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Contract theories have been proposed to formally support distributed and decentralized system design while ensuring safe system integration. We propose hypercontracts, a general model with a richer structure for its underlying model of components, subsuming simulation preorders. While general, the new model provides a richer algebra for its notions of refinement, parallel composition, and quotient. Further, it allows the introduction of new operations. Building on top of these foundations, we propose conic hypercontracts, which are still generic but come with a finite description.

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