T. Altenkirch, S. Boulier, A. Kaposi, and N. Tabareau, Setoid type theory -a syntactic translation, MPC 2019 -13th International Conference on Mathematics of Program Construction, vol.11825, pp.155-196, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02281225

A. Asperti, W. Ricciotti, C. Sacerdoti-coen, and E. Tassi, A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions, vol.8, p.1, 2012.

R. Atkey, N. Ghani, and P. Johann, A relationally parametric model of dependent type theory, The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '14, pp.503-516, 2014.

J. Bernardy, T. Coquand, and G. Moulin, A Presheaf Model of Parametric Type Theory, Electronic Notes in Theoretical Computer Science, vol.319, pp.67-82, 2015.

J. Bernardy, P. Jansson, and R. Paterson, Proofs for free: Parametricity for dependent types, Journal of Functional Programming, vol.22, issue.2, pp.107-152, 2012.

, Proceedings of the 43rd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, 2016.

S. Boulier, P. Pédrot, and N. Tabareau, The next 700 syntactical models of type theory, Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, pp.182-194, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01445835

G. Castagna, V. Lanvin, T. Petrucciani, and J. G. Siek, Gradual typing: a new perspective, vol.16, p.32, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02183382

E. Cavallo and R. Harper, Parametric Cubical Type Theory, 2019.

C. Cohen, T. Coquand, S. Huber, and A. Mörtberg, Cubical Type Theory: a constructive interpretation of the univalence axiom, vol.262, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01378906

T. Coquand and G. Huet, The Calculus of Constructions, Information and Computation, vol.76, issue.3, pp.95-120, 1988.
URL : https://hal.archives-ouvertes.fr/inria-00076024

P. Dagand, N. Tabareau, and É. Tanter, Foundations of Dependent Interoperability, Journal of Functional Programming, vol.28, p.44, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01629909

G. Dowek, Chapter 16 -Higher-Order Unification and Matching, Handbook of Automated Reasoning, 2001.

P. Dybjer and A. Setzer, Induction-recursion and initial algebras, Ann. Pure Appl. Log, vol.124, pp.96-105, 2003.

T. Ehrhard, A Categorical Semantics of Constructions, Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS '88), pp.264-273, 1988.

R. A. Eisenberg, Dependent Types in Haskell: Theory and Practice, 2016.

J. Eremondi, É. Tanter, and R. Garcia, Approximate Normalization for Gradual Dependent Types, vol.88, p.30, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02399594

R. Garcia, A. M. Clark, and É. Tanter, Abstracting Gradual Typing, See [Bodík and Majumdar, pp.429-442, 2016.

N. Ghani, L. Malatesta, and F. Forsberg, Positive Inductive-Recursive Definitions, Log. Methods Comput. Sci, vol.11, p.1, 2015.

E. Giménez, Structural Recursive Definitions in Type Theory, ICALP, pp.397-408, 1998.

M. Hofmann, Conservativity of Equality Reflection over Intensional Type Theory, Types for Proofs and Programs, International Workshop TYPES'95, vol.1158, pp.153-164, 1995.

K. Knowles and C. Flanagan, Hybrid type checking, ACM Transactions on Programming Languages and Systems, vol.32, 2010.

N. Lehmann and É. Tanter, Gradual Refinement Types, Proceedings of the 44th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp.775-788, 2017.

R. Daniel, R. Licata, and . Harper, 2-Dimensional Directed Type Theory, Twenty-seventh Conference on the Mathematical Foundations of Programming Semantics, vol.276, pp.263-289, 2011.

P. Martin-löf, On the Meanings of the Logical Constants and the Justifications of the Logical Laws, Nordic Journal of Philosophical Logic, vol.1, pp.11-60, 1996.

C. Mcbride, Basics of Bidirectionalism, vol.1, p.27, 2018.

C. Mcbride, Check the Box!, 25th International Conference on Types for Proofs and Programs, 2019.

, Graduality from Embedding-Projection Pairs, vol.73, pp.1-73, 2018.

M. S. New, D. R. Licata, and A. Ahmed, Gradual Type Theory. See[POPL, vol.15, p.31, 2019.

P. North, Towards a Directed Homotopy Type Theory, Proceedings of the Thirty-Fifth Conference on the Mathematical Foundations of Programming Semantics, vol.347, pp.223-239, 2019.

P. Osera, V. Sjöberg, and S. Zdancewic, Dependent Interoperability, Proceedings of the 6th workshop on Programming Languages Meets Program Verification, pp.3-14, 2012.

X. Ou, G. Tan, Y. Mandelbaum, and D. Walker, Dynamic Typing with Dependent Types, Proceedings of the IFIP International Conference on Theoretical Computer Science, pp.437-450, 2004.

E. Palmgren, On universes in type theory, Twenty Five Years of Constructive Type Theory, pp.191-204, 1998.

E. Palmgren and V. Stoltenberg-hansen, Domain Interpretations of Martin-Löf's Partial Type Theory, Ann. Pure Appl. Log, vol.48, pp.135-196, 1990.

C. Paulin-mohring, Introduction to the Calculus of Inductive Constructions, All About Proofs, Proofs for All, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01094195

P. , M. Pédrot, and N. Tabareau, Failure is Not an Option -An Exceptional Type Theory, Proceedings of the 27th European Symposium on Programming Languages and Systems, vol.10801, pp.245-271, 2018.

P. , M. Pédrot, and N. Tabareau, The fire triangle: how to mix substitution, dependent elimination, and effects, Proceedings of the ACM on Programming Languages, vol.4, p.28, 2020.

P. Pédrot, N. Tabareau, H. Fehrmann, and É. Tanter, A Reasonably Exceptional Type Theory, vol.108, pp.1-108, 2019.

J. Siek and W. Taha, Gradual Typing for Functional Languages, Proceedings of the Scheme and Functional Programming Workshop, pp.81-92, 2006.

J. Siek and W. Taha, Gradual Typing for Objects, Proceedings of the 21st European Conference on Objectoriented Programming, vol.4609, pp.2-27, 2007.

J. G. Siek, M. M. Vitousek, M. Cimini, and J. Boyland, Refined Criteria for Gradual Typing, Leibniz International Proceedings in Informatics (LIPIcs), vol.32, pp.274-293, 2015.

M. Sozeau, S. Boulier, Y. Forster, N. Tabareau, and T. Winterhalter, Coq Coq correct! verification of type checking and erasure for Coq, Coq. Proc. ACM Program. Lang. 4, POPL (2020), vol.8, pp.1-8, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02380196

M. Sozeau and N. Tabareau, Universe Polymorphism in Coq, pp.499-514, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00974721

N. Swamy, C. Hritcu, C. Keller, A. Rastogi, A. Delignat-lavaud et al., Dependent types and multi-effects in F ? , See, pp.256-270, 2016.

É. Tanter and N. Tabareau, Gradual Certified Programming in Coq, Proceedings of the 11th ACM Dynamic Languages Symposium, pp.26-40, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01238774

T. Winterhalter, M. Sozeau, and N. Tabareau, Eliminating reflection from type theory, Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs, pp.91-103, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01849166

H. Xi and F. Pfenning, Eliminating array bound checking through dependent types, Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI '98), pp.249-257, 1998.

B. Ziliani and M. Sozeau, A comprehensible guide to a new unifier for CIC including universe polymorphism and overloading, p.27, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01671925

M. Bertrand and K. Maillard, Nicolas Tabareau, and Éric Tanter 1669