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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>
18 lines
589 B
Idris
18 lines
589 B
Idris
||| Additional properties/lemmata of Nats
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module Data.Nat.Properties
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import Data.Nat
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%default total
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export
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multRightCancel : (a,b,r : Nat) -> Not (r = 0) -> a*r = b*r -> a = b
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multRightCancel a b 0 r_nz ar_eq_br = void $ r_nz Refl
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multRightCancel 0 0 r@(S predr) r_nz ar_eq_br = Refl
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multRightCancel 0 (S b) r@(S predr) r_nz ar_eq_br impossible
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multRightCancel (S a) 0 r@(S predr) r_nz ar_eq_br impossible
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multRightCancel (S a) (S b) r@(S predr) r_nz ar_eq_br =
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cong S $ multRightCancel a b r r_nz
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$ plusLeftCancel r (a*r) (b*r) ar_eq_br
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