Formalization of physics index notation in Lean 4

Joseph Tooby-Smith

Formalization of physics index notation in Lean 4

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arxiv 2411.07667 v1 pith:5EJIOZAI submitted 2024-11-12 cs.LO hep-phhep-th

Joseph Tooby-Smith This is my paper · ORCID

classification cs.LOhep-phhep-th

keywords notationindexleanphysicsimplementationproveresultstensors

verification ladder T0 review T1 audit T2 compute T3 formal

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The physics community relies on index notation to effectively manipulate types of tensors. This paper introduces the first formally verified implementation of index notation in the interactive theorem prover Lean 4. By integrating index notation into Lean, we bridge the gap between traditional physics notation and formal verification tools, making it more accessible for physicists to write and prove results within Lean. We also open up a new avenue through which AI tools can be used to prove results related to tensors in physics. Behind the scenes our implementation leverages a novel application of category theory.

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LeanTree: Accelerating White-Box Proof Search with Factorized States in Lean 4
cs.LG 2025-07 conditional novelty 6.0 of 10

White-box proof search with factorized Lean 4 goals reaches 18.4% on MiniF2F with Llemma-7B, outperforming black-box generation at 9.6%.

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