| We present a proof system for establishing the correctness of results produced by optimization algorithms, with a focus on mixed-integer programming (MIP). Our system generalizes the seminal work of Bogaerts, Gocht, McCreesh, and Nordström (2022) for binary programs to handle any additional difficulties arising from unbounded and continuous variables, and covers a broad range of solving techniques, including symmetry handling, cutting planes, and presolving reductions. Consistency across all decisions that affect the feasible region is achieved by a pair of transitive relations on the set of solutions, which relies on the newly introduced notion of consistent branching trees. Combined with a series of machine-verifiable derivation rules, the resulting framework offers practical solutions to enhance the trust in integer programming as a methodology for applications where reliability and correctness are key. |