TAPSOFT '89 2nd international joint conference on Theory and practice of software development
TACS '97 Proceedings of the Third International Symposium on Theoretical Aspects of Computer Software
Recursive Families of Inductive Types
TPHOLs '00 Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics
Nested General Recursion and Partiality in Type Theory
TPHOLs '01 Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics
Modelling general recursion in type theory
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Interactive Theorem Proving and Program Development: Coq'Art The Calculus of Inductive Constructions
Interactive Theorem Proving and Program Development: Coq'Art The Calculus of Inductive Constructions
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MKM'11 Proceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics
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ICTAC'06 Proceedings of the Third international conference on Theoretical Aspects of Computing
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Surreal Numbers form a totally ordered (commutative) Field, containing copies of the reals and (all) the ordinals. I have encoded most of the Ring structure of surreal numbers in Coq. This encoding relies on Aczel's encoding of set theory in type theory. This paper discusses in particular the definitional or proving points where I had to diverge from Conway's or the most natural way, like separation of simultaneous induction-recursion into two inductions, transforming the definition of the order into a mutually inductive definition of “at most” and “at least” and fitting the rather complicated induction/recursion schemes into the type theory of Coq.