Kleis – Executable mathematics with formal verification Kleis is a language for mathematical specification and verification. Define structures with axioms, write first-order logic (∀, ∃, →), and prove properties via SMT solvers. Solver abstraction layer — ships with Z3, pluggable for others. Documented applications: AI/ML: Neural network robustness, safe RL, compositional contracts (aligned with Berkeley's "Toward Verified AI") Hardware: Formal verification vs UVM simulation, bitvector proofs Physics: Tensor calculus, Einstein equations, Lagrangian mechanics, dimensional analysis Control: LQR, LQG, Kalman filters, stability proofs Quantum: State normalization, Hermitian observables, Grover/Deutsch-Jozsa Security: Zanzibar-style ReBAC, OAuth2, SQL injection detection Protocols: IPv4 validation, stop-and-wait ARQ Category Theory: Functors, natural transformations Meta: Ships with a LISP interpreter written in pure Kleis