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CoQMTU: A Higher-Order Type Theory with a Predicative Hierarchy of Universes Parametrized by a Decidable First-Order Theory

2011
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2011 IEEE 26th Annual Symposium on Logic in Computer Science
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We study

doi:10.1109/lics.2011.37
dblp:conf/lics/BarrasJSW11
fatcat:vpc5bawvmravlimvbzfe7ruyiu
*a*complex*type**theory*,*a*Calculus*of*Inductive Constructions*with**a**predicative**hierarchy**of**universes*and*a**first*-*order**theory*T built in its conversion relation. ... The*theory*T is specified abstractly,*by**a*set*of*constructors,*a*set*of*defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and*a*generic elimination ...*A**first*main contribution is*a*definition*of**a**parametrized*elimination principle for an abstract*first*-*order**theory*T constrained*by*three natural axioms: non-triviality, constructor-freeness, and completeness ...##
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Type Theory Unchained: Extending Agda with User-Defined Rewrite Rules

2020
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Types for Proofs and Programs
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In this paper I show how to extend

doi:10.4230/lipics.types.2019.2
dblp:conf/types/Cockx19
fatcat:2ieag5d5tnazviahqquzip3pca
*a*dependently*typed*language*with*user-defined*higher*-*order*non-linear rewrite rules. ... Rewrite rules are*a*form*of*equality reflection that is applied automatically*by*the typechecker. ... The source code*of*Agda is available on Github 2 , the code dealing*with*rewrite rules specifically can be found in the files Rewriting.hs 3 (418 lines), NonLinPattern.hs 4 (329 lines), NonLinMatch.hs ...##
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The taming of the rew: a type theory with computational assumptions

2021
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Proceedings of the ACM on Programming Languages (PACMPL)
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This paper introduces Rewriting

doi:10.1145/3434341
fatcat:nuswqtklpfgi3f6xmkzfksnudi
*Type**Theory*(RTT),*a**type**theory*where it is possible to add computational assumptions in the form*of*rewrite rules. ... On the other hand, the equality reflection rule from extensional*type**theory*solves these problems*by*collapsing computation and equality, at the cost*of*having no practical*type*checking algorithm. ... This means CoqMT only deals*with**decidable**first*-*order**theories*. ...