I'm working through the Idris book, and I'm doing the first exercises on proof.
With the exercise to prove same_lists
, I'm able to implement it like this, as matching Refl
forces x
and y
to unify:
total same_lists : {xs : List a} -> {ys : List a} ->
x = y -> xs = ys -> x :: xs = y :: ys
same_lists Refl Refl = Refl
However, when I try to prove something else in the same manner, I get mismatches. For example:
total allSame2 : (x, y : Nat) -> x = y -> S x = S y
allSame2 x y Refl = Refl
The compiler says:
Type mismatch between y = y (Type of Refl) and x = y (Expected type)
If I case-match after the =
, either explicitly or with a lambda, it works as expected:
total allSame2 : (x : Nat) -> (y : Nat) -> x = y -> S x = S y
allSame2 x y = \Refl => Refl
What's the difference here?
Another modification that works is making the problematic arguments implicit:
total allSame2 : {x : Nat} -> {y : Nat} -> x = y -> S x = S y
allSame2 Refl = Refl