[Haskell-cafe] using scoped type variables in proofs...

Mon May 11 12:43:33 UTC 2015

That worked!....

But doesn’t seem to be true of all the previous cases!

i.e.

The b1 here is bound quite happily

> theoremPlusAbelian (SZ :: SNat a) ((SS a) :: SNat b) = 
> -- forall 0 and a + 1
>   case theoremPlusAbelian (SZ :: SNat a) (a :: (b ~ 'S b1) => SNat b1) of
>     (Refl :: (a :+ b1) :== (b1 :+ a)) -> 
>       (Refl :: (a :+ b) :== (b :+ a))

yet

> theoremPlusAbelian ((SS a) :: (SNat a)) ((SS b) :: (SNat b))      = 
> -- forall a+1 and b+1
> -- 1st prove (a + 1) + b =  b + (a + 1)...from above
>   case theoremPlusAbelian ((SS a) :: (SNat a)) (b :: (b ~ 'S b1) => SNat b1) of
>      (Refl :: (a :+ b1) :== (b1 :+ a)) -> 

Complains

There seems little difference?

>GHC tells you that b1 is ambiguous, so that should tell you that it isn't getting
>bound in the case expression as you'd think.
>
>The reason is that type variables only get bound at value bindings; but you
>attached the signature to an expression.
>
>To fix it, move the annotation to the pattern like this:
>
>theoremPlusAbelian ((SS a) :: (SNat a)) ((SS (b :: SNat b1)) :: (SNat b)) =
>
>
>On 11/05/15 13:33, Nicholls, Mark wrote:
>> Hello,
>>
>>
>>
>> I have written the below proof as an exercise.
>>
>>
>>
>> I want to explicitly annotate the proof with type variables
..but I
>> cant get a line to work

>>
>>
>>
>>> {-# LANGUAGE DataKinds #-}
>>
>>> {-# LANGUAGE ExplicitForAll #-}
>>
>>> {-# LANGUAGE FlexibleContexts #-}
>>
>>> {-# LANGUAGE FlexibleInstances #-}
>>
>>> {-# LANGUAGE GADTs #-}
>>
>>> {-# LANGUAGE MultiParamTypeClasses #-}
>>
>>> {-# LANGUAGE PolyKinds #-}
>>
>>> {-# LANGUAGE StandaloneDeriving #-}
>>
>>> {-# LANGUAGE TypeFamilies #-}
>>
>>> {-# LANGUAGE TypeOperators #-}
>>
>>> {-# LANGUAGE UndecidableInstances #-}
>>
>>> {-# LANGUAGE ScopedTypeVariables #-}
>>
>>
>>
>>> import Prelude hiding (head, tail, (++), (+), replicate)
>>
>>> import qualified Prelude as P
>>
>>
>>
>>> data Nat where
>>
>>>   Z :: Nat
>>
>>>   S :: Nat -> Nat
>>
>>
>>
>>> data SNat (a :: Nat) where
>>
>>>   SZ :: SNat 'Z
>>
>>>   SS :: SNat a -> SNat ('S a)
>>
>>
>>
>> here a nice plus
>>
>>
>>
>>> type family   (n :: Nat) :+ (m :: Nat) :: Nat
>>
>>> type instance Z     :+ m = m
>>
>>> type instance (S n) :+ m = S (n :+ m)
>>
>>
>>
>> lets try to do
>>
>>
>>
>>> data val1 :== val2 where
>>
>>>   Refl :: val :== val
>>
>>
>>
>> If I prove its abelian then we get it...
>>
>>
>>
>> the "proof" works if we remove the type annotation
>>
>> but I want the annotation to convince myself I know whats going on.
>>
>>
>>
>>
>>
>>> theoremPlusAbelian :: SNat a -> SNat b -> (a :+ b) :== (b :+ a)
>>
>>> theoremPlusAbelian (SZ :: SNat a) (SZ :: SNat b) =
>>
>>> -- forall 0 and 0
>>
>>>   (Refl :: (a :+ b) :== (b :+ a))
>>
>>> theoremPlusAbelian ((SS a) :: SNat a) (SZ :: SNat b) =
>>
>>> -- forall a + 1 and 0
>>
>>>   case theoremPlusAbelian (a :: (a ~ 'S a1) => SNat a1) (SZ :: SNat
>>> b) of
>>
>>>     (Refl :: (a1 :+ b) :== (b :+ a1)) ->
>>
>>>       (Refl :: (a :+ b) :== (b :+ a))
>>
>>> theoremPlusAbelian (SZ :: SNat a) ((SS a) :: SNat b) =
>>
>>> -- forall 0 and a + 1
>>
>>>   case theoremPlusAbelian (SZ :: SNat a) (a :: (b ~ 'S b1) => SNat
>>> b1) of
>>
>>>     (Refl :: (a :+ b1) :== (b1 :+ a)) ->
>>
>>>       (Refl :: (a :+ b) :== (b :+ a))
>>
>>> -- cant seem to prove this...
>>
>>> theoremPlusAbelian ((SS a) :: (SNat a)) ((SS b) :: (SNat b))      =
>>
>>> -- forall a+1 and b+1
>>
>>> -- 1st prove (a + 1) + b =  b + (a + 1)...from above
>>
>>>   case theoremPlusAbelian ((SS a) :: (SNat a)) (b :: (b ~ 'S b1) =>
>> SNat b1) of
>>
>>
>>
>> THE COMMENTED LINE FAILS.
>>
>>
>>
>> Could not deduce ((b1 :+ 'S a1) ~ (a2 :+ 'S a1))
>>
>>     from the context (a ~ 'S a1)
>>
>>       bound by a pattern with constructor
>>
>>                  SS :: forall (a :: Nat). SNat a -> SNat ('S a),
>>
>>                in an equation for ‘theoremPlusAbelian’
>>
>>       at Cafe2.lhs:57:24-27
>>
>>     or from (b ~ 'S a2)
>>
>>       bound by a pattern with constructor
>>
>>                  SS :: forall (a :: Nat). SNat a -> SNat ('S a),
>>
>>                in an equation for ‘theoremPlusAbelian’
>>
>>       at Cafe2.lhs:57:45-48
>>
>>     NB: ‘:+’ is a type function, and may not be injective
>>
>>     The type variable ‘b1’ is ambiguous
>>
>>     Expected type: (a :+ a2) :== (a2 :+ a)
>>
>>       Actual type: (a :+ b1) :== (b1 :+ a)
>>
>>     Relevant bindings include
>>
>>       b :: SNat a2 (bound at Cafe2.lhs:57:48)
>>
>>       a :: SNat a1 (bound at Cafe2.lhs:57:27)
>>
>>     In the pattern: Refl :: (a :+ b1) :== (b1 :+ a)
>>
>>     In a case alternative:
>>
>>         (Refl :: (a :+ b1) :== (b1 :+ a))
>>
>>           -> case theoremPlusAbelian a (SS b) of {
>>
>>                Refl -> case theoremPlusAbelian a b of { Refl -> Refl }
>> }
>>
>>     In the expression:
>>
>>       case
>>
>>           theoremPlusAbelian ((SS a) :: SNat a) (b :: b ~ S b1 => SNat
>> b1)
>>
>>       of {
>>
>>         (Refl :: (a :+ b1) :== (b1 :+ a))
>>
>>           -> case theoremPlusAbelian a (SS b) of {
>>
>>                Refl -> case theoremPlusAbelian a b of { Refl -> ... }
>> } }
>>
>>
>>
>>
>>
>>
>>
>>>     -- (Refl :: (a :+ b1) :== (b1 :+ a)) ->
>>
>>>     Refl ->
>>
>>> -- now prove a + (b + 1) =  (b + 1) + a...from above
>>
>>>       case theoremPlusAbelian a (SS b) of
>>
>>>         Refl ->
>>
>>> -- now prove a + b = b + a
>>
>>>           case theoremPlusAbelian a b of
>>
>>> -- which seems to have proved a + b = b + a
>>
>>>             Refl -> Refl
>>
>>
>>
>>
>>
>>
>>
>>
>>
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This e-mail (and any attached files) is confidential and protected by copyright (and other intellectual property rights). If you are not the intended recipient please e-mail the sender and then delete the email and any attached files immediately. Any further use or dissemination is prohibited.

While MTV Networks Europe has taken steps to ensure that this email and any attachments are virus free, it is your responsibility to ensure that this message and any attachments are virus free and do not affect your systems / data.

Communicating by email is not 100% secure and carries risks such as delay, data corruption, non-delivery, wrongful interception and unauthorised amendment. If you communicate with us by e-mail, you acknowledge and assume these risks, and you agree to take appropriate measures to minimise these risks when e-mailing us.

MTV Networks International, MTV Networks UK & Ireland, Greenhouse, Nickelodeon Viacom Consumer Products, VBSi, Viacom Brand Solutions International, Be Viacom, Viacom International Media Networks and VIMN and Comedy Central are all trading names of MTV Networks Europe.  MTV Networks Europe is a partnership between MTV Networks Europe Inc. and Viacom Networks Europe Inc.  Address for service in Great Britain is 17-29 Hawley Crescent, London, NW1 8TT.