* contrib library unused import removal
* remove a few unused imports.
* another round of unused import removal
* another round of unused import deletion.
* another round of unused import deletion.
* [ breaking ] remove parsing of dangling binders
It used to be the case that
```
ID : Type -> Type
ID a = a
test : ID (a : Type) -> a -> a
test = \ a, x => x
```
and
```
head : List $ a -> Maybe a
head [] = Nothing
head (x :: _) = Just x
```
were accepted but these are now rejected because:
* `ID (a : Type) -> a -> a` is parsed as `(ID (a : Type)) -> a -> a`
* `List $ a -> Maybe a` is parsed as `List (a -> Maybe a)`
Similarly if you want to use a lambda / rewrite / let expression as
part of the last argument of an application, the use of `$` or parens
is now mandatory.
This should hopefully allow us to make progress on #1703
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.
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>
