In hot weather I like to drink my coffee in an iced latte. To make it, I have a very large Bialetti Moka Express. Recently when I got it going again after a winter of disuse, it took me a couple of attempts to get the technique right, so here are some notes as a reminder to my future self next year. It’s worth noting that I’m not fussy about my coffee: I usually drink pre-ground beans from the supermarket, with cream (in winter hot coffee) or milk and ice. basic principle When I was getting the hang of my moka pot, I learned from YouTube coffee geeks such as James Hoffmann that the main aim is for the water to be pushed through the coffee smoothly and gently. Better to err on the side of too little flow than too much. I have not had much success trying to make fine temperature adjustments while the coffee is brewing, because the big moka pot has a lot of thermal inertia: it takes a long time for any change in gas level to have any effect on on the coffee flow. routine fill the kettle and turn it on put the moka pot’s basket in a mug to keep it stable fill it with coffee (mine needs about 4 Aeropress scoops) tamp it down firmly [1] when the kettle has boiled, fill the base of the pot to just below the pressure valve (which is also just below the filter screen in the basket) insert the coffee basket, making sure there are no stray grounds around the edge where the seal will mate screw on the upper chamber firmly put it on a small gas ring turned up to the max [2] leave the lid open and wait for the coffee to emerge immediately turn the gas down to the minimum [3] the coffee should now come out in a steady thin stream without spluttering or stalling when the upper chamber is filled near the mouths of the central spout, it’ll start fizzing or spluttering [4] turn off the gas and pour the coffee into a carafe notes If I don’t tamp the grounds, the pot tends to splutter. I guess tamping gives the puck better integrity to resist channelling, and to keep the water under even pressure. Might be an effect of the relatively coarse supermarket grind? It takes a long time to get the pot back up to boiling point and I’m not sure that heating it up slower helps. The main risk, I think, is overshooting the ideal steady brewing state too much, but: With my moka pot on my hob the lowest gas flow on the smallest rings is just enough to keep the coffee flowing without stalling. The flow when the coffee first emerges is relatively fast, and it slows to the steady state several seconds after I turn the heat down, so I think the overshoot isn’t too bad. This routine turns almost all of the water into coffee, which Hoffmann suggests is a good result, and a sign that the pressure and temperature aren’t getting too high.

Ten years ago, Apple’s Phil Schiller surprised Apple enthusiasts and developers by walking out on stage at John Gruber’s The Talk Show Live WWDC event and giving an open, human, honest interview to a somewhat jaded community. I wrote this in response: Both Apple and Phil Schiller himself took a huge risk in doing this. That they agreed at all is a noteworthy gift to this community of long-time enthusiasts, many of whom have felt under-appreciated as the company has grown. […] Phil’s appearance on the show was warm, genuine, informative, and entertaining. It was human. And humanizing the company and its decisions, especially to developers — remember, developer relations is all under Phil — might be worth the PR risk. This started a ten-year run of interviews by Apple executives on The Talk Show every year at WWDC that proved to be great, surprisingly safe PR for Apple. No executive ever said something they shouldn’t have (they’re pros), no sensational or negative news stories ever resulted from them, and Apple’s enthusiastic fans and developers felt seen, heard, and appreciated. *     *     * For unspecified reasons, Apple has declined to participate this year, ending what had become a beloved tradition in our community — and I can’t help but suspect that it won’t come back. (A lot has changed in the meantime.) Maybe Apple has good reasons. Maybe not. We’ll see what their WWDC PR strategy looks like in a couple of weeks. In the absence of any other information, it’s easy to assume that Apple no longer wants its executives to be interviewed in a human, unscripted, unedited context that may contain hard questions, and that Apple no longer feels it necessary to show their appreciation to our community and developers in this way. I hope that’s either not the case, or it doesn’t stay the case for long. This will be the first WWDC I’m not attending since 2009 (excluding the remote 2020 one, of course). Given my realizations about my relationship with Apple and how they view developers, I’ve decided that it’s best for me to take a break this year, gain some perspective, and decide what my future relationship should look like. Maybe Apple’s leaders are doing that, too.

Thinking about moving to Japan? You’re not alone—Japan is a popular destination for those hoping to move abroad. What’s more, Japan actually needs more international developers. But how easy is it to immigrate to and work in Japan? Scores of videos on social media warn that living in Japan is quite different from holidaying here, and graphic descriptions of exploitative companies also create doubt. The truth is that Japan is not the easiest country to immigrate to, nor is it the hardest. Some Japanese tech companies and developer roles offer great work-life balance and good compensation; others do not. Based on other developers’ experiences, you’ll thrive here if you: Are an experienced developer Value safety, good food, and convenience over a high salary Are willing to invest time and effort into learning Japanese over the long term Read on to discover if Japan is a good fit for you, and the best ways to get a visa and begin your life here. What is it like working as a developer in Japan? TokyoDev conducts an annual survey of international developers living in Japan. Many of the questions in TokyoDev’s 2024 survey specifically addressed respondents’ work environments. Compensation When TokyoDev asked about “workplace difficulties” in the 2024 survey, 45% of respondents said that “compensation” was their number one problem at work. Overall, compensation for developers in Japan is far lower than the US developer median salary of 120,000 USD (currently 17.5 million yen), but higher than the Indian developer median salary of 640,000 rupees (currently around 1.1 million yen). Yet evaluating compensation for international developers in Japan, specifically, is trickier than you might expect. It’s hard to define an expected salary range because international developers tend to work in different companies and roles than the average Japanese developer. According to a 2024 survey conducted by the Japanese Ministry of Health, Labor and Welfare, the average annual salary of software engineers in Japan was 5.69 million yen. In a survey conducted that same year by TokyoDev, though, English-speaking international software developers in Japan enjoyed a median salary of 8.5 million yen. Of those international developers who responded, only 71% of them worked at a company headquartered in Japan, and almost 80% of them used English always or frequently, with 79% belonging to an engineering team with many other non-Japanese members. Wages, then, are heavily influenced by a range of factors, but particularly by whether you’re working for a Japanese or international company. In general, 75% of the international developers surveyed made 6 million yen or more. The real question is, is that enough for you to be comfortable in Japan? The answer is likely to be yes, if you don’t have overseas financial obligations or dependents. If you do, you’ll want to look carefully at rent, grocery, and education prices in your area of choice to guesstimate the expense of your Japanese lifestyle. Work-life balance Japan has a tradition of long hours and overtime. The Financial Times reports that the Japanese government has taken many measures to reduce the phenomenon of death from overwork (過労死, karoushi), from capping overtime to 100 hours a month, to setting up a national hotline for employees to report abusive companies. The results seem mixed. The Financial Times article adds that in 2024, employees at 26,000 organizations reported working illegal overtime at 44.5% of those businesses. On the other hand, average working hours for men fell to below 45 hours per week, and for women to below 35, which is similar to average working hours in the US. Still, 72% of the developers surveyed by TokyoDev worked for less than 40 hours a week. In addition, 70% of TokyoDev respondents cited work-life balance as their top workplace perk. The number of respondents happy with their working conditions came in just below that, at 69%. There was some correlation between hours worked and the type of employer, though. Employees at international subsidiaries were slightly more likely to enjoy shorter work weeks than those at Japanese companies. Remote work Remote work is still relatively new in Japan. Although more offices adopted the practice during Covid, many of them are now backtracking and requiring employees to return to the office, often with a hybrid schedule. While only 9% of TokyoDev respondents weren’t allowed any remote work, 76% of those required to work in-office were employed by Japan-headquartered companies. By contrast, of the 16% who worked fully remotely, only 57% worked for a Japanese company. Those with the option to work remotely really enjoy it. When asked what their most important workplace benefit was, 49% of respondents answered “remote work,” outstripping every other answer by far. Job security A major plus of working in Japan is job security—which, given the waves of layoffs at American tech companies, may now seem extra appealing. It’s overwhelmingly difficult to fire or lay off an employee with a permanent contract (正社員, seishain) in Japan, due to labor laws designed to protect the individual. This may be why 54% of TokyoDev survey respondents named “job security” as their most important workplace perk. Not every company will adhere to labor protection laws, and sometimes businesses pressure employees to “voluntarily” resign. Nonetheless, employees have significant legal recourse when companies attempt to fire them, change their contracts, or alter the current workplace conditions (sometimes, even if those conditions were never stated in writing). Developer stories TokyoDev regularly interviews developers working at our client companies, for information on both their specific positions and their general work environment. Our interviewees work with a variety of technology in many different roles, and at companies ranging from fintech enterprises like PayPay to game companies like Wizcorp. Why do developers choose Japan? In 2024 TokyoDev also asked developers, “What’s your favorite thing about Japan?” The results were: Safety: 21% Food: 13% Convenience: 11% Culture: 8% Peacefulness: 7% Nature: 5% Interestingly, there was a strong correlation between the amount of time someone had lived in Japan and their answer. Those who had been in Japan three years or less more frequently chose “food” or “culture.” Those who’d lived in Japan for four or more years were significantly more likely to answer “safety” or “peacefulness.” Safety It’s true that Japan enjoys a lower crime rate than many developed nations. The Security Journal UK ranked it ninth in a list of the world’s twenty safest countries. In 2024, World Population Review selected Tokyo as the safest city in the world. The homicide rate in 2023 was only 0.23 per 100,000 people, and has been steadily declining since the nineties. There are a few women-specific concerns, such as sexual violence. Nonetheless, the subjective experience of many women in the TokyoDev audience is that Japan feels safe; for example, they experience no trepidation walking around late at night. Of course, crime statistics don’t take into account natural disasters, of which Japan has more than its fair share. Thanks to being located on the Ring of Fire, Japan regularly copes with earthquakes and volcanic activity, and its location in the Pacific means that it is also affected by typhoons and tsunamis. To compensate, Japan also takes natural disaster countermeasures extremely seriously. It’s certainly the only country I’ve been to that posts large-scale evacuation maps on the side of the street, stores emergency supply stockpiles in public parks, and often requires schoolchildren to keep earthquake safety headgear at their desks. Food Food is another major draw. Many respondents simply wrote that “food” or “fresh, affordable food” was their favorite thing about Japan, but a few listed specific dishes. Favorite Japanese foods of the TokyoDev audience include: Yakiniku (self-grilled meat) Ramen Peaches Sushi Hiroshima-style okonomiyaki (savory pancake) Curry rice Onigiri (rice balls) Of those, sushi was mentioned most often. One respondent also answered the question with “drinking,” if you think that should count! Personal experiences Our contributors have also shared their personal experiences of moving to and working in Japan. We’ve got articles from Filipino, Indonesian, Australian, Vietnamese, and Mongolian developers, as well as others sharing what it’s like to work as a female software developer in Japan, or to live in Japan with a disability. Why shouldn’t you live in Japan? Safety, food, convenience, and culture are the most commonly-cited upsides of living in Japan. The downsides are the necessity of learning the language and some strict, yet often-unspoken, cultural expectations. Language Fluency in Japanese is not strictly necessary to live or work in Japan. Access to government services for you and your family, such as Japanese public school, is possible even if you speak little Japanese. (That doesn’t mean that most city hall clerks speak English; usually they’ll either locate a translator, or work with you via a translation app.) Nonetheless, TokyoDev’s 2024 survey showed that language ability was highly correlated to social success in Japan. In particular, 56% of all respondents were happy or very happy with their adjustment to Japanese culture. Breaking down that number, though, 76% of those with fluent or native Japanese ability reported being happy with their cultural adjustment, while only 34% of those with little or no Japanese ability were similarly happy. The same held true for social life satisfaction: 59% of those with fluent or native Japanese ability were happy or very happy with their social life, compared to 42% of those who don’t speak much Japanese. While English study is compulsory in Japan and starts in elementary school, as of 2025, only 28% of Japanese people speak English, and most of them can’t converse with high fluency. Living and working in Japan is possible without Japanese, but it’s hard to integrate, make friends, and participate in cultural activities if you can’t communicate with the locals. Cultural expectations As mentioned above, fluency in Japanese is closely allied to fluency in Japanese culture. At the same time, one does not necessarily imply the other. It’s possible to be fluent in Japanese, but still not grasp many of the unspoken rules your Japanese friends, neighbors, and coworkers operate by. Japan’s culture is both high-context and specifically averse to confrontation and outspokenness; if you get it “wrong,” people aren’t likely to tell you so. Japanese culture also values conformity: as the saying goes, “the nail that sticks up, gets hammered down.” While there are hints of things changing, with many Japanese companies saying support for greater diversity is necessary, minorities or those who are different may experience pressure to fit in. Introspection is required: are you the kind of person who’s adept at “reading the room,” a highly-valued quality in Japan? Conversely, are you self-confident enough to not sweat the small stuff? Either of these personality types may do well in Japan, but if social acceptance is very important to you, and you’re also uncomfortable with feeling occasionally awkward or uncertain, then you may struggle more to adjust. I want to go! How can I get there? If you’ve decided to immigrate to Japan, there are a number of ways to acquire a work visa. The simplest way is to get hired by a company operating in Japan. Alternatively, you can start your own business in Japan, come over on a Working Holiday, or even—if you’re very determined—arrive first as an English teacher. Let’s begin with the most straightforward route: getting hired as a developer. Getting a developer job in Japan As mentioned before, Japan needs more international developers. Some types of developers, though, will find it easier to get a job in Japan. In particular, companies in Japan are looking for the following: Senior developers. Companies are particularly interested in those with management experience and soft skills such as communication and leadership. Backend developers. This is one of the most widely-available roles for those who don’t speak Japanese. Developers who know Python. Python is one of Japan’s top in-demand languages. AI and Machine Learning Specialists. Japan is leaning hard on AI to help cope with demographic changes. Those who already know, or are willing to learn, Japanese. Combining those criteria, an experienced developer who speaks Japanese should have little difficulty finding a job! If you’re none of these things, you don’t need to give up—you just need to be patient, flexible, and willing to think outside the box. As Mercari Senior Technical Recruiter Clement Chidiac told me, “I know a bunch of people that managed to land a job because they’ve tried harder, going to meetups, reaching out to people, networking, that kind of thing.” Edmund Ho, Principal Consultant at Talisman Corporation, agreed that overseas candidates hoping to work in Japan for the first time face a tough road. He believes candidates should maintain a realistic, but optimistic, view of the process. “Keep a longer mindset,” he suggested. “Maybe you don’t get an offer the first year, but you do the second year.” “Stepping-stone” jobs Candidates from overseas do face a severe disadvantage: many companies, even those founded by non-Japanese people, are only open to developers who already live in Japan. Although getting a work visa for an overseas employee is cheaper and easier in Japan than in many countries, it still presents a barrier some organizations are reluctant to overcome. By contrast, once you’re already on the ground, more companies will be interested in your skills. This is why some developers settle on a “stepping-stone” position—in other words, a job that may not be all you hoped for, but that is willing to sponsor your visa and bring you into the country. Here’s where some important clarification on Japanese work visas is required. Work visas The most common visa for developers is the Engineer/Specialist in Humanities/International Services visa, a broad-category visa for foreign workers in those fields. To qualify, a developer must have a college degree, or have ten years of work experience, or have passed an approved IT exam. Another relatively common visa for high-level developers is the Highly-Skilled Professional (HSP) visa. To acquire it, applicants must score at least 70 points on an assessment scale that addresses age, education level, Japanese level, income, and more. The HSP visa has many advantages, but there is one important difference between it, and the more standard Engineer visa. The Engineer/Specialist in Humanities/International Services visa is not tied to a specific company. It grants you the legal right to work within those fields for a specific period of time in Japan. The Highly-Skilled Foreign Professional visa, on the other hand, is tied to a specific employer. If you want to change jobs, you’ll need to update your residency status with immigration. Some unscrupulous companies will try to claim that because they sponsored your Engineer/Specialist in Humanities/International Servicesvisa, you are obligated to remain with their company or risk being deported. This is not the case. If you do leave your job without another one lined up, you have three months to find another before you may be at risk for deportation. In addition, the fields of work covered by the Engineer/Specialist in Humanities/International Services visa are incredibly broad, and include everything from sales to product development to language instruction. As TokyoDev specifically confirmed with immigration, you can even come to Japan as an English instructor, then later work as a developer, without needing to alter your visa. Those with the HSP visa will need to go to immigration and alter their residency status each time they change roles. However, if you have the points and qualifications for an HSP visa, that means you’re also eligible for Permanent Residency within one to three years. Once you’ve obtained Permanent Residency, you’re free to pursue whatever sort of employment you like. International or Japanese company? As you begin your job hunt, you’ll hopefully receive responses from several sorts of companies: Japanese companies that also primarily hire Japanese people, Japanese companies with designated multinational developer teams, companies that were founded in Japan but nonetheless hire international developers for a variety of positions, and international subsidiaries. There are advantages and disadvantages to working with mostly-Japanese or mostly-international companies. Japanese companies The more Japanese a company is—both in philosophy and personnel—the more you’ll need Japanese language skills to thrive there. It’s true that a number of well-established Japanese tech companies are now creating developer teams designed to be multinational from the outset: typically, these are very English-language friendly. Some organizations, such as Money Forward, have even adopted English as the official company language. However, this often results in an institutional language barrier between development teams and the rest of the company, which is usually staffed by Japanese speakers. Developers are still encouraged to learn Japanese, particularly as they climb the promotional ladder, to help facilitate interdepartmental communication. Some companies, such as DeepX and Beatrust, either offer language classes themselves or provide a stipend for language learning. In addition to the language, you’ll also need to become “fluent” in Japanese business norms, which can be much more rigid and hierarchical than American or European company cultures. For example, at introductory drinking parties (themselves a potential surprise for many!), it is customary for new employees or women employees to go around with a bottle of beer and pour glasses for their managers and the company’s senior management. As mentioned in the cultural expectations section, most Japanese people won’t correct you even if you’re doing it all wrong, which leaves foreigners to discover their gaffes via trial-and-error. The advantage here is that you’ll be pressured, hopefully in a good way, to adapt swiftly to the Japanese language and business culture. There’s a sink-or-swim element to this approach, but if you’re serious about settling in Japan, then this “downside” could benefit you in the long run. Finally, there is the above-mentioned issue of compensation. On average, international companies pay more than Japanese ones; the median salary difference is around three million yen per year. Specific roles may be paid at higher rates, though, and most Japanese companies do offer bonuses. Many Japanese companies also offer other perks, such as housing stipends, spouse and child allowances, etc. If you receive an offer, it’s worth examining the whole compensation package before you make a decision. International companies The advantages of working either for an international company, or for a Japanese company that already employs many non-Japanese people, are straightforward: you can usually communicate in English, you already understand most of the business norms, and such companies typically pay developers more. You do run the risk of getting stuck in a rut, though. As mentioned earlier, TokyoDev found in its own survey that the correlation between Japanese language skills and social life satisfaction is high. You can of course study Japanese in your free time—and many do—but the more your work environment and social life revolve around English, the more difficult acquiring Japanese becomes. Want a job? Start here! If you’re ready to begin your job hunt, you can start with the TokyoDev job board. TokyoDev only works with companies we feel good about sending applicants to, and the job board includes positions that don’t require Japanese and that accept candidates from abroad. Other alternatives These visas don’t lead directly to working as a software developer in Japan, but can still help you get your foot in the door. DIY options If you prefer to be your own boss, there are several visas that allow you to set up a business in Japan. The Business Manager visa is typically good for one year, although repeated applicants may get longer terms. Applicants should have five million yen in a bank account when they apply, and there are some complicated requirements for getting and keeping the visa, such as maintaining an office, paying yourself a minimum salary, following proper accounting procedures, etc. The Startup visa is another option if the Business Manager visa appeals to you, but you don’t yet have the funds or connections to make it happen. You’ll be granted the equivalent of a Business Manager visa for up to one year so that you can launch your business in Japan. Working Holiday visa This is the path our own founder Paul McMahon took to get his first developer job in Japan. If you meet various qualifications, and you belong to a country that has a Working Holiday visa agreement with Japan, you can come to Japan for a period of one year and do work that is “incidental” to your holiday. In practice, this means you can work almost any job except for those that are considered “immoral” (bars, clubs, gambling, etc.). The Working Holiday visa is a great opportunity for those who have the option. It allows you to experience living and working in Japan without any long-term commitments, and also permits you to job-hunt freely without time or other visa constraints. J-Find visa The J-Find visa is a one-year visa, intended to let graduates of top universities job-hunt or prepare to found a start-up in Japan. To qualify, applicants should have: A degree from a university ranked in the top 100 by at least two world university rankings, or completed a graduate course there Graduated within five years of the application date At least 200,000 yen for initial living expenses TokyoDev contributor Oguzhan Karagözoglu received a J-Find visa, though he did run into some difficulties, particularly given immigration’s unfamiliarity with this relatively new type of visa. Digital Nomad visa This is another new visa category that allows foreigners from specific countries, who must make over 10 million yen or more a year, to work remotely from Japan for six months. Given that the application process alone can take months, the visa isn’t extendable or renewable, and you’re not granted residency, it’s questionable whether the pay-off is worth the effort. Still, if you have the option to work remotely and want to test out living in Japan before committing long-term, this is one way to do that. TokyoDev contributor Christian Mack was not only one of the first to acquire the Digital Nomad visa, but has since opened a consultancy to help others through the process. Conclusion If your takeaway from this article is, “Japan, here I come!” then there are more TokyoDev articles that can help you on your way. For example, if you want to bring your pets with you, you should know that you need to start preparing the import paperwork up to seven months in advance. If you’re ready now to start applying for jobs, check out the TokyoDev job board. You’ll also want to look at how to write a resume for a job in Japan, and our industry insider advice on passing the resume screening process. These tips for interviewing at Japanese tech companies would be useful, and when you’re ready for it, see this guide to salary negotiations. Once you’ve landed that job, we’ve got articles on everything from bringing your family with you, to getting your first bank account and apartment. In addition, the TokyoDev Discord hosts regular discussions on all these topics and more. It’s a great chance to make developer friends in Japan before you ever set foot in the country. Once you are here, you can join some of Japan’s top tech meetups, including many organized by TokyoDev itself. We look forward to seeing you soon!

We go over the "Wake up, Remix!" article by the remix team and talk about their decisions moving forward and also speculate on what is coming next.

TIL (or this week-ish I learned) why big-sigma and big-pi turn up in the notation of dependent type theory. I’ve long been aware of the zoo of more obscure Greek letters that turn up in papers about type system features of functional programming languages, μ, Λ, Π, Σ. Their meaning is usually clear from context but the reason for the choice of notation is usually not explained. I recently stumbled on an explanation for Π (dependent functions) and Σ (dependent pairs) which turn out to be nicer than I expected, and closely related to every-day algebraic data types. sizes of types The easiest way to understand algebraic data types is by counting the inhabitants of a type. For example: the unit type () has one inhabitant, (), and the number 1 is why it’s called the unit type; the bool type hass two inhabitants, false and true. I have even seen these types called 1 and 2 (cruelly, without explanation) in occasional papers. product types Or pairs or (more generally) tuples or records. Usually written, (A, B) The pair contains an A and a B, so the number of possible values is the number of possible A values multiplied by the number of possible B values. So it is spelled in type theory (and in Standard ML) like, A * B sum types Or disjoint union, or variant record. Declared in Haskell like, data Either a b = Left a | Right b Or in Rust like, enum Either<A, B> { Left(A), Right(B), } A value of the type is either an A or a B, so the number of possible values is the number of A values plus the number of B values. So it is spelled in type theory like, A + B dependent pairs In a dependent pair, the type of the second element depends on the value of the first. The classic example is a slice, roughly, struct IntSlice { len: usize, elem: &[i64; len], } (This might look a bit circular, but the idea is that an array [i64; N] must be told how big it is – its size is an explicit part of its type – but an IntSlice knows its own size. The traditional dependent “vector” type is a sized linked list, more like my array type than my slice type.) The classic way to write a dependent pair in type theory is like,      Σ len: usize . Array(Int, len) The big sigma binds a variable that has a type annotation, with a scope covering the expression after the dot – similar syntax to a typed lambda expression. We can expand a simple example like this into a many-armed sum type: either an array of length zero, or an array of length 1, or an array of length 2, … but in a sigma type the discriminant is user-defined instead of hidden. The number of possible values of the type comes from adding up all the alternatives, a summation just like the big sigma summation we were taught in school. ∑ a ∈ A B a When the second element doesn’t depend on the first element, we can count the inhabitants like, ∑ A B = A*B And the sigma type simplifies to a product type. telescopes An aside from the main topic of these notes, I also recently encountered the name “telescope” for a multi-part dependent tuple or record. The name “telescope” comes from de Bruijn’s AUTOMATH, one of the first computerized proof assistants. (I first encountered de Bruijn as the inventor of numbered lambda bindings.) dependent functions The return type of a dependent function can vary according to the argument it is passed. For example, to construct an array we might write something like, fn repeat_zero(len: usize) -> [i64; len] { [0; len] } The classic way to write the type of repeat_zero() is very similar to the IntSlice dependent pair, but with a big pi instead of a big sigma:      Π len: usize . Array(Int, len) Mmm, pie. To count the number of possible (pure, total) functions A ➞ B, we can think of each function as a big lookup table with A entries each containing a B. That is, a big tuple (B, B, … B), that is, B * B * … * B, that is, BA. Functions are exponential types. We can count a dependent function, where the number of possible Bs depends on which A we are passed, ∏ a ∈ A B a danger I have avoided the terms “dependent sum” and “dependent product”, because they seem perfectly designed to cause confusion over whether I am talking about variants, records, or functions. It kind of makes me want to avoid algebraic data type jargon, except that there isn’t a good alternative for “sum type”. Hmf.