Proving properties of type-level natural numbers obtained from user input

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Bas van Dijk

25 Nov 2014 25 Nov '14

2:51 a.m.

Hi, I have another type-level programming related question:

...

{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...

p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...

main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...

f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p? Obviously, applying it directly gives the type error:

...

f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...

g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...

data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...

c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g? Cheers, Bas

Show replies by date

Hi, Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...

{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...

newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...

fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...

fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...

f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...

main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...

Hi,

I have another type-level programming related question:

...
{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...
p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...
f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

...
f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...
g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...
data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...
c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

Richard Eisenberg

1:34 p.m.

Hi Bas, I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free. Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits. If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway. The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce. Richard On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...

Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...
{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...
newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...
fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...
fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...
f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...
Hi,

I have another type-level programming related question:

...
{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...
p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...
f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

...
f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...
g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...
data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...
c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

Alexander V Vershilov

1:55 p.m.

Hi, Richard, Bas. Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor. Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...

lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better. -- Alexander On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...

Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...
Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...
{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...
newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...
fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...
fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...
f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...
Hi,

I have another type-level programming related question:

...
{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...
p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...
f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

...
f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...
g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...
data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...
c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

Bas van Dijk

26 Nov 26 Nov

2:15 a.m.

Hi Alexander, Thanks for your answer! This provides a lot of ideas how to proceed. I'm unsure about the following though:

...

lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Doesn't this mean lessThen returns a Proxy n for all (n :: Nat) and not just the Nat inside the SomeNat? I also see that p is only used for comparing it to k. It's not used to produce the return value. Cheers, Bas On 25 November 2014 at 19:55, Alexander V Vershilov wrote:

...

Hi, Richard, Bas.

Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor.

Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better.

-- Alexander

On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...
Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...
Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...
{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...
newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...
fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...
fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...
f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...
Hi,

I have another type-level programming related question:

...
{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...
p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...
f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

...
f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...
g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...
data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...
c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

Alexander V Vershilov

28 Nov 28 Nov

5:27 p.m.

Hi, Bas, Richard. I've played a bit with example, obvously first approach contained bugs, but seems that I have fixed it and here I have 2 approaches, one uses unsafeCoerce (as Richard suggested) another is safe but with bad complexity: Full file is quite big, so I'm not inlining it in mail, but here is a link: https://github.com/qnikst/haskell-fun/blob/master/typelevel-literals/Proof.l... I wonder how far it's possible to go with singletons approach that I have not tried yet. -- Alexander On 26 November 2014 at 10:15, Bas van Dijk wrote:

...

Hi Alexander,

Thanks for your answer! This provides a lot of ideas how to proceed.

I'm unsure about the following though:

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Doesn't this mean lessThen returns a Proxy n for all (n :: Nat) and not just the Nat inside the SomeNat?

I also see that p is only used for comparing it to k. It's not used to produce the return value.

Cheers,

Bas

On 25 November 2014 at 19:55, Alexander V Vershilov wrote:

...
Hi, Richard, Bas.

Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor.

Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better.

-- Alexander

On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...
Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...
Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...
{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...
newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...
fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...
fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...
f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...
Hi,

I have another type-level programming related question:

...
{-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE KindSignatures #-}

import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

...
p :: forall (n :: Nat). Proxy n p = Proxy

Imagine I get p from user input like this:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

...
f :: forall (n :: Nat). (n <= 255) => proxy n -> () f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

...
f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

...
g :: forall (n :: Nat). proxy n -> Proof g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

...
data Proof where NoProof :: Proof Proof :: forall (n :: Nat). (n <= 255) => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

...
c :: () c = case g p of NoProof -> error "Input is bigger than 255!" Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

-- Alexander

Richard Eisenberg

30 Nov 30 Nov

3:26 p.m.

Hi Alexander, Nice idea to test against the set of known values. That's more type-safe than anything I've thought of. I agree that it's a bit of a painful construction, but I don't think we can do better with type-lits, as there is limited reasoning ability that GHC can manage. If you want to switch to unary naturals (`data Nat = Zero | Succ Nat`), then this can be built somewhat nicely with singletons and without unsafeCoerce. But, of course, unary naturals are very slow. By the way, the bug in the Proof2 version is a bug in GHC 7.8.3 (only in .3 -- not in .2 or in the soon-to-be .4) that allows you to write unsaturated type families that don't work. Saying `LessThan255` without a parameter should be a syntax error, but that check was accidentally turned off for 7.8.3, leading to a bogus type error. Thanks for sharing this work! Richard On Nov 28, 2014, at 5:27 PM, Alexander V Vershilov wrote:

...

Hi, Bas, Richard.

I've played a bit with example, obvously first approach contained bugs, but seems that I have fixed it and here I have 2 approaches, one uses unsafeCoerce (as Richard suggested) another is safe but with bad complexity:

Full file is quite big, so I'm not inlining it in mail, but here is a link:

https://github.com/qnikst/haskell-fun/blob/master/typelevel-literals/Proof.l...

I wonder how far it's possible to go with singletons approach that I have not tried yet.

-- Alexander

On 26 November 2014 at 10:15, Bas van Dijk wrote:

...
Hi Alexander,

Thanks for your answer! This provides a lot of ideas how to proceed.

I'm unsure about the following though:

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Doesn't this mean lessThen returns a Proxy n for all (n :: Nat) and not just the Nat inside the SomeNat?

I also see that p is only used for comparing it to k. It's not used to produce the return value.

Cheers,

Bas

On 25 November 2014 at 19:55, Alexander V Vershilov wrote:

...
Hi, Richard, Bas.

Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor.

Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better.

-- Alexander

On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...
Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...
Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

...
{-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE PolyKinds #-} {-# LANGUAGE UndecidableInstances #-} import GHC.TypeLits import GHC.Exts import Data.Proxy import Data.Tagged import System.Environment

New constraint carrying data type

...
newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

...
fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) fromSome (SomeNat p) | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Compiletime converter for known naturals

...
fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) fromKnown n = Proof $ Tagged n

Function to test:

...
f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () f2 _ = ()

Example of use:

...
main :: IO () main = do [arg] <- getArgs let n = read arg :: Integer

case someNatVal n of Nothing -> error "Input is not a natural number!" Just sn -> case fromSome sn of Just p -> return $ f2 p _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote:

...
Hi,

I have another type-level programming related question:

> {-# LANGUAGE GADTs #-} > {-# LANGUAGE TypeOperators #-} > {-# LANGUAGE ScopedTypeVariables #-} > {-# LANGUAGE KindSignatures #-} > > import GHC.TypeLits

Say I have a Proxy p of some type-level natural number:

> p :: forall (n :: Nat). Proxy n > p = Proxy

Imagine I get p from user input like this:

> main :: IO () > main = do > [arg] <- getArgs > let n = read arg :: Integer > > case someNatVal n of > Nothing -> error "Input is not a natural number!" > Just (SomeNat (p :: Proxy n)) -> ...

I also have a function f which takes a proxy of a natural number but it has the additional constraint that the number should be lesser than or equal to 255:

> f :: forall (n :: Nat). (n <= 255) => proxy n -> () > f _ = ()

How do I apply f to p?

Obviously, applying it directly gives the type error:

> f p <interactive>:179:1: Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ The type variable ‘n0’ is ambiguous In the expression: f p In an equation for ‘it’: it = f p

I imagine I somehow need to construct some Proof object using a function g like:

> g :: forall (n :: Nat). proxy n -> Proof > g _ = ...

Where the Proof constructor encapsulates the (n <= 255) constraint:

> data Proof where > NoProof :: Proof > Proof :: forall (n :: Nat). (n <= 255) > => Proxy n -> Proof

With g in hand I can construct c which patterns matches on g p and when there's a Proof the (n <= 255) constraint will be in scope which allows applying f to p:

> c :: () > c = case g p of > NoProof -> error "Input is bigger than 255!" > Proof p -> f p

But how do I define g?

Cheers,

Bas _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

-- Alexander

Alexander V Vershilov

4 Dec 4 Dec

5:50 p.m.

Hi, Richard Can you give some ideas or where to read how to properly use signletons and unary naturals in order to be able to express such constraints? Thanks -- Alexander On 30 November 2014 at 23:26, Richard Eisenberg wrote:

...

Hi Alexander,

Nice idea to test against the set of known values. That's more type-safe than anything I've thought of. I agree that it's a bit of a painful construction, but I don't think we can do better with type-lits, as there is limited reasoning ability that GHC can manage. If you want to switch to unary naturals (`data Nat = Zero | Succ Nat`), then this can be built somewhat nicely with singletons and without unsafeCoerce. But, of course, unary naturals are very slow.

By the way, the bug in the Proof2 version is a bug in GHC 7.8.3 (only in .3 -- not in .2 or in the soon-to-be .4) that allows you to write unsaturated type families that don't work. Saying `LessThan255` without a parameter should be a syntax error, but that check was accidentally turned off for 7.8.3, leading to a bogus type error.

Thanks for sharing this work! Richard

On Nov 28, 2014, at 5:27 PM, Alexander V Vershilov wrote:

...
Hi, Bas, Richard.

I've played a bit with example, obvously first approach contained bugs, but seems that I have fixed it and here I have 2 approaches, one uses unsafeCoerce (as Richard suggested) another is safe but with bad complexity:

Full file is quite big, so I'm not inlining it in mail, but here is a link:

https://github.com/qnikst/haskell-fun/blob/master/typelevel-literals/Proof.l...

I wonder how far it's possible to go with singletons approach that I have not tried yet.

-- Alexander

On 26 November 2014 at 10:15, Bas van Dijk wrote:

...
Hi Alexander,

Thanks for your answer! This provides a lot of ideas how to proceed.

I'm unsure about the following though:

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Doesn't this mean lessThen returns a Proxy n for all (n :: Nat) and not just the Nat inside the SomeNat?

I also see that p is only used for comparing it to k. It's not used to produce the return value.

Cheers,

Bas

On 25 November 2014 at 19:55, Alexander V Vershilov wrote:

...
Hi, Richard, Bas.

Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor.

Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better.

-- Alexander

On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...
Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

...
Hi,

Following approach can work, the idea is to define a type that will carry a proof (constraint) that we want to check. Here I have reused Data.Tagged, but it's possible to introduce your own with concrete constraints.

> {-# LANGUAGE DataKinds #-} > {-# LANGUAGE GADTs #-} > {-# LANGUAGE TypeOperators #-} > {-# LANGUAGE KindSignatures #-} > {-# LANGUAGE PolyKinds #-} > {-# LANGUAGE UndecidableInstances #-} > import GHC.TypeLits > import GHC.Exts > import Data.Proxy > import Data.Tagged > import System.Environment

New constraint carrying data type

> newtype Proof a b = Proof { unProof :: Tagged a b }

Runtime check for unknown naturals

> fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) > fromSome (SomeNat p) > | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) > | otherwise = Nothing

Compiletime converter for known naturals

> fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) > fromKnown n = Proof $ Tagged n

Function to test:

> f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () > f2 _ = ()

Example of use:

> main :: IO () > main = do > [arg] <- getArgs > let n = read arg :: Integer > > case someNatVal n of > Nothing -> error "Input is not a natural number!" > Just sn -> case fromSome sn of > Just p -> return $ f2 p > _ -> error "Input if larger than 255"

On 25 November 2014 at 10:51, Bas van Dijk wrote: > Hi, > > I have another type-level programming related question: > >> {-# LANGUAGE GADTs #-} >> {-# LANGUAGE TypeOperators #-} >> {-# LANGUAGE ScopedTypeVariables #-} >> {-# LANGUAGE KindSignatures #-} >> >> import GHC.TypeLits > > Say I have a Proxy p of some type-level natural number: > >> p :: forall (n :: Nat). Proxy n >> p = Proxy > > Imagine I get p from user input like this: > >> main :: IO () >> main = do >> [arg] <- getArgs >> let n = read arg :: Integer >> >> case someNatVal n of >> Nothing -> error "Input is not a natural number!" >> Just (SomeNat (p :: Proxy n)) -> ... > > I also have a function f which takes a proxy of a natural number but > it has the additional constraint that the number should be lesser than > or equal to 255: > >> f :: forall (n :: Nat). (n <= 255) => proxy n -> () >> f _ = () > > How do I apply f to p? > > Obviously, applying it directly gives the type error: > >> f p > <interactive>:179:1: > Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ > The type variable ‘n0’ is ambiguous > In the expression: f p > In an equation for ‘it’: it = f p > > I imagine I somehow need to construct some Proof object using a function g like: > >> g :: forall (n :: Nat). proxy n -> Proof >> g _ = ... > > Where the Proof constructor encapsulates the (n <= 255) constraint: > >> data Proof where >> NoProof :: Proof >> Proof :: forall (n :: Nat). (n <= 255) >> => Proxy n -> Proof > > With g in hand I can construct c which patterns matches on g p and > when there's a Proof the (n <= 255) constraint will be in scope which > allows applying f to p: > >> c :: () >> c = case g p of >> NoProof -> error "Input is bigger than 255!" >> Proof p -> f p > > But how do I define g? > > Cheers, > > Bas > _______________________________________________ > Glasgow-haskell-users mailing list > Glasgow-haskell-users@haskell.org > http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander _______________________________________________ Glasgow-haskell-users mailing list Glasgow-haskell-users@haskell.org http://www.haskell.org/mailman/listinfo/glasgow-haskell-users

-- Alexander

-- Alexander

-- Alexander

Richard Eisenberg

8 Dec 8 Dec

1:09 p.m.

Sadly, I don't have a go-to place for this stuff. Perhaps it will be helpful to see an example of this in action? Here is my unary version of this:

...

{-# LANGUAGE TemplateHaskell, DataKinds, PolyKinds, TypeFamilies, ScopedTypeVariables, TypeOperators, UndecidableInstances, GADTs, RankNTypes #-} {-# OPTIONS_GHC -ftype-function-depth=300 -fcontext-stack=300 #-}

import Data.Singletons.TH import GHC.TypeLits hiding ( Nat )

$(singletons [d| data Nat = Zero | Succ Nat

leNat :: Nat -> Nat -> Bool leNat Zero _ = True leNat (Succ _) Zero = False leNat (Succ a) (Succ b) = a `leNat` b |])

-- | Singletons's 'withSomeSing' is what we want, but a bug in 7.8.3 doesn't -- let it work without a specialized type for 'Nat's withSomeNat :: Nat -> (forall (n :: Nat). Sing n -> r) -> r withSomeNat = withSomeSing

-- | Conveniently generate unary naturals type family U n where U 0 = Zero U n = Succ (U (n-1))

toNat :: Integer -> Maybe Nat toNat n | n < 0 = Nothing | otherwise = Just $ go n where go 0 = Zero go n = Succ (go (n-1))

type Bound = U 255

-- easier to test in GHCi than a proper 'main' go :: Integer -> IO () go n = case toNat n of Nothing -> putStrLn "Input is not a natural number!" Just nat -> putStrLn $ withSomeNat nat $ \ snat -> case snat `sLeNat` (sing :: Sing Bound) of STrue -> f snat SFalse -> "Didn't work"

f :: forall proxy (n :: Nat). (n `LeNat` Bound) ~ True => proxy n -> String f _ = "It worked!"

I'm happy to answer questions, but it's hard to know how much detail to use in a description of the code. I hope this helps, Richard On Dec 4, 2014, at 5:50 PM, Alexander V Vershilov wrote:

...

Hi, Richard

Can you give some ideas or where to read how to properly use signletons and unary naturals in order to be able to express such constraints?

Thanks -- Alexander

On 30 November 2014 at 23:26, Richard Eisenberg wrote:

...
Hi Alexander,

Nice idea to test against the set of known values. That's more type-safe than anything I've thought of. I agree that it's a bit of a painful construction, but I don't think we can do better with type-lits, as there is limited reasoning ability that GHC can manage. If you want to switch to unary naturals (`data Nat = Zero | Succ Nat`), then this can be built somewhat nicely with singletons and without unsafeCoerce. But, of course, unary naturals are very slow.

By the way, the bug in the Proof2 version is a bug in GHC 7.8.3 (only in .3 -- not in .2 or in the soon-to-be .4) that allows you to write unsaturated type families that don't work. Saying `LessThan255` without a parameter should be a syntax error, but that check was accidentally turned off for 7.8.3, leading to a bogus type error.

Thanks for sharing this work! Richard

On Nov 28, 2014, at 5:27 PM, Alexander V Vershilov wrote:

...
Hi, Bas, Richard.

I've played a bit with example, obvously first approach contained bugs, but seems that I have fixed it and here I have 2 approaches, one uses unsafeCoerce (as Richard suggested) another is safe but with bad complexity:

Full file is quite big, so I'm not inlining it in mail, but here is a link:

https://github.com/qnikst/haskell-fun/blob/master/typelevel-literals/Proof.l...

I wonder how far it's possible to go with singletons approach that I have not tried yet.

-- Alexander

On 26 November 2014 at 10:15, Bas van Dijk wrote:

...
Hi Alexander,

Thanks for your answer! This provides a lot of ideas how to proceed.

I'm unsure about the following though:

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Doesn't this mean lessThen returns a Proxy n for all (n :: Nat) and not just the Nat inside the SomeNat?

I also see that p is only used for comparing it to k. It's not used to produce the return value.

Cheers,

Bas

On 25 November 2014 at 19:55, Alexander V Vershilov wrote:

...
Hi, Richard, Bas.

Maybe I didn't spell it properly but my point was to create a data type that carry a proof without exposing it's constructor and having clever constructor only, then the only place where you need to check will be that constructor.

Also it's possible to write in slightly clearer, but again it's possible to make a mistake here and it will be a false proof.

...
lessThen :: KnownNat m => SomeNat -> Proxy m -> Maybe (Proof (n <= m) (Proxy n)) lessThen (SomeNat p) k | natVal p <= natVal k = Just (Proof $ Tagged (Proxy :: Proxy n)) | otherwise = Nothing

Of cause solution using singletons could solve this problem much better.

-- Alexander

On 25 November 2014 at 21:34, Richard Eisenberg wrote:

...
Hi Bas,

I believe to do this "right", you would need singleton types. Then, when you discover that the number is bounded by 255, you would also discover that the type is bounded by 255, and you'd be home free.

Unfortunately, I there isn't currently a way to do comparison on GHC.TypeLits Nats with singletons. (There is a module Data.Singletons.TypeLits in the `singletons` package, but there's a comment telling me TODO in the part where comparison should be implemented.) If it were implemented, it would have to use unsafeCoerce, as there's no built-in mechanism connecting runtime numbers with TypeLits.

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

The solution Alexander provides below doesn't quite build a proof, I think. Tellingly, if we omit the `natVal p <= 255` check, everything else still compiles. Thus, the `Proof` type he uses can be built even if the fact proven is false. That said, I don't know if my solution is any better, crucially relying on unsafeCoerce.

Richard

On Nov 25, 2014, at 4:52 AM, Alexander V Vershilov wrote:

> Hi, > > Following approach can work, the idea is to define a type that will > carry a proof (constraint) that we want to check. Here I have reused > Data.Tagged, but it's possible to introduce your own with concrete > constraints. > >> {-# LANGUAGE DataKinds #-} >> {-# LANGUAGE GADTs #-} >> {-# LANGUAGE TypeOperators #-} >> {-# LANGUAGE KindSignatures #-} >> {-# LANGUAGE PolyKinds #-} >> {-# LANGUAGE UndecidableInstances #-} >> import GHC.TypeLits >> import GHC.Exts >> import Data.Proxy >> import Data.Tagged >> import System.Environment > > New constraint carrying data type > >> newtype Proof a b = Proof { unProof :: Tagged a b } > > Runtime check for unknown naturals > >> fromSome :: SomeNat -> Maybe (Proof (n <= 255) (Proxy n)) >> fromSome (SomeNat p) >> | natVal p <= 255 = Just (Proof $ Tagged (Proxy :: Proxy n)) >> | otherwise = Nothing > > Compiletime converter for known naturals > >> fromKnown :: (KnownNat n, n <= 255) => Proxy n -> Proof (n <= 255) (Proxy n) >> fromKnown n = Proof $ Tagged n > > Function to test: > >> f2 :: (c ~ (n <= 255)) => Proof c (Proxy n) -> () >> f2 _ = () > > Example of use: > >> main :: IO () >> main = do >> [arg] <- getArgs >> let n = read arg :: Integer >> >> case someNatVal n of >> Nothing -> error "Input is not a natural number!" >> Just sn -> case fromSome sn of >> Just p -> return $ f2 p >> _ -> error "Input if larger than 255" > > > On 25 November 2014 at 10:51, Bas van Dijk wrote: >> Hi, >> >> I have another type-level programming related question: >> >>> {-# LANGUAGE GADTs #-} >>> {-# LANGUAGE TypeOperators #-} >>> {-# LANGUAGE ScopedTypeVariables #-} >>> {-# LANGUAGE KindSignatures #-} >>> >>> import GHC.TypeLits >> >> Say I have a Proxy p of some type-level natural number: >> >>> p :: forall (n :: Nat). Proxy n >>> p = Proxy >> >> Imagine I get p from user input like this: >> >>> main :: IO () >>> main = do >>> [arg] <- getArgs >>> let n = read arg :: Integer >>> >>> case someNatVal n of >>> Nothing -> error "Input is not a natural number!" >>> Just (SomeNat (p :: Proxy n)) -> ... >> >> I also have a function f which takes a proxy of a natural number but >> it has the additional constraint that the number should be lesser than >> or equal to 255: >> >>> f :: forall (n :: Nat). (n <= 255) => proxy n -> () >>> f _ = () >> >> How do I apply f to p? >> >> Obviously, applying it directly gives the type error: >> >>> f p >> <interactive>:179:1: >> Couldn't match expected type ‘'True’ with actual type ‘n0 <=? 255’ >> The type variable ‘n0’ is ambiguous >> In the expression: f p >> In an equation for ‘it’: it = f p >> >> I imagine I somehow need to construct some Proof object using a function g like: >> >>> g :: forall (n :: Nat). proxy n -> Proof >>> g _ = ... >> >> Where the Proof constructor encapsulates the (n <= 255) constraint: >> >>> data Proof where >>> NoProof :: Proof >>> Proof :: forall (n :: Nat). (n <= 255) >>> => Proxy n -> Proof >> >> With g in hand I can construct c which patterns matches on g p and >> when there's a Proof the (n <= 255) constraint will be in scope which >> allows applying f to p: >> >>> c :: () >>> c = case g p of >>> NoProof -> error "Input is bigger than 255!" >>> Proof p -> f p >> >> But how do I define g? >> >> Cheers, >> >> Bas >> _______________________________________________ >> Glasgow-haskell-users mailing list >> Glasgow-haskell-users@haskell.org >> http://www.haskell.org/mailman/listinfo/glasgow-haskell-users > > > > -- > Alexander > _______________________________________________ > Glasgow-haskell-users mailing list > Glasgow-haskell-users@haskell.org > http://www.haskell.org/mailman/listinfo/glasgow-haskell-users >

-- Alexander

-- Alexander

-- Alexander

Ranjit Jhala

1:45 p.m.

Dear Alexander, FWIW, this sort of thing is quite straightforward with LiquidHaskell: http://goto.ucsd.edu:8090/index.html#?demo=permalink%2F1418064183.hs or, the code below: ----- {-@ LIQUID "--no-termination" @-} module Nat255 where -- | Define a predicate for valid integers {-@ predicate IsValid X = 0 <= X && X < 255 @-} -- | Use the predicate to define a refinement type (subset) of valid integers {-@ type Valid = {v:Int | IsValid v} @-} -- | A function that checks whether a given Int is indeed valid {-@ isValid :: n:Int -> {v:Bool | Prop v <=> IsValid n} @-} isValid n = 0 <= n && n < (255 :: Int) -- | A function that can only be called with Valid Ints. {-@ workWithValidNumber :: Valid -> IO () @-} workWithValidNumber n = putStrLn $ "This is a valid number" ++ show (n :: Int) -- | This is fine... ok = workWithValidNumber 12 -- | ... But this is not. notOk = workWithValidNumber 257 -- | Finally the top level loop that inputs a number, tests it -- and calls `workWithValidNumber` if the number is valid. loop = do putStrLn "Enter Number between 0 and 255" n <- readLn :: IO Int if isValid n then workWithValidNumber n else putStrLn "Humph, bad input, try again!" >> loop

Alexander V Vershilov

19 Dec 19 Dec

5:53 p.m.

Richard, Ranjit, Thanks for providing your solutions. -- Alexander. On 8 December 2014 at 21:45, Ranjit Jhala wrote:

...

Dear Alexander,

FWIW, this sort of thing is quite straightforward with LiquidHaskell:

http://goto.ucsd.edu:8090/index.html#?demo=permalink%2F1418064183.hs

or, the code below:

-----

{-@ LIQUID "--no-termination" @-}

module Nat255 where

-- | Define a predicate for valid integers

{-@ predicate IsValid X = 0 <= X && X < 255 @-}

-- | Use the predicate to define a refinement type (subset) of valid integers

{-@ type Valid = {v:Int | IsValid v} @-}

-- | A function that checks whether a given Int is indeed valid

{-@ isValid :: n:Int -> {v:Bool | Prop v <=> IsValid n} @-} isValid n = 0 <= n && n < (255 :: Int)

-- | A function that can only be called with Valid Ints.

{-@ workWithValidNumber :: Valid -> IO () @-} workWithValidNumber n = putStrLn $ "This is a valid number" ++ show (n :: Int)

-- | This is fine... ok = workWithValidNumber 12

-- | ... But this is not. notOk = workWithValidNumber 257

-- | Finally the top level loop that inputs a number, tests it -- and calls `workWithValidNumber` if the number is valid.

loop = do putStrLn "Enter Number between 0 and 255" n <- readLn :: IO Int if isValid n then workWithValidNumber n else putStrLn "Humph, bad input, try again!" >> loop

-- Alexander

Bas van Dijk

26 Nov 26 Nov

2:19 a.m.

On 25 November 2014 at 19:34, Richard Eisenberg wrote:

...

If I were you, I would just write `g` using unsafeCoerce in the right spot, instead of bothering with all the singletons, which would have to use unsafety anyway.

Thanks, I hadn't considered this yet. Cheers, Bas

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Alexander V Vershilov
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Richard Eisenberg