Extensions of Pure Type Systems
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
Fixed point semantics and partial recursion in Coq
Proceedings of the 10th international ACM SIGPLAN conference on Principles and practice of declarative programming
Modules over monads and initial semantics
Information and Computation
Structured formal development with quotient types in Isabelle/HOL
AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Quotients revisited for Isabelle/HOL
Proceedings of the 2011 ACM Symposium on Applied Computing
Formalizing projective plane geometry in Coq
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
A design structure for higher order quotients
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
Pragmatic quotient types in coq
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
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This note studies quotient types in the Calculus of Inductive Constructions (CIC), implemented in the proof assistant coq, and compares their expressivity to that of mathematical quotients. In [Hof95], Martin Hofmann proposes an extension of the Calculus of Constructions (CC) with quotient types which he shows consistent, but notices that they are not sufficient to account for the natural isomorphism θ which exists in set theory between functional spaces E → F/R and E→F/S where fSg iff ∀x ∈ F, f(x)Rg(x). One can thus ask the question to know if it is possible to extend these quotient types to be able to show injectivity and surjectivity of this morphism. We show here that any extension of this kind in Coq with the impredicative sort Set would be contradictory.