This also changes the return type of `char` and `string`. They
previously returned `()`, they now return `Char` and `String`
repectively.
Signed-off-by: Alex Humphreys <alex.humphreys@here.com>
Rather than translating the constraints to a Dybjer-Setzer IR code
we can produce an ad-hoc definition of a `Domain` that we will be
able to make runtime irrelevant.
This means that compiled code will never need to construct a proof
that a value is in the domain of the function: it will simply run
the function!
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
