
Cubical informal type theory: the higher groupoid structure
Following a project of developing conventions and notations for informal...
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Impredicative Encodings of (Higher) Inductive Types
Postulating an impredicative universe in dependent type theory allows Sy...
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Parametric Cubical Type Theory
We exhibit a computational type theory which combines the higherdimensi...
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Homotopy canonicity for cubical type theory
Cubical type theory provides a constructive justification of homotopy ty...
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CostAware Type Theory
Although computational complexity is a fundamental aspect of program beh...
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Frege's theory of types
There is a widespread assumption in type theory that the discipline begi...
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A HoTT Quantum Equational Theory (Extended Version)
This paper presents an equational theory for the QRAM model of quantum c...
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Naive cubical type theory
This paper proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper and Licata as the implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types and the EckmmanHilton duality.
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