Jonas Oberhauser: Bachelor Thesis
Formalizations of Metatheoretic Properties of Type Theories
Author: Jonas Oberhauser
Advisor: Chad E. Brown
Supervisor: Gert Smolka
Abstract
Confluence and termination are powerful properties of type theories that together prove that for every wellformed term of that type theory, all reduction paths are finite and ultimately lead to the same normal form.
In conjunction with subject reduction, they can be used to prove the consistency of powerful type thoeries such as ECC.
In this thesis, we studied various proofs for confluence and termination and formalized many of them.
We started by formalizing two proofs of confluence for a simply typed lambda calculus: one similar to that by Takahashi, and the canonical proof by Tait and MartinLof.
Then we considered a formalization by Barras of a termination proof for CC.
The proof closely follows a proof of termination presented by Geuvers, who also gave proof sketches on how to extend his approach to some extensions of CC, e.g., small sigma types.
We extended Barras' formalization to incorporate small sigma types, following the proof sketch of Geuvers.
Thesis Document
In the thesis we give a mathematical presentation of the extended proof, and use it to describe and explain the intricacies of the formal development.
We also give a short discussion on what decisions we faced and how other choices could have made our life easier.
Finally, we render our thoughts on to whether this approach can be extended to stronger type theories or not.
Talks

15.05.2012, 16:15: Proofs for Confluence of Simple Type Theory (Building E1 3, Room 528)
In this talk we will describe two proof techniques for confluence we have formalized already: One is the common argument using parallel reduction due to Tait and MartinLof. The second technique is similar to the "complete development" method due to Takahashi.

17.08.2012, 14:15: Proofs for Strong Normalization (Building E1 3, Room 528)
We will introduce the notion of termination and explain logical relations, a technique to prove termination of various type theories. We have already formalized such a proof for simply typed lambda calculus, and show the issues richer type theories introduce. Finally we will give an idea of how to address those additional issues within the framework of logical relations.

03.05.2013, 14:15: FINAL TALK: Formalizing Termination of CC (Building E1 3, Room 528)
Termination is the property that for any wellformed term of CC, all
reduction paths are finite.
This is an extremely powerful property and is a key theorem in proving
consistency of CC.
Various proofs of Termination have been proposed; we consider that by
Geuvers.
Geuvers proof makes use of the fact called Classification, i.e., that
every wellformed term of CC can be classified into one of three
classes: kinds, constructors, and objects. Key functions are then often
defined on only one or two of these classes and hence are partial on the
set of the terms of CC.
We consider a formalization of one of the key functions by Barras, which
makes use of advanced features of Coq's type theory in order to deal
with the partiality in an elegant way. We also show a proof and
formalization of Classification, which we deem more elegant than the one
presented by Barras.
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