Formal verification using proof assistants, such as Coq or Lean, is an effective way of ensuring software correctness. Recent research has shown that machine learning can automate proof synthesis, but such approaches are only successful a fraction of the time. Methods for improving such proof synthesis include training more precise models, improving the model-driven synthesis search mechanisms, and effectively combining multiple models into synthesis search. This paper focuses on the latter and develops ProofCoop, for the first time demonstrating collaborative cooperation between models. Diva, the state-of-the-art method for combining models, uses a disparate search for each model. On the CoqGym benchmark of 68.5K theorems from 124 open-source Coq projects a Diva combination of five models automatically proves 3,287 (27.5%) of CoqGym’s 11.9K test set theorems, while the best individual model proves only 2,604 (21.8%). By contrast, our method, ProofCoop can use the same five models to fully automatically prove 3,932 (33.0%) theorems, meaning that ProofCoop is 19.6% more likely to prove a theorem, on average, than the prior state of the art combination method. ProofCoop allows its models to build on each other’s contributions and demonstrates that such collaboration significantly increases synthesis success. ProofCoop enables six different types of collaboration: joint model next-step prediction at each search step; preferential next-step prediction via voting, bidding, and stacking; sharing lemmas proven across models mid-search, and models completing each other’s partial proofs. Together with CoqHammer, ProofCoop synthesizes proofs for 36.0% of the theorems. Our research demonstrates that creative uses of learned models can lead to collaborative synthesis that is more effective than prior approaches, suggesting a powerful new research direction in automated formal verification.