From 5f0704739ee99a6f64283494056ad8b9c5f9a07b Mon Sep 17 00:00:00 2001 From: rnhmjoj Date: Sat, 23 Apr 2016 18:22:37 +0200 Subject: [PATCH] Initial commit --- Interlude.cabal | 27 ++ LICENSE | 674 +++++++++++++++++++++++++++++++++++++++ Setup.hs | 2 + default.nix | 24 ++ src/Data/Bool.hs | 48 +++ src/Data/Function.hs | 39 +++ src/Data/Integer.hs | 5 + src/Data/Nat.hs | 146 +++++++++ src/Data/TypeClass.hs | 18 ++ src/Data/Vec.hs | 144 +++++++++ src/Relation/Equality.hs | 47 +++ 11 files changed, 1174 insertions(+) create mode 100644 Interlude.cabal create mode 100644 LICENSE create mode 100644 Setup.hs create mode 100644 default.nix create mode 100644 src/Data/Bool.hs create mode 100644 src/Data/Function.hs create mode 100644 src/Data/Integer.hs create mode 100644 src/Data/Nat.hs create mode 100644 src/Data/TypeClass.hs create mode 100644 src/Data/Vec.hs create mode 100644 src/Relation/Equality.hs diff --git a/Interlude.cabal b/Interlude.cabal new file mode 100644 index 0000000..e19ccd6 --- /dev/null +++ b/Interlude.cabal @@ -0,0 +1,27 @@ +name: Interlude +version: 0.1.0.0 +synopsis: My take on Prelude +description: My take on Prelude +license: GPL-3 +license-file: LICENSE +author: rnhmjoj +maintainer: micheleguerinirocco@me.com +copyright: ยฉ Michele Guerini Rocco +category: Math +build-type: Simple +-- extra-source-files: +cabal-version: >=1.10 + +library + exposed-modules: Relation.Equality, Data.TypeClass, + Data.Bool, Data.Nat, Data.Integer + Data.Function, Data.Vec + -- other-modules: + other-extensions: GADTs, DataKinds, TypeFamilies, TypeOperators, + StandaloneDeriving, ScopedTypeVariables, + UndecidableInstances, NoImplicitPrelude, + TemplateHaskell, QuasiQuotes, PolyKinds, + UnicodeSyntax + build-depends: base ==4.*, singletons, integer-gmp + hs-source-dirs: src + default-language: Haskell2010 diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..45644ff --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/Setup.hs b/Setup.hs new file mode 100644 index 0000000..9a994af --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/default.nix b/default.nix new file mode 100644 index 0000000..a1e4dcd --- /dev/null +++ b/default.nix @@ -0,0 +1,24 @@ +{ nixpkgs ? import {}, compiler ? "default" }: + +let + + inherit (nixpkgs) pkgs; + + f = { mkDerivation, base, singletons, stdenv }: + mkDerivation { + pname = "Interlude"; + version = "0.1.0.0"; + src = ./.; + libraryHaskellDepends = [ base singletons ]; + license = stdenv.lib.licenses.gpl3; + }; + + haskellPackages = if compiler == "default" + then pkgs.haskellPackages + else pkgs.haskell.packages.${compiler}; + + drv = haskellPackages.callPackage f {}; + +in + + if pkgs.lib.inNixShell then drv.env else drv diff --git a/src/Data/Bool.hs b/src/Data/Bool.hs new file mode 100644 index 0000000..70dc100 --- /dev/null +++ b/src/Data/Bool.hs @@ -0,0 +1,48 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE TypeOperators #-} +{-# LANGUAGE StandaloneDeriving #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE TemplateHaskell #-} +{-# LANGUAGE QuasiQuotes #-} +{-# LANGUAGE PolyKinds #-} +{-# LANGUAGE UnicodeSyntax #-} + +module Data.Bool where + +import Data.TypeClass (Eq, Enum, Show) +import Data.Singletons.TH +import Data.Singletons.Prelude.Enum + + +singletons [d| + + -- | Boolean Type + data ๐”น โˆท โ˜… where + T โˆท ๐”น -- ^ True + F โˆท ๐”น -- ^ False + deriving (Eq, Show) + + -- Logical operatos + + -- | Conjuction + (โˆง) โˆท ๐”น โ†’ ๐”น โ†’ ๐”น + T โˆง T = T + _ โˆง F = F + F โˆง _ = F + + -- | Disjuction + (โˆจ) โˆท ๐”น โ†’ ๐”น โ†’ ๐”น + T โˆจ _ = T + _ โˆจ T = T + F โˆจ F = F + + -- | Negation + (ยฌ) โˆท ๐”น โ†’ ๐”น + (ยฌ) T = F + (ยฌ) F = T + + |] + diff --git a/src/Data/Function.hs b/src/Data/Function.hs new file mode 100644 index 0000000..f685c31 --- /dev/null +++ b/src/Data/Function.hs @@ -0,0 +1,39 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE TypeOperators #-} +{-# LANGUAGE StandaloneDeriving #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE NoImplicitPrelude #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE TemplateHaskell #-} +{-# LANGUAGE QuasiQuotes #-} +{-# LANGUAGE PolyKinds #-} +{-# LANGUAGE UnicodeSyntax #-} + + +module Data.Function where + +-- | Identity Function +id โˆท โˆ€ (a โˆท โ˜…). a โ†’ a +id x = x + +-- | Constant function +const โˆท โˆ€ (a โˆท โ˜…) (b โˆท โ˜…). a โ†’ b โ†’ a +const k _ = k + +-- | Undefined +(โŠฅ) โˆท โˆ€ a. a +(โŠฅ) = (โŠฅ) + +infixr 0 $ +infixr 9 โˆ˜ + +-- | Function application operator +($) โˆท โˆ€ a b. (a โ†’ b) โ†’ a โ†’ b +f $ x = f x + +-- | Function composition operator +(โˆ˜) :: โˆ€ a b c. (b โ†’ c) โ†’ (a โ†’ b) โ†’ a โ†’ c +f โˆ˜ g = \x โ†’ f (g x) + diff --git a/src/Data/Integer.hs b/src/Data/Integer.hs new file mode 100644 index 0000000..ca0cfb8 --- /dev/null +++ b/src/Data/Integer.hs @@ -0,0 +1,5 @@ +module Data.Integer + ( module GHC.Integer + ) where + +import GHC.Integer diff --git a/src/Data/Nat.hs b/src/Data/Nat.hs new file mode 100644 index 0000000..4a58d76 --- /dev/null +++ b/src/Data/Nat.hs @@ -0,0 +1,146 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE TypeOperators #-} +{-# LANGUAGE StandaloneDeriving #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE NoImplicitPrelude #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE TemplateHaskell #-} +{-# LANGUAGE QuasiQuotes #-} +{-# LANGUAGE PolyKinds #-} +{-# LANGUAGE UnicodeSyntax #-} + +module Data.Nat where + +import Relation.Equality +import Data.Bool +import Data.Integer (Integer) +import Data.Function (id, const, (โˆ˜)) +import Data.TypeClass (Eq, Show, Num, Enum, Bounded) +import Data.Singletons.TH hiding ((:<), (:>), (%:<), Refl) + +import qualified Data.TypeClass as T + +singletons [d| + + -- | Natural numbers type + data โ„• โˆท โ˜… where + Z โˆท โ„• -- ^ Zero + S โˆท โ„• โ†’ โ„• -- ^ Successor + deriving (Eq, Show) + + infixl 6 + + infixl 7 ร— + + -- | Predecessor of n + pred โˆท โ„• โ†’ โ„• + pred Z = Z + pred (S n) = n + + -- | Addition + (+) โˆท โ„• โ†’ โ„• โ†’ โ„• + Z + m = m + (S n) + m = S (n + m) + + -- | Subtraction + (-) โˆท โ„• โ†’ โ„• โ†’ โ„• + Z - _ = Z + n - Z = n + (S n) - (S m) = n - m + + -- | Multiplication + (ร—) โˆท โ„• โ†’ โ„• โ†’ โ„• + Z ร— m = Z + (S n) ร— m = (n ร— m) + m + + -- | Ordering relation < + (<) โˆท โ„• โ†’ โ„• โ†’ ๐”น + Z < Z = F + Z < (S _) = T + (S _) < Z = F + (S n) < (S m) = n < m + + -- | Ordering relation > + (>) โˆท โ„• โ†’ โ„• โ†’ ๐”น + n > m = (ยฌ) (n < m) + + |] + + +instance Num โ„• where + (+) = (+) + (-) = (-) + (*) = (ร—) + abs = id + signum = const 1 + negate = id + fromInteger = fromInteger + +instance Enum โ„• where + toEnum = fromInteger โˆ˜ T.toInteger + fromEnum = T.fromInteger โˆ˜ toInteger + +instance Bounded โ„• where + minBound = Z + maxBound = (โˆž) + + +-- | Convert a natural into an "Integer" +toInteger โˆท โ„• โ†’ Integer +toInteger Z = 0 +toInteger (S n) = 1 T.+ toInteger n + + +-- | Convert na "Integer" into a natural +fromInteger โˆท Integer โ†’ โ„• +fromInteger 0 = Z +fromInteger n + | n T.<= 0 = 0 + | n T.> 0 = S (fromInteger (T.pred n)) + + +-- | Equality for natural numbers +(โ‰ก) โˆท โ„• โ†’ โ„• โ†’ ๐”น +Z โ‰ก Z = T +(S n) โ‰ก (S m) = n โ‰ก m +_ โ‰ก _ = F + + +-- | Infinity +(โˆž) โˆท โ„• +(โˆž) = S (โˆž) + + +-- Proofs + +-- | Lemma +switch โˆท Sโ„• n โ†’ Sโ„• m โ†’ (n :+ S m) :โ‰ก (S n :+ m) +switch SZ m = Refl +switch (SS n) m = cong SS (switch n m) + +-- | Lemma +succ โˆท Sโ„• n โ†’ Sโ„• m โ†’ S (n :+ m) :โ‰ก (n :+ S m) +succ SZ m = Refl +succ (SS n) m = cong SS (succ n m) + +-- | Relation between successor and addition +succ_plus โˆท Sโ„• n โ†’ (n :+ S Z) :โ‰ก S n +succ_plus n = switch n SZ โˆพ plus_right_id (SS n) + +-- | Right identity of addition +plus_right_id โˆท Sโ„• n โ†’ (n :+ Z) :โ‰ก n +plus_right_id SZ = Refl +plus_right_id (SS n) = cong SS (plus_right_id n) + +-- | Associative property of addition +plus_assoc โˆท Sโ„• a โ†’ Sโ„• b โ†’ Sโ„• c โ†’ (a :+ b) :+ c :โ‰ก a :+ (b :+ c) +plus_assoc SZ _ _ = Refl +plus_assoc (SS a) b c = cong SS (plus_assoc a b c) + +-- | Commutative property of addition +plus_commut โˆท Sโ„• n โ†’ Sโ„• m โ†’ (n :+ m) :โ‰ก (m :+ n) +plus_commut n SZ = plus_right_id n +plus_commut n (SS m) = + (sym (succ n m) โˆพ cong SS (plus_commut n m)) โˆพ + (succ m n โˆพ switch m n) diff --git a/src/Data/TypeClass.hs b/src/Data/TypeClass.hs new file mode 100644 index 0000000..4c0d28d --- /dev/null +++ b/src/Data/TypeClass.hs @@ -0,0 +1,18 @@ +module Data.TypeClass + ( Eq(..) + , Ord(..) + , Enum(..) + , Bounded(..) + , Num(..) + , Integral(..) + , Show(..) + , Read(..) + , IsList(..) + , IsString(..) + ) where + +import GHC.Enum +import GHC.Show +import GHC.Read +import GHC.Num +import GHC.Exts diff --git a/src/Data/Vec.hs b/src/Data/Vec.hs new file mode 100644 index 0000000..e68c96e --- /dev/null +++ b/src/Data/Vec.hs @@ -0,0 +1,144 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE TypeOperators #-} +{-# LANGUAGE StandaloneDeriving #-} +{-# LANGUAGE FlexibleInstances #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE NoImplicitPrelude #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE TemplateHaskell #-} +{-# LANGUAGE QuasiQuotes #-} +{-# LANGUAGE PolyKinds #-} +{-# LANGUAGE UnicodeSyntax #-} + +module Data.Vec where + +import Relation.Equality (gcastWith) +import Data.Nat +import Data.Bool +import Data.Char (Char) +import Data.Function ((โˆ˜)) +import Data.TypeClass (Show, IsList, IsString, Eq) +import Data.Singletons (SingI, sing) + +import qualified Data.TypeClass as T + + +infixr 5 :- + +data Vec (a โˆท โ˜…) โˆท โ„• โ†’ โ˜… where + Nil โˆท Vec a Z + (:-) โˆท a โ†’ Vec a n โ†’ Vec a (S n) + + +type String n = Vec Char n + + +deriving instance Eq a โ‡’ Eq (Vec a n) + +instance Show a โ‡’ Show (Vec a n) where + showsPrec d = T.showsPrec d โˆ˜ toList + +instance SingI n โ‡’ IsList (Vec a n) where + type Item (Vec a n) = a + fromList = fromList + toList = toList + +instance SingI n โ‡’ IsString (String n) where + fromString = fromList + + +toList โˆท Vec a n โ†’ [a] +toList Nil = [] +toList (x :- xs) = x : toList xs + + +fromList โˆท SingI n โ‡’ [a] โ†’ Vec a n +fromList = f sing + where f โˆท Sโ„• n โ†’ [a] โ†’ Vec a n + f SZ _ = Nil + f (SS n) (x:xs) = x :- f n xs + + +(โงบ) โˆท Vec a n โ†’ Vec a m โ†’ Vec a (n :+ m) +(โงบ) (x :- xs) ys = x :- xs โงบ ys +(โงบ) Nil ys = ys + + +head โˆท Vec a (S n) โ†’ a +head (x :- _) = x + + +last โˆท Vec a (S n) โ†’ a +last (x :- Nil) = x +last (x :- y :- ys) = last (y :- ys) + + +tail โˆท Vec a (S n) โ†’ Vec a n +tail (_ :- xs) = xs + + +init โˆท Vec a (S n) โ†’ Vec a n +init (x :- Nil) = Nil +init (x :- y :- ys) = x :- init (y :- ys) + + +uncons โˆท Vec a (S n) โ†’ (a, Vec a n) +uncons (x :- xs) = (x, xs) + + +null โˆท Vec a n โ†’ ๐”น +null Nil = T +null _ = F + + +length โˆท Vec a n โ†’ โ„• +length Nil = Z +length (_ :- xs) = S (length xs) + + +sLength โˆท Vec a n โ†’ Sโ„• n +sLength Nil = SZ +sLength (x :- xs) = SS (sLength xs) + + +map โˆท (a โ†’ b) โ†’ Vec a n โ†’ Vec b n +map _ Nil = Nil +map f (x :- xs) = f x :- map f xs + + +foldl โˆท (a โ†’ b โ†’ a) โ†’ a โ†’ Vec b n โ†’ a +foldl _ x Nil = x +foldl f x (y :- ys) = foldl f (f x y) ys + + +foldr โˆท (a โ†’ b โ†’ b) โ†’ b โ†’ Vec a n โ†’ b +foldr f x Nil = x +foldr f x (y:-ys) = f y (foldr f x ys) + + +foldlโ‚ โˆท (a โ†’ a โ†’ a) โ†’ Vec a (S n) โ†’ a +foldlโ‚ f (x :- Nil) = x +foldlโ‚ f (x :- y :- ys) = foldlโ‚ f (f x y :- ys) + + +foldrโ‚ โˆท (a โ†’ a โ†’ a) โ†’ Vec a (S n) โ†’ a +foldrโ‚ f (x :- Nil) = x +foldrโ‚ f (x :- y :- ys) = f x (foldrโ‚ f (y :- ys)) + + +reverse โˆท Vec a n โ†’ Vec a n +reverse Nil = Nil +reverse (y :- ys) = gcastWith proof (reverse ys โงบ (y :- Nil)) + where proof = succ_plus (sLength ys) + + +replicate โˆท Sโ„• n โ†’ a โ†’ Vec a n +replicate SZ _ = Nil +replicate (SS n) a = a :- replicate n a + + +take โˆท Sโ„• n โ†’ Vec a m โ†’ Vec a n +take SZ _ = Nil +take (SS n) (x :- xs) = x :- take n xs diff --git a/src/Relation/Equality.hs b/src/Relation/Equality.hs new file mode 100644 index 0000000..19e4332 --- /dev/null +++ b/src/Relation/Equality.hs @@ -0,0 +1,47 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE TypeOperators #-} +{-# LANGUAGE StandaloneDeriving #-} +{-# LANGUAGE NoImplicitPrelude #-} +{-# LANGUAGE PolyKinds #-} +{-# LANGUAGE RankNTypes #-} +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE ExplicitNamespaces #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE UnicodeSyntax #-} + +module Relation.Equality where + +import Data.Singletons (Sing) + +infix 4 :โ‰ก + +-- | Equality relation for types +data (a :: k) :โ‰ก (b :: k) where + Refl โˆท โˆ€ a. a :โ‰ก a + +-- | Operator version of "trans" +(โˆพ) โˆท โˆ€ a b c. (a :โ‰ก b) โ†’ (b :โ‰ก c) โ†’ (a :โ‰ก c) +(โˆพ) = trans + +-- | Congruence property +cong โˆท โˆ€ a b c f. (Sing a โ†’ Sing b) โ†’ (a :โ‰ก c) โ†’ (f a :โ‰ก f c) +cong _ Refl = Refl + +-- | Transitive property +trans โˆท โˆ€ a b c. (a :โ‰ก b) โ†’ (b :โ‰ก c) โ†’ (a :โ‰ก c) +trans Refl Refl = Refl + +-- | Symmetric property +sym โˆท โˆ€ a b. (a :โ‰ก b) โ†’ (b :โ‰ก a) +sym Refl = Refl + +-- | Type-safe cast using propositional equality +castWith โˆท โˆ€ a b. (a :โ‰ก b) โ†’ a โ†’ b +castWith Refl x = x + +-- | Generalized form of cast +gcastWith โˆท โˆ€ a b r. (a :โ‰ก b) โ†’ ((a ~ b) โ‡’ r) โ†’ r +gcastWith Refl x = x +