google-deepmind/alphageometry

Coq is an interactive theorem prover and proof assistant used for formal mathematical verification and verified software development. It utilizes the Gallina functional language to define computable functions and logical propositions, which are then verified through a machine-checked kernel. The system employs a dependent type system and a Caldicott-style proof engine to automate proof search and tactic execution. These capabilities allow for the creation of formal specifications and the development of algorithms that are mathematically proven to meet specific requirements. The toolset inclu

Lean 4 is a functional programming language and interactive proof assistant used to formalize mathematics and verify software correctness. It functions as a dependent type theorem prover and a formal verification tool that allows users to construct mathematical proofs and ensure program correctness. Additionally, it serves as a logic-based source for generating verified datasets used to train and benchmark artificial intelligence reasoning systems. The system distinguishes itself through a small-kernel verification model, where all proofs are verified by a trusted core of basic logical rules.

SymPy is a Python computer algebra system and symbolic mathematics library. It performs algebraic manipulations, calculus, and equation solving using symbolic representations to achieve exact computations rather than numerical approximations. The library includes a LaTeX expression parser that converts mathematical strings into symbolic representations for computation and formula manipulation. It also incorporates a mathematical benchmarking suite to measure execution speed and detect performance regressions across different software versions. The system provides capabilities for automated m

Z3 is an automated theorem prover and satisfiability modulo theories solver designed to determine the validity of complex logical formulas. It functions as a formal verification framework, enabling the systematic checking of hardware and software system specifications against defined logical constraints to identify inconsistencies or design flaws. The engine distinguishes itself through a combination of theory-specific decision procedures and symbolic execution capabilities. It employs conflict-driven clause learning and backtracking search algorithms to prune search spaces, while maintaining