Guarded Cubical Type Theory

This paper improves the treatment of equality in guarded dependent type theory ( GDTT ), by combining it with cubical type theory ( CTT ). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with c...

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Bibliographic Details
Published in:Journal of automated reasoning Vol. 63; no. 2; pp. 211 - 253
Main Authors: Birkedal, Lars, Bizjak, Aleš, Clouston, Ranald, Grathwohl, Hans Bugge, Spitters, Bas, Vezzosi, Andrea
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.08.2019
Springer Nature B.V
Subjects:
Artificial Intelligence
Computation
Computer Science
Cubical type theory
Guarded recursion
Homotopy type theory
Mathematical Logic and Formal Languages
Mathematical Logic and Foundations
Number theory
Semantics
Symbolic and Algebraic Manipulation
Guarded recursion
Homotopy type theory
Cubical type theory
ISSN:0168-7433, 1573-0670, 1573-0670
Online Access:Get full text
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Summary:This paper improves the treatment of equality in guarded dependent type theory ( GDTT ), by combining it with cubical type theory ( CTT ). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin–Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory ( GCTT ), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation. We show that CTT can be given semantics in presheaves on C × D , where C is the cube category, and D is any small category with an initial object. We then show that the category of presheaves on C × ω provides semantics for GCTT .
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ISSN:0168-7433
1573-0670
1573-0670
DOI:10.1007/s10817-018-9471-7