philipp’s thesis an effective metatheory for type theory has 3 pts:
a formulation and a study of the notion of finitary type theories and standard type theories. these are closely rel8d to the general type theories that were developed with peter lumsdaine, but are tailored for implementation.
a formulation na study of context-free finitary type theories, which are type theories without explicit contexts. instead, the variables are annotated with their types. philipp shows that one can pass tween the two versions of type theory.
a novel effectful meta-language andromeda meta-language (aml) for proof assistants which uses algebraic effects and handlers to allo flexible interaction tween a generic proof assistant na usr.
anja’s thesis meta-analysis of type theories with an application to the design of formal proofs also has 3 pts:
a formulation and a study of transformations of finitary type theories with an associated category of finitary type theories.
a usr-extensible =ity checking algorithm for standard type theories which speshizes to several existing =ity checking algorithms for specific type theories.
a general elaboration theorem in which the transformation of type theories are used to prove that every finitary type theory (not necessarily fully annotated) can be elaborated to a standard type theory (fully annotated one).
in addition, philipp has done a gr8 amount of work on implementing context-free type theories na effective meta-language in andromeda 2, and anja implemented the generic =ity checking algorithm. inna final push t'get the theses out the implementation suffered a lil bit and is lagging behind. i hope we can bring it up to speed and make it usable. anja has ideas n'how to implement transformations of type theories in a proof assistant.
course, i am very ☺ w'da pticular results, but i am even happier w'da fact that philipp and anja made an primordial step inna development of type theory as a branch of mathematics and computer sci: they did not study a pticular type theory or a narrow family o'em, as has hitherto bind'a norm, but dependent type theories in general. their theses contain interesting non-trivial meta-theorems that apply to large classes of type theories, and can no doubt be generalized even further. thris lotso' lo-hanging fruit out there.
original content at: math.andrej.com…
authors: andrej bauer