Theory and Applications of Type-Theoretic Universes

Future work: extensional propositions

In EMTT, we have a monotonically increasing knowledge base about equality that we use for conversion checking.

Why not a monotonically increasing knowledge base for other things?

Ancestry :\equiv (P : U) \times (parent : P \to P \to Prop) \times (ancestor : P \to P \to Prop) \times (_: (x y : P) \to parent x y \to ancestor x y) \times (_: (x y z : P) \to parent x y \to ancestor y z \to ancestor x z) AncestryDB : {open A : Ancestry} \to U AncestryDB :\equiv {∣ bob : P, sarah : P, sam : P,_: parent bob sarah,_: parent sarah sam ∣}