R. Massimo-bartoletti, E. Tuosto, and R. Zunino, Contract-Oriented Computing in CO2, Scientific Annals of Computer Science, vol.22, issue.1, pp.5-60, 2012.
DOI : 10.7561/SACS.2012.1.5

M. Bartoletti, A. Scalas, E. Tuosto, and R. Zunino, Honesty by Typing, Proceedings of FMOODS/FORTE'13, pp.305-320, 2013.
DOI : 10.1007/978-3-642-38592-6_21
URL : https://hal.archives-ouvertes.fr/hal-01515248

M. Blume and D. A. Mcallester, Sound and complete models of contracts, Journal of Functional Programming, vol.16, issue.4&5, pp.375-414, 2006.
DOI : 10.1017/S0956796806005971

L. Bocchi, K. Honda, E. Tuosto, and N. Yoshida, A Theory of Design-by-Contract for Distributed Multiparty Interactions, Proceedings of CONCUR'10, pp.162-176, 2010.
DOI : 10.1007/978-3-642-15375-4_12

L. Bocchi, R. Tzu-chun-chen, K. Demangeon, N. Honda, and . Yoshida, Monitoring networks through multiparty session types, Proceedings FMOODS/FORTE'13, pp.50-65, 2013.
DOI : 10.1016/j.tcs.2017.02.009
URL : https://hal.archives-ouvertes.fr/hal-01213683

L. Tzu-chun-chen, P. Bocchi, K. Deniélou, N. Honda, and . Yoshida, Asynchronous distributed monitoring for multiparty session enforcement, Proceedings of TGC'11, pp.25-45, 2011.

C. Dimoulas and M. Felleisen, On contract satisfaction in a higher-order world, ACM Transactions on Programming Languages and Systems, vol.33, issue.5, p.16, 2011.
DOI : 10.1145/2039346.2039348

C. Dimoulas, R. B. Findler, C. Flanagan, and M. Felleisen, Correct blame for contracts: no more scapegoating, Proceedings of POPL'11, pp.215-226, 2011.
DOI : 10.1145/1925844.1926410

C. Dimoulas, S. Tobin-hochstadt, and M. Felleisen, Complete Monitors for Behavioral Contracts, Proceedings of ESOP'12, pp.214-233, 2012.
DOI : 10.1007/978-3-642-28869-2_11
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.230.8298

T. Disney, C. Flanagan, and J. Mccarthy, Temporal higher-order contracts, ACM SIGPLAN Notices, pp.176-188, 2011.
DOI : 10.1145/2034574.2034800
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.8880

R. Bruce, F. , and M. Blume, Contracts as pairs of projections, Proceedings of FLOPS'06, pp.226-241, 2006.

R. Bruce, F. , and M. Felleisen, Contracts for higher-order functions, Proceedings of ICFP'02, pp.48-59, 2002.

J. Garrigue, J. L. , and N. , Adding GADTs to OCaml: the direct approach, Proceedings of ACM SIGPLAN Workshop on ML, 2011.

J. Simon, M. Gay, and . Hole, Subtyping for Session Types in the ?-calculus, Acta Informatica, vol.42, issue.2-3, pp.191-225, 2005.

J. Simon, V. Gay, and . Vasconcelos, Linear type theory for asynchronous session types, Journal of Functional Programming, vol.20, issue.1, pp.19-50, 2010.

S. J. Gay, A. Vasco-thudichum-vasconcelos, N. Ravara, A. Z. Gesbert, and . Caldeira, Modular session types for distributed object-oriented programming, Proceedings of POPL'10, pp.299-312, 2010.
DOI : 10.1145/1707801.1706335
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.5609

M. Greenberg, B. C. Pierce, and S. Weirich, Contracts made manifest, Journal of Functional Programming, vol.22, issue.03, pp.225-274, 2012.
DOI : 10.1145/581478.581484
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.155.4383

J. Gronski and C. Flanagan, Unifying hybrid types and contracts, Proceedings of TFP'07, pp.54-70, 2007.

R. Hinze, J. Jeuring, and A. Löh, Typed Contracts for Functional Programming, Proceedings of FLOPS'06, pp.208-225, 2006.
DOI : 10.1007/11737414_15
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.8217

K. Honda, Types for dyadic interaction, Proceedings of CONCUR'93, pp.509-523, 1993.
DOI : 10.1007/3-540-57208-2_35
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.1355

K. Honda, V. T. Vasconcelos, and M. Kubo, Language primitives and type discipline for structured communication-based programming, Proceedings of ESOP'98, pp.122-138, 1998.
DOI : 10.1007/BFb0053567
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.3650

L. Jia, H. Gommerstadt, and F. Pfenning, Monitors and blame assignment for higherorder session types, Proceedings of POPL'16, pp.582-594, 2016.
DOI : 10.1145/2914770.2837662

K. Knowles and C. Flanagan, Hybrid type checking, ACM Transactions on Programming Languages and Systems, vol.32, issue.2, p.2010
DOI : 10.1145/1667048.1667051

N. Kobayashi, B. C. Pierce, and D. N. Turner, Linearity and the pi-calculus, ACM Transactions on Programming Languages and Systems, vol.21, issue.5, pp.914-947, 1999.
DOI : 10.1145/330249.330251
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.9195

B. Meyer, Design by contract, Advances in Object-oriented Software Engineering, pp.1-50, 1992.

C. Phúc, S. Nguyen, D. Tobin-hochstadt, and . Van-horn, Soft contract verification, In ACM SIGPLAN Notices, vol.49, pp.139-152, 2014.

L. Padovani, Abstract, Journal of Functional Programming, vol.6, 2017.
DOI : 10.1016/0304-3975(83)90059-2

A. Ravara and V. Vasconcelos, Typing Non-uniform Concurrent Objects, Proceedings of CONCUR'00, pp.474-488, 2000.
DOI : 10.1007/3-540-44618-4_34
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.4337

C. Scholliers, E. Tanter, and W. D. Meuter, Computational contracts, Science of Computer Programming, vol.98, pp.360-375, 2015.
DOI : 10.1016/j.scico.2013.09.005
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.309.1972

T. , S. Strickland, S. Tobin-hochstadt, R. B. Findler, and M. Flatt, Chaperones and impersonators: run-time support for reasonable interposition, Proceedings of OOPSLA'12, pp.943-962, 2012.
DOI : 10.1145/2398857.2384685

C. Swords, A. Sabry, and S. Tobin-hochstadt, Expressing contract monitors as patterns of communication, ACM SIGPLAN Notices, vol.50, issue.9, pp.387-399, 2015.
DOI : 10.1007/3-540-58184-7_118

P. Thiemann, Session Types with Gradual Typing, Proceedings of TGC'14, pp.144-158, 2014.
DOI : 10.1007/978-3-662-45917-1_10
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.431.4685

B. Toninho and N. Yoshida, Certifying data in multiparty session types. In A List of Successes That Can Change the World -Essays Dedicated to Philip Wadler on the Occasion of His 60th Birthday, LNCS, vol.9600, pp.433-458, 2016.
DOI : 10.1007/978-3-319-30936-1_23

B. Toninho, L. Caires, and F. Pfenning, Dependent session types via intuitionistic linear type theory, Proceedings of the 13th international ACM SIGPLAN symposium on Principles and practices of declarative programming, PPDP '11, pp.161-172, 2011.
DOI : 10.1145/2003476.2003499
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.206.2721

J. A. Tov and R. Pucella, Stateful Contracts for Affine Types, Proceedings of ESOP'10, pp.550-569, 2010.
DOI : 10.1007/978-3-642-11957-6_29
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.185.2322

J. A. Tov and R. Pucella, Practical affine types, Proceedings of POPL'11, pp.447-458, 2011.
DOI : 10.1145/1925844.1926436

P. Wadler, A complement to blame, Proceedings of SNAPL'15, LIPIcs 32, pp.309-320, 2015.

P. Wadler and R. B. Findler, Well-Typed Programs Can???t Be Blamed, Proceedings of ESOP'09, pp.1-16, 2009.
DOI : 10.1007/978-3-642-00590-9_1
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.156.6379

P. Assume, e1,r,s . Since P is balanced, e 1 e 2 and p = s and q = r. As in the previous case, we conclude that E [, p.1

P. Assume and P. , Hence, a = c. Then, the proof is completed by using Proposition 10 to conclude c dual c

_. , _. , and _. @bullet, flat_c w,p,_ and v and w are user expressions and e ? v if and only if wv ? v , ?, ?

=. Case-e, Since e 2 is a user expression, erase(e 2 ) = e 2

. Proof, We proceed by induction on the derivation of erase(e) ? e

?. Hence, Since e 0 = flat_c w, it trivially holds that wv ? v if and only if wv ? v . Hence, e is well-stacked and e = erase(e )

_. Case, _. , and _. ?. , erase(e) = v p false ? blame p = e . Since e is well-stacked, e =