# Questions tagged [type-theory]

Type structure is a syntactic discipline for enforcing levels of abstraction.

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### What is the role of “universe” types in intuitionistic type theory?

According to the Wikipedia article on intuitionsitic type theory: The universe types allow proofs to be written about all the types created with the other type constructors. Every term in the ...
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### Implementations of Dependent Type Theory

I am trying to find a minimal implementation of dependent type theory that supports Pi Types (obviously) Modules containing records Inductive data types Universe Hierarchy A notion of equality ...
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### Extended Church's thesis and internal parametricity

I am wondering if there is any known relationship between these 2 concepts in intensional MLTT as formulated here. Does $Internal\ parametricity \implies ECT$ hold? For forumlation of ECT see https://...
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### Lack of atomic propositions in the Calculus of Constructions from ATTAPL textbook

I am working through the Dependent Types chapter from Advanced Topics in Types and Programming Languages (ATTAPL) by Benjamin Pierce et al. I am confused with the calculus presented Fig 2-7 (Calculus ...
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### Proposition terms vs types in Coq

Consider the following div function written in Coq. It takes in a proof that the divider is non-zero. Definition div (n d:nat) (pf: ~(d = 0)) := n/d. Focus on <...
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### How to define list zipping categorically/inductively?

Lists and fixpoints The type of $A$-lists is defined as $\mu F_A$, where $F_A(X) = 1 + A \times X$ is the "cons-or-nil"-functor and $\mu$ is the least fixpoint operator. In Haskell syntax, this would ...
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### Differences in the type signatures using dependent types

What is the difference between the following types for the $head$ function on a vector of integers. ($head$ takes a natural number $n$ and a vector $v$ of length $n$ or $(n+1)$ (depending on the ...
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### PHOAS with extrinsic typing?

Parameterized Higher Order Abstract Syntax (PHOAS) is a representation of syntax trees that allows the host language's binding to be used to represent binding in the language being modelled, while ...
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### What's the point of stack judgement in CBPV?

Call-by-push-value (CBPV) introduces two main families of types, values and computations, and their corresponding judgements. However, in some extensions/variants/adaptation of CBPV, there is a third ...
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### Is System-F with higher-kinded newtypes equivalent in computational power to System-F omega?

If we have System-F with higher-kinded types and newtypes, then we can express everything (I think) of System-F omega, except we have to manually (un)pack. For example: ...
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### Fixed set of type constructors to simulate all intensional inductive families?

I'm wondering, are there small dependent calculi that can simulate a language with inductive families (that is, has a type isomorphic to each inductive family, at least as powerful of induction ...
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### Structural equality of Pi Types with heterogeneous equality?

I'm trying to implement a proof of the following type: ...
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### Is my understanding regarding how to implement Quotient Types correct?

I was trying to understand Quotient Types, and determine if Self-Types can be used to implement them. From a Reddit post, Here is an example and explanation that may be more familiar to non-...
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### Type-theoretic interpretation of Skolemization

What is the type-theoretic interpretation / equivalent of Skolemization? Skolemization converts some formula into Skolem normal form. The two formulae are equisatisfiable with each other. Or, to say ...
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### Model of Coq (pCuIC) in higher toposes?

Can the type theory of Coq (pCuIC) be modeled in all higher Grothendieck toposes? First of all, even the set theoretical model is not complete (e.g. inductive types in Prop). Although, this is ...
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