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