Final Project Suggestions

Below are some suggestions for a final project for 15–815. Please discuss your project with us before you start working on it in earnest. That way, we can offer advice and guidance, and try to anticipate some of the difficulties you will face. (Also, it would be nice to avoid a lot of duplication.)

A good strategy is to look for elementary gaps in Lean’s library. It is perfectly fair game (and a good idea) to look for developments in other libraries and try to “port” them to Lean; for example, you can look at things that have been formalized in Isabelle, SSReflect, Coq, or HOL light. Here are some suggestions that seem to be within reach:

• Lists: Lean has a rudimentary library for lists. There are many additional operations on lists that you can define, and many additional properties you can prove.

• Formalize n! and (m choose n), and prove some of the basic identities. Similarly, you can define the fibonacci numbers and derive some of their properties.

• Use the factorial function to prove that there are infinitely many prime numbers, that is, for every n, there is a prime number greater than n. A slightly more involved argument shows that there are infinitely many primes of the form 4*n + 1.

• Elementary number theory: we have defined gcd and lcm for the natural numbers, and these definitions will be “lifted” to the integers soon. The next step is to define the “extended gcd” function, and use it to prove Bezout’s theorem: if m and n are nonzero natural numbers, there are integers a and b such that a * m

• b * n = gcd m n. (Many basic number-theoretic facts follow from this.)

• Naive set theory: in Lean, a subset of a type A is just a predicate on A. The library has only rudimentary properties of sets, and set operations like binary union and intersection. You can define notions of union and intersection for arbitrary collections of sets and/or for indexed sequences of sets, and derive some of the basic properties of these operations.

• Along the same lines, you can formalize basic properties of functions, like being injective and surjective, define the notion of the image of a function, and prove some basic facts about these notions. Similarly, you can define properties of relations and operations like the composition of relations and the transitive closure of a relation.

• Finiteness: the property of being a finite set (that is, a finite subset of a type) can be made into a type class, with instances corresponding to closure operations on the class of finite sets. That would make it possible for Lean to infer automatically that various sets are finite. (With a good theory of finite sets, we will be able prove e.g. that card (a union b) = card a + card b - card (a intersect b), the inclusion / exclusion principle, etc.)

• Multisets: a multiset on A is best modeled as a function from A to nat. Define the notion, and prove some basic properties of multisets.

• Red-black trees: for a more ambitious project, define red-black trees and prove some of their key properties.

Lean’s library has algebraic structures like groups, rings, fields, and ordered rings. (The integers, for example, are an instance of an ordered ring.) We have defined various types of orderings, but not yet the notion of a lattice:

• Define the notion of a lattice. Show that the integers with min and max form a lattice, as do the booleans with “and” and “or”, or the collection of subsets of a type, with union and intersection.

We also have not yet done a lot of abstract algebra. If you are adventurous, you could try any of the following:

• Define the notion of a subgroup, and prove Lagrange’s theorem: the order of a subgroup divides the order of a group.

• Alternatively, define the notions of orbit and stabilizer, and prove the orbit - stabilizer theorem.

• Alternatively, define the notion of a subgroup and a homomorphism, and show that the kernel of a homomorphism is a subgroup. If you are ambitious, you can work towards the isomorphism theorems.

• We still need a library for “big operations” on a semigroup, like sums and products. Doing this generically is very ambitious, but we can talk about how to make a start along these lines.

Combinatorial properties of finite structures like graphs are probably out of reach until we have a theory of finite sets and cardinality (see above). But here is one combinatorial notion that does not require arithmetic:

• Define the notion of a well-quasi-order, and prove some basic facts about well-quasiorders. For example, show that the product of two well-quasi-orders is well-quasiordered. (This notion is not only important in combinatorics, but also in the theory of term rewriting.)

Lean has a special library for homotopy type theory that is already populated with basic properties of paths and transports, etc. We also have basic category-theoretic notions in that framework. But there are many topics developed in the HoTT book that are ripe for formalization. Here is a project that is ambitious, but (we hope) within reach:

• Show that the fundamental group of the circle is the integers.

For this, you will need to encode the corresponding higher inductive type (HIT) in Lean. Talk to us about how to do this.

Benjamin Peirce et al. have written a book on interactive theorem proving and program verification, based on Coq:

You can try to carry out some of the exercises from that book in Lean. For example:

• Carry out some of the exercises in the chapter “Imp: Simple Imperative Programs”.

Definitely talk to us if you want to follow this route, and we can help you consider what is involved. Here is a similar project:

• Write a “tautology checker” in Lean, as follows. Define an inductive datatype representing propositional logic formulas. Define a “denotation” function, define a simplification function, and prove (denote (simplify P) <-> denote P). Finally write a tautology checker and show it is correct: denote P <-> (checker P = true).

Finally, Freek Wiedijk has pointed us to a list of suggested projects used for a similar course that is taught in the Netherlands based on Coq.

Do any of the suggested projects on that list.