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
Handbook of logic in computer science (vol. 1)
Handbook of logic in computer science (vol. 2)
Computation and reasoning: a type theory for computer science
Computation and reasoning: a type theory for computer science
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
Using Reflection to Build Efficient and Certified Decision Procedures
TACS '97 Proceedings of the Third International Symposium on Theoretical Aspects of Computer Software
A Two-Level Approach Towards Lean Proof-Checking
TYPES '95 Selected papers from the International Workshop on Types for Proofs and Programs
Computational Metatheory in Nuprl
Proceedings of the 9th International Conference on Automated Deduction
Exploring Abstract Algebra in Constructive Type Theory
CADE-12 Proceedings of the 12th International Conference on Automated Deduction
A Set Constructor for Inductive Sets in Martin-Löf's Type Theory
Category Theory and Computer Science
COLOG '88 Proceedings of the International Conference on Computer Logic
Recursive Families of Inductive Types
TPHOLs '00 Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics
Type Isomorphisms and Proof Reuse in Dependent Type Theory
FoSSaCS '01 Proceedings of the 4th International Conference on Foundations of Software Science and Computation Structures
Formalizing in Coq Hidden Algebras to Specify Symbolic Computation Systems
Proceedings of the 9th AISC international conference, the 15th Calculemas symposium, and the 7th international MKM conference on Intelligent Computer Mathematics
A Type of Partial Recursive Functions
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Dependently-typed formalisation of relation-algebraic abstractions
RAMICS'11 Proceedings of the 12th international conference on Relational and algebraic methods in computer science
Developing the algebraic hierarchy with type classes in coq
ITP'10 Proceedings of the First international conference on Interactive Theorem Proving
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We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. types endowed with an arbitrary equivalence relation, as carriers for algebras. In this way it is possible to define the quotient of an algebra by a congruence. Standard constructions over algebras are defined and their basic properties are proved formally. To overcome the problem of defining term algebras in a uniform way, we use types of trees that generalize wellorderings. Our implementation gives tools to define new algebraic structures, to manipulate them and to prove their properties.