Axiom J in Homotopy Type Theory (HoTT)

Preface Happy new years! Notation ∀ may be used in place of a Pi-type in some cases. A nested Π (a b : A). ... (or sim. for Σ ) represents Π (a : A). Π (b : A). ... . J Axiom J is the following: Given: A : Type x y : A P : Π (a b : A). a = b -> Type with P x x refl and eq : x = y we can conclude P x y eq where refl : ∀x. x = x . In short, for any property P , if P is true for x x refl , and we…