In this work, we introduce several novel concepts and capabilities to address obstacles faced by machine-assisted theorem proving: Atomic Tactics which are a small finite, and complete set of tactics; Transposing Atomization which turns arbitrary proofs into atomic tactic traces; ExprGraph, which is a graph representation of expression; Nazrin Prover, which is a GNN-based theorem proving agent.

In this paper, we introduce Pantograph, a tool that provides a versatile interface to the Lean 4 proof assistant and enables efficient proof search via powerful search algorithms such as Monte Carlo Tree Search. In addition, Pantograph enables high-level reasoning by enabling a more robust handling of Lean 4's inference steps.

Music theory obeys a rich set of mathematical rules and symmetries. These symmetries follow mathematical structures which can be verified and expressed in the precise language of a proof assistant. In this paper, we present Prismriver, a formalization library of music theory in Lean 4. We use Prismriver to generalize beyond existing work that assumes equal temperament tuning. We also discuss modelling counterpoint music theory with Prismriver. By formalizing music theory in Lean 4, we open the door to verifiable algorithmic composition and accompaniment generation. Prismriver also has a custom DSL integrated with MusicXML exports to interoperate with other music software. Prismriver can be used to compose music with Lean, using monadic composition primitives.