Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Inductively defined types in the calculus of constructions
Proceedings of the fifth international conference on Mathematical foundations of programming semantics
Handbook of logic in computer science (vol. 2)
Representing inductively defined sets by wellorderings in Martin-Löf's type theory
Theoretical Computer Science
Using Reflection to Build Efficient and Certified Decision Procedures
TACS '97 Proceedings of the Third International Symposium on Theoretical Aspects of Computer Software
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
Universal Algebra in Type Theory
TPHOLs '99 Proceedings of the 12th International Conference on Theorem Proving in Higher Order Logics
TPHOLs '99 Proceedings of the 12th International Conference on Theorem Proving in Higher Order Logics
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
COLOG '88 Proceedings of the International Conference on Computer Logic
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
A Type of Partial Recursive Functions
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
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Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong positivity, that characterizes these families. Then we investigate its solutions. First, we construct a model using wellorderings. Second, we use an extension of type theory, implemented in the proof tool Coq, to construct another model that does not have extensionality problems. Finally, we apply the two level approach: We internalize inductive definitions, so that we can manipulate them and reason about them inside type theory.