Seven Years of Thulium

I’ve now been running my personal server, Thulium, for seven years and four-and-a-half months, and three of those years have elapsed since my previous post. Not much had changed in those intervening years until very recently, so I thought I’d map them out here.

The Philadelphia Restaurant Graveyard

This was originally an updating post on types.pl.

Since moving to Philadelphia in 2022, a number of beloved restaurants I’ve discovered have since closed in my mere years here, so I’m keeping this table as a remembrance of restaurants past.

HTML Day 2025 ❇️

Zhuyin table with examples

This is a table similar to Wikipedia’s zhuyin table, but with slightly different sorting and an example for each possible combination of initials and finals.

Impredicative Inductives Need Not be Strictly Positive

If you look around for why inductive datatypes need to be strictly positive, you’ll probably end up at Vilhelm Sjöberg’s blog post Why must inductive types be strictly positive?, which gets passed around a lot as an accessible and modernized description of the inconsistency that arises from a certain large, impredicative inductive datatype that is positive but not strictly positive. This example originally comes from Coquand and Paulin-Mohring’s COLOG-88 paper Inductively defined types. A key component of the inconsistency relies on the injectivity of its constructor, but since the inductive is large, even if Rocq were to permit nonstrictly positive inductives, it would still disallow its strong elimination and therefore injectivity!

U+237C ⍼ is (also) S9576 ⍼

Part 1: U+237C ⍼ RIGHT ANGLE WITH DOWNWARDS ZIGZAG ARROW
Part 2: update: U+237C ⍼ angzarr;
Part 3: Monotype Mathematical Sorts
This is part 4.
Many thanks to Sallie Morris at the Science Museum Group, Claire Welford-Elkin at St Bride Library, and Brian Corrigan for their help.
This update was originally posted on cohost.

Impredicative Type Theories

Models of predicative type theory are well-studied and have been mechanized, ranging from proofs of consistency for a minimal type theory with a predicative hierarchy in 1000 lines of Rocq to proofs all the way up to decidability of conversion in 10 000 lines of Agda.

In contrast, the story for impredicative type theory is not so clear. Incorporating different features alongside an impredicative Prop may require substantially different proof methods. This post catalogues these various models, what type theories they model, and what proof technique is used. Most proofs fall into one of three techniques: proof-irrelevant set-theoretic models, reducibility methods, and realizabililty semantics.

Overview

Specialty Coffee Roasters in Philadelphia

Get your beans here! This is a list of specialty coffee roasters local to Philly that do single-origin light roasts whose coffeeshops are in or close to Centre City. This list is tailored to my preferences: I make coffee with an Aeropress at home, and I enjoy getting pourover coffee when I’m out.

ECIC is Inconsistent

…probably. To be clear, ECIC¹ refers to Monnier and Bos’ Erasable CIC², with erasable in the sense of erasable pure type systems (EPTS)³. I’ll argue that even with erased impredicative fields, Coquand’s paradox of trees⁴ is still typeable.

Respooled Film Stocks

This was originally posted on cohost.

While looking up the various colour films my local photo shops carry, I found that a lot of them are actually respools and repackagings of other film (most Kodak), so I’ve tried to compile that information here to keep track of them all.

Monotype Mathematical Sorts

Part 1: U+237C ⍼ RIGHT ANGLE WITH DOWNWARDS ZIGZAG ARROW
Part 2: update: U+237C ⍼ angzarr;
This is part 3.
Part 4: U+237C ⍼ is (also) S9576 ⍼
Many thanks to Alicia Chilcott and Sophie Hawkey-Edwards at St Bride for their help.

Designing Stratified Type Theory

I’ve wanted to write an informal (but technical!) post¹ about my current research in progress on Stratified Type Theory (StraTT), but we’ve been occupied with writing up a paper draft for submission to CPP 2024, then writing a talk for NJPLS Nov 2023, then being rejected from CPP, then I’ve just been randomly distracted for two weeks but I’m so back now.

That paper draft along with the supplementary material will have all the details, but I’ve decided that what I want to focus on in this post is all the other variations on the system we’ve tried that are either inconsistent or less expressive. This means I won’t cover a lot of motivation or examples (still a work in progress), or mention any metatheory unless where relevant; those can be found in the paper. Admittedly, these are mostly notes for myself, and I go at a pace that sort of assumes enough familiarity with the system to be able to verify well-typedness mentally, but this might be interesting to someone else too.