# Questions tagged [homotopy-type-theory]

For questions about the field of study known as homotopy type theory.

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### Formalizing Homotopy Type theory in Idris

Looking at the homotopy type theory blog one can easily find a lot of library formalizing most of Homotopy Type Theory in Agda and Coq. Is there anyone aware if there is any similar attempt to ...
831 views

### In the Hott book, are the most of the type formers redundant? And if so, why?

In chapter 1 and Appendix A of the Hott book, several primitive type families are presented (universe types, dependent function types, dependent pair types, Coproduct types, Empty Type, Unit type, ...
1k views

### Proof relevance vs. proof irrelevance

I want to use use Agda to help me write proofs, but I am getting contradictory feedback about the value of proof relevance. Jacques Carette wrote a Proof-relevant Category Theory in Agda library. But ...
770 views

### What are the negative consequences of extending CIC with axioms?

Is it true that adding axioms to the CIC might have negative influences in the computational content of definitions and theorems? I understand that, in the theory's normal behavior, any closed term ...
755 views

### Can we derive Cubical Type Theory from Self-Types?

Self Types are known for being a simple extension to the Calculus of Constructions that allow it to derive all inductive datatypes of a proof assistant like Coq and Agda, without a "hardcoded&...
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### Homotopy type theory and Gödel's incompleteness theorems

Kurt Gödel's incompleteness theorems establish the "inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic". Homotopy Type Theory provides an alternative ...
In my current understanding, regularity in cubical type theory is the following definitional equality (I'm using $A~\textbf{type}$ to emphasize the fact that $i \notin FV(A)$):  \cfrac{A~\textbf{...