Introduction to higher order categorical logic
Introduction to higher order categorical logic
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Computation and reasoning: a type theory for computer science
Computation and reasoning: a type theory for computer science
Completing the rationals and metric spaces in LEGO
Papers presented at the second annual Workshop on Logical environments
Elimination of extensionality in Martin-Lo¨f type theory
TYPES '93 Proceedings of the international workshop on Types for proofs and programs
Typing algorithm in type theory with inheritance
Proceedings of the 24th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Proof, language, and interaction
TACS '97 Proceedings of the Third International Symposium on Theoretical Aspects of Computer Software
A Simple Model for Quotient Types
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
Extensions of Pure Type Systems
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
Implementing Constructive Real Analysis: Preliminary Report
Constructivity in Computer Science, Summer Symposium
Extensional Equality in Intensional Type Theory
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Normalization and the Yoneda embedding
Mathematical Structures in Computer Science
A computational view of implicit coercions in type theory
Mathematical Structures in Computer Science
Interfaces as functors, programs as coalgebras: a final coalgebra theorem in intensional type theory
Theoretical Computer Science
Implementing the cylindrical algebraic decomposition within the Coq system
Mathematical Structures in Computer Science
Quotients over Minimal Type Theory
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Finite Groups Representation Theory with Coq
Calculemus '09/MKM '09 Proceedings of the 16th Symposium, 8th International Conference. Held as Part of CICM '09 on Intelligent Computer Mathematics
Packaging Mathematical Structures
TPHOLs '09 Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics
Invariants for the FoCaL language
Annals of Mathematics and Artificial Intelligence
Zermelo's well-ordering theorem in type theory
TYPES'06 Proceedings of the 2006 international conference on Types for proofs and programs
A modular formalisation of finite group theory
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
A computer-verified monadic functional implementation of the integral
Theoretical Computer Science
Towards Measurable Types for Dynamical Process Modeling Languages
Electronic Notes in Theoretical Computer Science (ENTCS)
Dependently-typed formalisation of relation-algebraic abstractions
RAMICS'11 Proceedings of the 12th international conference on Relational and algebraic methods in computer science
Point-free, set-free concrete linear algebra
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Proving formally the implementation of an efficient gcd algorithm for polynomials
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Proving equalities in a commutative ring done right in coq
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
A machine-checked formalization of the random oracle model
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
A semi-reflexive tactic for (sub-)equational reasoning
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
Pragmatic quotient types in coq
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
Abstract effects and proof-relevant logical relations
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
Combining proofs and programs in a dependently typed language
Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
Normalization by Evaluation and Algebraic Effects
Electronic Notes in Theoretical Computer Science (ENTCS)
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.