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0c1a124704
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.idr
https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr
https://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>
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| .. | ||
| Bool | ||
| Fin | ||
| Fun | ||
| Linear | ||
| List | ||
| Logic | ||
| Morphisms | ||
| Nat | ||
| Stream | ||
| String | ||
| Vect | ||
| HVect.idr | ||
| SortedMap.idr | ||
| SortedSet.idr | ||
| Void.idr | ||