, but until we do, add the
flags. Chez seems to spot these anyway, but again it makes the generated
code easier to look at, and it removes some needless abstraction over
interfaces at runtime.
We've had these for a while, used for interface specialisation, but
they're not yet used anywhere else or properly documented. We should
document them soon, but for now, it's a useful performance boost to
always use the fast versions of pack/unpack/concat at runtime.
Also moves a couple to the prelude, to ensure that the fast versions are
defined in the same place as the 'normal' version so that the
transformation will always fire (that is, no need to import Data.String
for the transformation to work).
This has a much better behaviour with respect to proof search and
the coverage checker realising we don't need to consider the Z case
than the `Not (x = Z)` we used earlier.
Ideally, liftIO would always be linear, but that has lots of knock-on
effects for other monads which we might want to put in HasIO, now that
subtyping is gone. We'll have to revisit this when we have some kind of
multiplicity polymorphism.
Snoc add an element at the end of the vector. The main use case
for such a function is to get the expected type signature
Vect n a -> a -> Vect (S n) a instead of
Vect n a -> a -> Vect (n + 1) a which you get by using `++ [x]`
Snoc gets is name from `cons` by reversin each letter, indicating
tacking on the element at the end rather than the begining.
`append` would also be a suitable name.
It's disappointing to have to do this, but I think necessary because
various issue reports have shown it to be unsound (at least as far as
inference goes) and, at the very least, confusing. This patch brings us
back to the basic rules of QTT.
On the one hand, this makes the 1 multiplicity less useful, because it
means we can't flag arguments as being used exactly once which would be
useful for optimisation purposes as well as precision in the type. On
the other hand, it removes some complexity (and a hack) from
unification, and has the advantage of being correct! Also, I still
consider the 1 multiplicity an experiment.
We can still do interesting things like protocol state tracking, which
is my primary motivation at least.
Ideally, if the 1 multiplicity is going to be more generall useful,
we'll need some kind of way of doing multiplicity polymorphism in the
future. I don't think subtyping is the way (I've pretty much always come
to regret adding some form of subtyping).
Fixes#73 (and maybe some others).
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>
