In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology.

We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library \textsf{mathlib}, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.

We describe a formal proof of the independence of the continuum hypothesis (CH) in the Lean theorem prover. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of ¬ CH and a σ-closed forcing for the consistency of CH.

Sequential colimits are an important class of higher inductive types. We present a self-contained and fully formalized proof of the conjecture that in homotopy type theory sequential colimits appropriately commute with Σ-types. This result allows us to give short proofs of a number of useful corollaries, some of which were conjectured in other works: the commutativity of sequential colimits with identity types, with homotopy fibers, loop spaces, and truncations, and the preservation of the properties of truncatedness and connectedness under sequential colimits. Our entire development carries over to (∞, 1)-toposes using Shulman's recent interpretation of homotopy type theory into these structures.

We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space 2^{omega_2 x omega} and formally verify the failure of the continuum hypothesis in the resulting model.

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an n-type can be delooped n + 2 times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.

In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. This construction gives a characterization of functions from the propositional truncation to an arbitrary type, extending the universal property of the propositional truncation. We have fully formalized all the results in a new proof assistant, Lean.