[Haskell-cafe] Mild confusion around type family

[Nicholls, Mark]

Below is a very simple almost canonical natural number model.

I'm slightly confused by how type families work

Wikipedia says....

"Type families are a feature of some type systems<https://en.wikipedia.org/wiki/Type_system> that allow PARTIAL functions between types to be defined by pattern matching<https://en.wikipedia.org/wiki/Pattern_matching>"

So...

> {-# LANGUAGE DataKinds #-}
> {-# LANGUAGE ExplicitForAll #-}
> {-# LANGUAGE FlexibleContexts #-}
> {-# LANGUAGE FlexibleInstances #-}
> {-# LANGUAGE GADTs #-}
> {-# LANGUAGE MultiParamTypeClasses #-}
> {-# LANGUAGE PolyKinds #-}
> {-# LANGUAGE StandaloneDeriving #-}
> {-# LANGUAGE TypeFamilies #-}
> {-# LANGUAGE TypeOperators #-}
> {-# LANGUAGE UndecidableInstances #-}
> {-# LANGUAGE ScopedTypeVariables #-}

lets do something simple

> data Z
> data S n

> type family Sub a b

> -- a - 0 == a
> type instance Sub a Z = a
> -- (a + 1) - (b + 1) == a - b
> type instance Sub (S a) (S b) = Sub a b

> oneMinusOne :: Z
> oneMinusOne = undefined :: Sub (S Z) (S Z)

I want this to be a type error!...but the above type family appears to not be partial...this IS defined...."Sub Z (S Z)" is a type! (I thought it wasn't)

> oneMinusTwo :: Sub Z (S Z)
> oneMinusTwo = undefined :: Sub (S Z) (S (S Z))


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