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

Bruno Barras, Jean-Pierre Jouannaud, Pierre-Yves Strub, Qian Wang
2011 2011 IEEE 26th Annual Symposium on Logic in Computer Science  
We study 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  ... 
doi:10.1109/lics.2011.37 dblp:conf/lics/BarrasJSW11 fatcat:vpc5bawvmravlimvbzfe7ruyiu

Type Theory Unchained: Extending Agda with User-Defined Rewrite Rules

Jesper Cockx, Assia Mahboubi, Marc Bezem
2020 Types for Proofs and Programs  
In this paper I show how to extend 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  ... 
doi:10.4230/lipics.types.2019.2 dblp:conf/types/Cockx19 fatcat:2ieag5d5tnazviahqquzip3pca

The taming of the rew: a type theory with computational assumptions

Jesper Cockx, Nicolas Tabareau, Théo Winterhalter
2021 Proceedings of the ACM on Programming Languages (PACMPL)  
This paper introduces Rewriting 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.  ... 
doi:10.1145/3434341 fatcat:nuswqtklpfgi3f6xmkzfksnudi