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<p class="MsoNormal">Hello,<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">I have written the below proof as an exercise.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">I want to explicitly annotate the proof with type variables…..but I cant get a line to work…<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> {-# LANGUAGE DataKinds #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE ExplicitForAll #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleContexts #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleInstances #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE GADTs #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE MultiParamTypeClasses #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE PolyKinds #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE StandaloneDeriving #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE TypeFamilies #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE TypeOperators #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE UndecidableInstances #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE ScopedTypeVariables #-}<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> import Prelude hiding (head, tail, (++), (+), replicate)<o:p></o:p></p>
<p class="MsoNormal">> import qualified Prelude as P<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data Nat where<o:p></o:p></p>
<p class="MsoNormal">> Z :: Nat<o:p></o:p></p>
<p class="MsoNormal">> S :: Nat -> Nat<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data SNat (a :: Nat) where<o:p></o:p></p>
<p class="MsoNormal">> SZ :: SNat 'Z<o:p></o:p></p>
<p class="MsoNormal">> SS :: SNat a -> SNat ('S a)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">here a nice plus<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> type family (n :: Nat) :+ (m :: Nat) :: Nat<o:p></o:p></p>
<p class="MsoNormal">> type instance Z :+ m = m<o:p></o:p></p>
<p class="MsoNormal">> type instance (S n) :+ m = S (n :+ m)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">lets try to do<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data val1 :== val2 where<o:p></o:p></p>
<p class="MsoNormal">> Refl :: val :== val<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">If I prove its abelian then we get it...<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">the "proof" works if we remove the type annotation<o:p></o:p></p>
<p class="MsoNormal">but I want the annotation to convince myself I know whats going on.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> theoremPlusAbelian :: SNat a -> SNat b -> (a :+ b) :== (b :+ a)<o:p></o:p></p>
<p class="MsoNormal">> theoremPlusAbelian (SZ :: SNat a) (SZ :: SNat b) = <o:p></o:p></p>
<p class="MsoNormal">> -- forall 0 and 0<o:p></o:p></p>
<p class="MsoNormal">> (Refl :: (a :+ b) :== (b :+ a))<o:p></o:p></p>
<p class="MsoNormal">> theoremPlusAbelian ((SS a) :: SNat a) (SZ :: SNat b) = <o:p>
</o:p></p>
<p class="MsoNormal">> -- forall a + 1 and 0<o:p></o:p></p>
<p class="MsoNormal">> case theoremPlusAbelian (a :: (a ~ 'S a1) => SNat a1) (SZ :: SNat b) of<o:p></o:p></p>
<p class="MsoNormal">> (Refl :: (a1 :+ b) :== (b :+ a1)) -><o:p></o:p></p>
<p class="MsoNormal">> (Refl :: (a :+ b) :== (b :+ a))<o:p></o:p></p>
<p class="MsoNormal">> theoremPlusAbelian (SZ :: SNat a) ((SS a) :: SNat b) = <o:p>
</o:p></p>
<p class="MsoNormal">> -- forall 0 and a + 1<o:p></o:p></p>
<p class="MsoNormal">> case theoremPlusAbelian (SZ :: SNat a) (a :: (b ~ 'S b1) => SNat b1) of<o:p></o:p></p>
<p class="MsoNormal">> (Refl :: (a :+ b1) :== (b1 :+ a)) -> <o:p></o:p></p>
<p class="MsoNormal">> (Refl :: (a :+ b) :== (b :+ a))<o:p></o:p></p>
<p class="MsoNormal">> -- cant seem to prove this...<o:p></o:p></p>
<p class="MsoNormal">> theoremPlusAbelian ((SS a) :: (SNat a)) ((SS b) :: (SNat b)) =
<o:p></o:p></p>
<p class="MsoNormal">> -- forall a+1 and b+1<o:p></o:p></p>
<p class="MsoNormal">> -- 1st prove (a + 1) + b = b + (a + 1)...from above<o:p></o:p></p>
<p class="MsoNormal">> case theoremPlusAbelian ((SS a) :: (SNat a)) (b :: (b ~ 'S b1) => SNat b1) of<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">THE COMMENTED LINE FAILS.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Could not deduce ((b1 :+ 'S a1) ~ (a2 :+ 'S a1))<o:p></o:p></p>
<p class="MsoNormal"> from the context (a ~ 'S a1)<o:p></o:p></p>
<p class="MsoNormal"> bound by a pattern with constructor<o:p></o:p></p>
<p class="MsoNormal"> SS :: forall (a :: Nat). SNat a -> SNat ('S a),<o:p></o:p></p>
<p class="MsoNormal"> in an equation for ‘theoremPlusAbelian’<o:p></o:p></p>
<p class="MsoNormal"> at Cafe2.lhs:57:24-27<o:p></o:p></p>
<p class="MsoNormal"> or from (b ~ 'S a2)<o:p></o:p></p>
<p class="MsoNormal"> bound by a pattern with constructor<o:p></o:p></p>
<p class="MsoNormal"> SS :: forall (a :: Nat). SNat a -> SNat ('S a),<o:p></o:p></p>
<p class="MsoNormal"> in an equation for ‘theoremPlusAbelian’<o:p></o:p></p>
<p class="MsoNormal"> at Cafe2.lhs:57:45-48<o:p></o:p></p>
<p class="MsoNormal"> NB: ‘:+’ is a type function, and may not be injective<o:p></o:p></p>
<p class="MsoNormal"> The type variable ‘b1’ is ambiguous<o:p></o:p></p>
<p class="MsoNormal"> Expected type: (a :+ a2) :== (a2 :+ a)<o:p></o:p></p>
<p class="MsoNormal"> Actual type: (a :+ b1) :== (b1 :+ a)<o:p></o:p></p>
<p class="MsoNormal"> Relevant bindings include<o:p></o:p></p>
<p class="MsoNormal"> b :: SNat a2 (bound at Cafe2.lhs:57:48)<o:p></o:p></p>
<p class="MsoNormal"> a :: SNat a1 (bound at Cafe2.lhs:57:27)<o:p></o:p></p>
<p class="MsoNormal"> In the pattern: Refl :: (a :+ b1) :== (b1 :+ a)<o:p></o:p></p>
<p class="MsoNormal"> In a case alternative:<o:p></o:p></p>
<p class="MsoNormal"> (Refl :: (a :+ b1) :== (b1 :+ a))<o:p></o:p></p>
<p class="MsoNormal"> -> case theoremPlusAbelian a (SS b) of {<o:p></o:p></p>
<p class="MsoNormal"> Refl -> case theoremPlusAbelian a b of { Refl -> Refl } }<o:p></o:p></p>
<p class="MsoNormal"> In the expression:<o:p></o:p></p>
<p class="MsoNormal"> case<o:p></o:p></p>
<p class="MsoNormal"> theoremPlusAbelian ((SS a) :: SNat a) (b :: b ~ S b1 => SNat b1)<o:p></o:p></p>
<p class="MsoNormal"> of {<o:p></o:p></p>
<p class="MsoNormal"> (Refl :: (a :+ b1) :== (b1 :+ a))<o:p></o:p></p>
<p class="MsoNormal"> -> case theoremPlusAbelian a (SS b) of {<o:p></o:p></p>
<p class="MsoNormal"> Refl -> case theoremPlusAbelian a b of { Refl -> ... } } }<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> -- (Refl :: (a :+ b1) :== (b1 :+ a)) -> <o:p></o:p></p>
<p class="MsoNormal">> Refl -> <o:p></o:p></p>
<p class="MsoNormal">> -- now prove a + (b + 1) = (b + 1) + a...from above<o:p></o:p></p>
<p class="MsoNormal">> case theoremPlusAbelian a (SS b) of<o:p></o:p></p>
<p class="MsoNormal">> Refl -> <o:p></o:p></p>
<p class="MsoNormal">> -- now prove a + b = b + a<o:p></o:p></p>
<p class="MsoNormal">> case theoremPlusAbelian a b of<o:p></o:p></p>
<p class="MsoNormal">> -- which seems to have proved a + b = b + a<o:p></o:p></p>
<p class="MsoNormal">> Refl -> Refl<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
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