Implementing mathematics with the Nuprl proof development system
Implementing mathematics with the Nuprl proof development system
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
The Internal Type Theory of a Heyting Pretopos
TYPES '96 Selected papers from the International Workshop on Types for Proofs and Programs
Elementary Choiceless Constructive Analysis
Proceedings of the 14th Annual Conference of the EACSL on Computer Science Logic
Quotients over Minimal Type Theory
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
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We extend Martin-Löf's constructive set theory with effective quotient sets and the rule of uniqueness of propositional equality proofs. We prove that in the presence of at least two universes U0 and U1 the principle of excluded middle holds for small sets. The key point is the combination of uniqueness of propositional equality proofs with the effectiveness condition that allows us to recover information on the equivalence relation from the equality on the quotient set.