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. Proof, By a simulateous induction on 1 and 2. We look at the last used rule in ?

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K. Nour and K. , A semantical proof of the strong normalization theorem for full propositional classical natural deduction, Archive for Mathematical Logic, vol.45, issue.3, pp.357-364, 2005.
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D. Ph and . Groote, Strong normalization of classical natural deduction with disjunction, 5th International Conference on Typed Lambda Calculi and Applications, TLCA'01, pp.182-196, 2001.

G. Gentzen, Recherches sur la déduction logique. Press Universitaires de France, 1955.

J. Krivine, Lambda calcul, types et modèle, 1990.

R. Matthes, Non-strictly positive fixed points for classical natural deduction, Annals of Pure and Applied Logic, vol.133, issue.1-3, pp.205-230, 2005.
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K. Nakazawa, Confluency and strong normalizability of call-by-value ????-calculus, Theoretical Computer Science, vol.290, issue.1, pp.429-463, 2003.
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K. Nakazawa and M. Tatsuta, Abstract, The Journal of Symbolic Logic, vol.902, issue.03, pp.851-859, 2003.
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=. , =. U?, and ?. , u? : B ; ?, a : C, this implies that ?, y : A u? :? ; ?, a : C, b : B. By the induction hypothesis, ? u :? ; ?, a : A ? C, b : B, therefore ? µb

R. David and K. Nour, A short proof of the strong normalization of the simply typed ?µ-calculus, pp.27-33, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00382688

R. David, Une preuve simple de résultats classiques en ?-calcul. Compte Rendu de l'Académie des Sciences, pp.1401-1406, 1995.

S. Farkh and K. Nour, Types Complets dans une extension du système AF2, Informatique Théorique et Application, pp.31-37, 1998.

J. Girard, Y. Lafont, and P. Taylor, Proofs and types, 1986.

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J. R. Hindley, The simple semantics for Coppe-Dezani-Sallé types, Proceeding of the 5th Colloquium on International Symposium on Programming, pp.212-226, 1982.

J. R. Hindley, The completeness theorem for typing ??-terms, Theoretical Computer Science, vol.22, issue.1-2, pp.1-17, 1983.
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J. R. Hindley, Curry's type-rules are complete with respect to the F-semantics too, Theoretical Computer Science, vol.22, issue.1-2, pp.127-133, 1983.
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K. Nour, Op??rateurs de mise en m??moire et types $\forall $-positifs, RAIRO - Theoretical Informatics and Applications, vol.30, issue.3, pp.261-293, 1996.
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K. Nour, Mixed logic and storage operators, Archive for Mathematical Logic, vol.39, issue.4, pp.261-280, 2000.
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URL : https://hal.archives-ouvertes.fr/hal-00381633

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D. Ph and . Groote, Strong normalization of classical natural deduction with disjunction, 5th International Conference on typed lambda calculi and applications , TLCA'01, pp.182-196, 2001.

G. Gentzen, Recherches sur la déduction logique. Press Universitaires de France, 1955.

R. Matthes, Non-strictly positive fixed points for classical natural deduction, Annals of Pure and Applied Logic, vol.133, issue.1-3, pp.205-230, 2005.
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K. Nakazawa, Confluency and strong normalizability of call-by-value ????-calculus, Theoretical Computer Science, vol.290, issue.1, pp.429-463, 2003.
DOI : 10.1016/S0304-3975(01)00380-2

K. Nakazawa and M. Tatsuta, Abstract, The Journal of Symbolic Logic, vol.902, issue.03, pp.851-859, 2003.
DOI : 10.1016/S0304-3975(01)00380-2

K. Nour and K. Saber, A semantical proof of the strong normalization theorem for full propositional classical natural deduction, Archive for Mathematical Logic, vol.45, issue.3, 2005.
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