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

[Roman Cheplyaka]

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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