mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-22 19:21:39 +03:00
1576a578a0
* 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.
82 lines
2.6 KiB
Idris
82 lines
2.6 KiB
Idris
module Data.Monoid.Exponentiation
|
|
|
|
import Data.Nat.Views
|
|
import Syntax.PreorderReasoning
|
|
|
|
%default total
|
|
|
|
------------------------------------------------------------------------
|
|
-- Implementations
|
|
|
|
public export
|
|
linear : Monoid a => a -> Nat -> a
|
|
linear v Z = neutral
|
|
linear v (S k) = v <+> linear v k
|
|
|
|
public export
|
|
modularRec : Monoid a => a -> HalfRec n -> a
|
|
modularRec v HalfRecZ = neutral
|
|
modularRec v (HalfRecEven _ acc) = let e = modularRec v acc in e <+> e
|
|
modularRec v (HalfRecOdd _ acc) = let e = modularRec v acc in v <+> e <+> e
|
|
|
|
public export
|
|
modular : Monoid a => a -> Nat -> a
|
|
modular v n = modularRec v (halfRec n)
|
|
|
|
infixr 10 ^
|
|
public export
|
|
(^) : Num a => a -> Nat -> a
|
|
(^) = modular @{Multiplicative}
|
|
|
|
------------------------------------------------------------------------
|
|
-- Properties
|
|
|
|
-- Not using `MonoidV` because it's cursed
|
|
export
|
|
linearPlusHomo : (mon : Monoid a) =>
|
|
-- good monoid
|
|
(opAssoc : {0 x, y, z : a} -> ((x <+> y) <+> z) === (x <+> (y <+> z))) ->
|
|
(neutralL : {0 x : a} -> (neutral @{mon} <+> x) === x) ->
|
|
-- result
|
|
(v : a) -> {m, n : Nat} ->
|
|
(linear v m <+> linear v n) === linear v (m + n)
|
|
linearPlusHomo opAssoc neutralL v = go m where
|
|
|
|
go : (m : Nat) -> (linear v m <+> linear v n) === linear v (m + n)
|
|
go Z = neutralL
|
|
go (S m) = Calc $
|
|
|~ (v <+> linear v m) <+> linear v n
|
|
~~ v <+> (linear v m <+> linear v n) ...( opAssoc )
|
|
~~ v <+> (linear v (m + n)) ...( cong (v <+>) (go m) )
|
|
|
|
export
|
|
modularRecCorrect : (mon : Monoid a) =>
|
|
-- good monoid
|
|
(opAssoc : {0 x, y, z : a} -> ((x <+> y) <+> z) === (x <+> (y <+> z))) ->
|
|
(neutralL : {0 x : a} -> (neutral @{mon} <+> x) === x) ->
|
|
-- result
|
|
(v : a) -> {n : Nat} -> (p : HalfRec n) ->
|
|
modularRec v p === linear v n
|
|
modularRecCorrect opAssoc neutralL v acc = go acc where
|
|
|
|
aux : {m, n : Nat} -> (linear v m <+> linear v n) === linear v (m + n)
|
|
aux = linearPlusHomo opAssoc neutralL v
|
|
|
|
go : {n : Nat} -> (p : HalfRec n) -> modularRec v p === linear v n
|
|
go HalfRecZ = Refl
|
|
go (HalfRecEven k acc) = rewrite go acc in aux
|
|
go (HalfRecOdd k acc) = rewrite go acc in Calc $
|
|
|~ (v <+> linear v k) <+> linear v k
|
|
~~ v <+> (linear v k <+> linear v k) ...( opAssoc )
|
|
~~ v <+> (linear v (k + k)) ...( cong (v <+>) aux )
|
|
|
|
export
|
|
modularCorrect : (mon : Monoid a) =>
|
|
-- good monoid
|
|
(opAssoc : {0 x, y, z : a} -> ((x <+> y) <+> z) === (x <+> (y <+> z))) ->
|
|
(neutralL : {0 x : a} -> (neutral @{mon} <+> x) === x) ->
|
|
-- result
|
|
(v : a) -> {n : Nat} -> modular v n === linear v n
|
|
modularCorrect opAssoc neutralL v
|
|
= modularRecCorrect opAssoc neutralL v (halfRec n)
|