  In homotopy type theory (HoTT), all constructions are necessarily stable
under homotopy equivalence. This has shortcomings: for example, it is believed
that it is impossible to define a type of semi-simplicial types. More
generally, it is difficult and often impossible to handle towers of coherences.
To address this, we propose a 2-level theory which features both strict and
weak equality. This can essentially be represented as two type theories: an
"outer" one, containing a strict equality type former, and an "inner" one,
which is some version of HoTT. Our type theory is inspired by Voevosky's
suggestion of a homotopy type system (HTS) which currently refers to a range of
ideas. A core insight of our proposal is that we no not need any form of
equality reflection in order to achieve what HTS was suggested for. Instead,
having unique identity proofs in the outer type theory is sufficient, and it
also has the meta-theoretical advantage of not breaking decidability of type
checking. The inner theory can be an easily justifiable extensions of HoTT,
allowing the construction of "infinite structures" which are considered
impossible in plain HoTT. Alternatively, we can set the inner theory to be
exactly the current standard formulation of HoTT, in which case our system can
be thought of as a type-theoretic framework for working with "schematic"
definitions in HoTT. As demonstrations, we define semi-simplicial types and
formalise constructions of Reedy fibrant diagrams.
