A. F. Alessi, F. Barbanera, and M. Dezani-ciancaglini, Intersection types and lambda models, Theoretical Computer Science, vol.355, issue.2, pp.108-126, 2006.
DOI : 10.1016/j.tcs.2006.01.004
URL : http://doi.org/10.1016/j.tcs.2006.01.004

P. Bai02 and . Baillot, Checking polynomial time complexity with types, IFIP TCS of IFIP Conference Proceedings, pp.370-382, 2002.

B. H. Barendregt, M. Coppo, and M. Dezani-ciancaglini, A filter lambda model and the completeness of type assignment, The Journal of Symbolic Logic, vol.37, issue.04, pp.931-940, 1983.
DOI : 10.1002/malq.19800261902

B. G. Boudol, P. Curien, and C. Lavatelli, A semantics for lambda calculi with resources, Mathematical Structures in Computer Science, vol.9, issue.4, pp.437-482, 1999.
DOI : 10.1017/S0960129599002893

A. Bec01 and . Beckmann, Exact bounds for lengths of reductions in typed lambda-calculus, J. of Symbolic Logic, vol.66, issue.3, pp.1277-1285, 2001.

A. Bl11a, S. Bernadet, and . Lengrand, Complexity of strongly normalising ?-terms via nonidempotent intersection types, Proc. of the 14th Int. Conf. on Foundations of Software Science and Computation Structures (FOSSACS'11), 2011.

A. Bl11b, S. Bernadet, L. Lengrand, and C. Polytechnique, Non-idempotent intersection types, orthogonality and filter models -long version, 2011.

B. P. Baillot and V. Mogbil, Soft lambda-Calculus: A Language for Polynomial Time Computation, p.312015, 2003.
DOI : 10.1007/978-3-540-24727-2_4
URL : https://hal.archives-ouvertes.fr/hal-00085129

C. M. Coppo and M. Dezani-ciancaglini, A new type assignment for lambda-terms. Archiv für mathematische Logik und Grundlagenforschung, pp.139-156, 1978.

C. T. Coquand and A. Spiwack, A proof of strong normalisation using domain theory. Logic, Methods Comput. Science, vol.3, issue.4, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00432490

D. De-carvalho, Intersection Types for Light Affine Lambda Calculus, Electronic Notes in Theoretical Computer Science, vol.136, pp.133-152, 2005.
DOI : 10.1016/j.entcs.2005.06.011

D. De-carvalho, Execution time of lambda-terms via denotational semantics and intersection types, p.4251, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00319822

D. M. Dezani-ciancaglini, F. Honsell, and Y. Motohama, Compositional characterizations of lambda-terms using intersection types (extended abstract), MFCS: Symp. on Mathematical Foundations of Computer Science, 2000.

D. V. Danos and J. Krivine, Disjunctive Tautologies as Synchronisation Schemes, Proc. of the 9th Annual Conf. of the European Association for Computer Science Logic (CSL'00), volume 1862 of LNCS, pp.292-301, 2000.
DOI : 10.1007/3-540-44622-2_19
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.2792

E. T. Ehrhard and L. Regnier, The differential lambda-calculus, Theoretical Computer Science, vol.309, issue.1-3, pp.1-41, 2003.
DOI : 10.1016/S0304-3975(03)00392-X
URL : https://hal.archives-ouvertes.fr/hal-00150572

S. Ghi96 and . Ghilezan, Strong normalization and typability with intersection types, Notre Dame J. Formal Loigc, vol.37, issue.1, pp.44-52, 1996.

J. Gir72 and . Girard, Interprétation fonctionelle et élimination des coupures de l'arithmétique d'ordre supérieur, Thèse d'état, 1972.

J. Gir87 and . Girard, Linear logic, Theoret. Comput. Sci, vol.50, issue.1, pp.1-101, 1987.

G. M. Gaboardi and S. R. Rocca, A soft type assignment system for lambda -calculus, Proc. of the 16th Annual Conf. of the European Association for Computer Science Logic (CSL'07), pp.253-267, 2007.

J. Kri01 and . Krivine, Typed lambda-calculus in classical Zermelo-Fraenkel set theory, Arch. Math. Log, vol.40, issue.3, pp.189-205, 2001.

K. A. Kfoury and J. B. Wells, Principality and decidable type inference for finite-rank intersection types, Proceedings of the 26th ACM SIGPLAN-SIGACT symposium on Principles of programming languages , POPL '99, pp.161-174, 1999.
DOI : 10.1145/292540.292556

Y. Laf04 and . Lafont, Soft linear logic and polynomial time, Theoret. Comput. Sci, vol.318, issue.12, pp.163-180, 2004.

D. Lei86 and . Leivant, Typing and computational properties of lambda expressions, Theoretical Computer Science, vol.44, issue.1, pp.51-68, 1986.

L. S. Lengrand and A. Miquel, Classical <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>??</mml:mi></mml:mrow></mml:msub></mml:math>, orthogonality and symmetric candidates, Annals of Pure and Applied Logic, vol.153, issue.1-3, pp.3-20, 2008.
DOI : 10.1016/j.apal.2008.01.005

M. G. Munch-maccagnoni, Focalisation and Classical Realisability, Proc. of the 18th Annual Conf. of the European Association for Computer Science Logic (CSL'09), pp.409-423, 2009.
DOI : 10.1016/0304-3975(87)90045-4
URL : https://hal.archives-ouvertes.fr/inria-00409793

M. Melliès and J. Vouillon, Recursive Polymorphic Types and Parametricity in an Operational Framework, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05), pp.82-91, 2005.
DOI : 10.1109/LICS.2005.42

N. P. Neergaard and H. G. Mairson, Types, potency, and idempotency: why nonlinearity and amnesia make a type system work, Proc. of the ACM Intern. Conf. on Functional Programming, pp.138-149, 2004.
DOI : 10.1145/1016850.1016871

M. Par97 and . Parigot, Proofs of strong normalisation for second order classical natural deduction, J. of Symbolic Logic, vol.62, issue.4, pp.1461-1479, 1997.

H. Sch82 and . Schwichtenberg, Complexity of normalization in the pure typed lambda calculus, The L. E. J. Brouwer Centenary Symposium. North-Holland, 1982.

W. W. Tai75 and . Tait, A realizability interpretation of the theory of species, Logic Colloquium vB95 S. van Bakel. Intersection Type Assignment Systems, pp.240-251385, 1975.

Y. H. Yokouchi, Embedding a Second-Order Type System into an Intersection Type System, Information and Computation, vol.117, issue.2, pp.206-220, 1995.
DOI : 10.1006/inco.1995.1040

F. ?. Let and F. So, Hence, there exist ? 1 , ? 2 and A such that ? 1 ? M : A ? F and ? 2 ? N : A and ? = ? 1 ? ? 2 . So ? ? ? 1 , and ? 1 ? ?. Hence, A ? F ? M ? . We also have A ? N ? . So we have F ? M ? @N ? . Conversely, let F ? M ? @N ? . There exists A such that A ? F ? M ? and A ? N ? . So there exists ? ? ? such that ? ? M : A ? F and there exists ? ? ? such that

F. Let, . M. ?x, and . @u, There exists A ? u such that A ? F ? ?x.M ? . So there exists ? ? ? such that ? ? ?x.M : A ? F . Hence, there exists U such that

F. So-we-have, F. Conversely, and . ?u, There exists ? ? (?, x ? u) such that then there exist ? 1 ? ? and A ? u such that ? = ? 1 , x : A (using Lemma 10.1) So we have ? 1 ? ?x.M : A ? F , and then A ? F ? ?x.M ? . Hence, we have F ? ?x.M ? @u, then for all y ? Dom(?), y = x (using Lemma 10.1). So ? ? ? and ?, x : ? ? M : F . Since u is a value

. Proof, By induction on the derivation of G S M : A. Let ? be a valuation