  This is the technical report accompanying the paper "A Sound and Complete
Substitution Algorithm for Multimode Type Theory" [Ceulemans, Nuyts and
Devriese, 2024]. It contains a full definition of Well-Scoped Multimode Type
Theory (WSMTT) in Section 2, including many rules for $\sigma$-equivalence and
a description of all rules that have been omitted. Furthermore, we present
completeness and soundness proofs of the substitution algorithm in full detail.
These can be found in Sections 4 and 5 respectively. In order to make this
document relatively self-contained, we also include a description of
Substitution-Free Multimode Type Theory (SFMTT) in Section 3.
