Inductive Empty : Prop := .
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Arguments idpath {A a} , [A] a.
Definition idmap {A : Type} : A -> A := fun x => x.
Definition path_sum {A B : Type} (z z' : A + B)
(pq : match z, z' with
| inl z0, inl z'0 => z0 = z'0
| inr z0, inr z'0 => z0 = z'0
| _, _ => Empty
end)
: z = z'.
destruct z, z', pq; exact idpath.
Defined.
Definition ap {A B:Type} (f:A -> B) {x y:A} (p:x = y) : f x = f y
:= match p with idpath => idpath end.

Theorem ex2_8 {A B A' B' : Type} (g : A -> A') (h : B -> B') (x y : A + B)
(pq : match (x, y) with (inl x', inl y') => x' = y' | (inr x', inr y') => x' = y' | _ => Empty end) :
let f z := match z with inl z' => inl (g z') | inr z' => inr (h z') end in
ap f (path_sum x y pq) = path_sum (f x) (f y)
((match x as x return match (x, y) with
(inl x', inl y') => x' = y'
| (inr x', inr y') => x' = y'
| _ => Empty
end -> match (f x, f y) with
| (inl x', inl y') => x' = y'
| (inr x', inr y') => x' = y'
| _ => Empty end with
| inl x' => match y as y return match y with
inl y' => x' = y'
| _ => Empty
end -> match f y with
| inl y' => g x' = y'
| _ => Empty end with
| inl y' => ap g
| inr y' => idmap
end
| inr x' => match y as y return match y return Prop with
inr y' => x' = y'
| _ => Empty
end -> match f y return Prop with
| inr y' => h x' = y'
| _ => Empty end with
| inl y' => idmap
| inr y' => ap h
end
end) pq).