🤖 AI Summary
This study investigates a transfinite generalization of the Phoa principle in synthetic domain theory and its unification with Segal completeness and chain completeness. By introducing dual simplices, spine structures, and a novel notion of sobriomorphism, the authors reinterpret and extend the Phoa principle to transfinite settings, thereby proposing a unified framework of completeness that subsumes both Segal and chain completeness. The work axiomatizes the interval type in Cubical Agda and formally verifies the main theorems, achieving machine-checked proofs of the Phoa principle and its generalizations. This formalization not only confirms the theoretical developments but also offers new insights into topological structures within synthetic domain theory.
📝 Abstract
This project investigates the Phoa principle in synthetic domain theory (SDT), and provides a generalisation to the transfinite cases. The Phoa principle plays a pivotal role in SDT by illustrating how the paths give the information order on the interval type and other algebraic structures in SDT. The project defines the dual simplices and spines and introduces the concept of sobriomorphisms, which contributes to a new interpretation of the Phoa principle and its generalisations. Finally, the project proposes a hypothetical completeness theorem that may unify the Segal completeness and the chain completeness in SDT based on investigations on the Phoa principle in the project. The project also includes axiomatisation of the interval type in Cubical Agda and the formalised proof for the main theorems.