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Explore a groundbreaking conference presentation that introduces SynthLean, a novel proof assistant that bridges the gap between synthetic mathematical theories and their classical interpretations. Learn how this innovative tool combines reasoning within synthetic theories like homotopy type theory with reasoning about their concrete models, addressing a long-standing challenge in formal mathematics. Discover how SynthLean embeds Martin-Löf type theory as a domain-specific language within Lean, enabling a bidirectional workflow where constructions can be made both internally within the synthetic theory and externally in its models. Understand the technical implementation featuring a certifying normalization-by-evaluation typechecker that automatically proves the soundness of internal definitions across any model, while allowing semantic entities to be axiomatized in the syntax. Examine the system's support for universes, Σ-types, Π-types, identity types, and arbitrary axiomatized constants, all while maintaining familiar Lean syntax and tactic language for user accessibility. See how the formalization builds upon Mathlib's natural model semantics and takes a generic approach that enables mechanization of various interpretations including groupoid, cubical, and simplicial models of homotopy type theory within HoTTLean. Gain insights into how this work opens new possibilities for formalizing classical mathematical results using synthetic methods, with practical implications for both theoretical computer science and mathematical formalization.
