An Adequacy Theorem for Dependent Type Theory
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DOI: 10.1007/s00224-018-9879-9
Abstract: We present a domain model of dependent type theory and use it to prove basic metatheoretic properties. In particular, we prove that two convertible terms have the same Böhm tree. The method used is reminiscent… read more here.
Keywords: adequacy theorem; theory; type theory; theorem dependent ... See more keywords
Expressing ‘the structure of’ in homotopy type theory
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DOI: 10.1007/s11229-017-1569-7
Abstract: As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy… read more here.
Keywords: type theory; homotopy type; type; mathematics ... See more keywords
The Justification of Identity Elimination in Martin-Löf’s Type Theory
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DOI: 10.1007/s11245-017-9509-1
Abstract: On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect… read more here.
Keywords: theory; type theory; justification; martin ... See more keywords
Finiteness and rational sequences, constructively*
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DOI: 10.1017/s0956796817000041
Abstract: Abstract Rational sequences are possibly infinite sequences with a finite number of distinct suffixes. In this paper, we present different implementations of rational sequences in Martin–Löf type theory. First, we literally translate the above definition… read more here.
Keywords: finiteness rational; lists backpointers; type theory; sequences constructively ... See more keywords
Cubical Agda: A dependently typed programming language with univalence and higher inductive types
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DOI: 10.1017/s0956796821000034
Abstract: Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such… read more here.
Keywords: dependently typed; higher inductive; univalence; type theory ... See more keywords
Gradual type theory
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DOI: 10.1017/s0956796821000125
Abstract: Abstract Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. Sound gradually typed languages dynamically check types at runtime at the boundary between… read more here.
Keywords: type theory; gradual type; semantics; eta equality ... See more keywords
Model structure on the universe of all types in interval type theory
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DOI: 10.1017/s0960129520000213
Abstract: Abstract Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a… read more here.
Keywords: model; type theory; universe types; model structure ... See more keywords
Studying models of balancing selection using phase-type theory.
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DOI: 10.1093/genetics/iyab055
Abstract: Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyse models with BLS acting on a biallelic… read more here.
Keywords: bls; type theory; balancing selection; selection ... See more keywords
Hypersingular Integral Equations of Prandtl's Type: Theory, Numerical Methods, and Applications
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DOI: 10.3390/axioms11120705
Abstract: In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method… read more here.
Keywords: hypersingular integral; type theory; equations prandtl; prandtl type ... See more keywords