In this appendix, we list the rules for dot-types as given in Luo (2009c, 2012b), except that we have added signatures in judgments. Note that dot-types are not ordinary inductive types and they are also governed by special rules that involve coercive subtyping.

Formation rule:

Introduction rule:

Elimination rules:

Computation rules:

Projections as coercions:

Coercion propagation:

where the coercions d₁, d₂ and d are defined as: d₁[c₁](x) = c₁(p₁(x)), p₂(x), d₂[c₂](x) = p₁(x), c₂(p₂(x)) and d[c₁, c₂](x) = c₁(p₁(x)), c₂(p₂(x)).