Getting path induction to work in Agda

Question

I can't figure out why my path induction isn't type checking correctly. It says "C x should be a function type, but it isn't" when referring to C (refl x). Perhaps my definition of refl is wrong or is there something wrong with my {}'s and ()'s?

data _≡_ {A : Set}(a : A) : A → Set where
      refl : a ≡ a
infix 4 _≡_

pathInd : ∀ {u} → {A : Set} → 
          (C : {x y : A} → x ≡ y → Set u) → 
          (c : (x : A) → C (refl x)) → 
          ({x y : A} (p : x ≡ y) → C p)
pathInd C c (refl x) = c x

Answer 1

refl is not a function. Here is the definition you need:

pathInd : ∀ {u} → {A : Set} → 
          (C : {x y : A} → x ≡ y → Set u) → 
          (c : (x : A) → C {x} refl) → 
          ({x y : A} (p : x ≡ y) → C p)
pathInd C c {x} refl = c x

Also, your pathInd works properly with this definition of _≡_:

data _≡_ {A : Set} : A → A → Set where
      refl : ∀ a -> a ≡ a