E. Lemma, 5.1. For any term t and base term b in ? CA , ?(t

. Proof, We proceed by structural induction over t ? ? CA ? t = x. Then ?(x[b/x]) = ?(b) = x[?(b)

@. and =. ?y, (?y : U.t ? [b/x]) and this is equal to ?y : U.?(t ? [b/x]) which by the induction hypothesis is equal to ?y, (t ? ))[?(b)/x] = ?(?y : U.t ? )[?(b)/x]

@. and =. ?x, r[b/x]) = ?X.?(r[b/x]) which by the induction hypothesis is equal to ?X.?(r)[?(b)

@. and =. R@u, Then ?(r@U = ?(r[b/x]@U ) = ?(r[b/x])@U which by the induction hypothesis is equal to ?(r)[?(b), @U = ?(r)@U [?(b)/x] = ?(r@U )[?(b)/x]

@. and =. R+u, = ?(r[b/x])+?(u[b/x]) which by the induction hypothesis is equal to ?(r)[?(b)

E. Lemma, For any terms t 1 , t 2 in ? add

. Proof, Notice that the only case where these terms are different is when one of the addends reduces to 0, in such case, their normal forms coincides, making it possible to compare in this way

. Proof, We proceed by structural induction over t ?? CA

=. ?x, By the induction hypothesis ?(r)? A ?(r?), so ?(?X.r)? A = ?X.?(r)? A ?X.?(r?) = ?(?X.r?)

. Proof, Rule by rule analysis. Elementary rules ? Rule 0

@. Application and . Rule, ? (t) u + (r) u. (t + r) u = (t + r) u ? v (t) u + (r)

@. Rule, Analogous to previous case Beta reductions ? Rule (?x : U.t) b ? t[b/x]. (?x : U.t) b = (?x.t) b. Since base vectors are translated into base vectors

T. Bibliography, J. J. Altenkirch, and . Grattage, A functional quantum programming language, Proceedings of LICS-2005, pp.249-258, 2005.

P. Arrighi and A. Díaz-caro, A System F accounting for scalars, Logical Methods in Computer Science, vol.8, issue.1, 2011.
DOI : 10.2168/LMCS-8(1:11)2012

URL : https://hal.archives-ouvertes.fr/hal-00924944

P. Arrighi and A. Díaz-caro, Scalar System F for Linear-Algebraic ??-Calculus: Towards a Quantum Physical Logic, Proceedings of QPL-2009
DOI : 10.1016/j.entcs.2011.01.033

URL : https://hal.archives-ouvertes.fr/hal-00924890

P. Arrighi and G. Dowek, A Computational Definition of the Notion of Vectorial Space, Proceedings of WRLA-2004, pp.249-261, 2004.
DOI : 10.1016/j.entcs.2004.06.013

URL : https://hal.archives-ouvertes.fr/hal-00940931

P. Arrighi and G. Dowek, Linear-algebraic ??-calculus: higher-order, encodings, and confluence., Proceedings of RTA-2008, pp.17-31, 2008.
DOI : 10.1007/978-3-540-70590-1_2

URL : http://arxiv.org/pdf/quant-ph/0612199v1.pdf

P. Arrighi, A. Díaz-caro, M. Gadella, and J. J. Grattage, Measurements and confluence in quantum lambda calculi with explicit qubits, Proceedings of QPL/DCM-2008, pp.59-74, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00924875

P. Arrighi, A. Díaz-caro, and B. Valiron, A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus, Proceedings of the 7th International Workshop on Developments of Computational Methods (DCM 2011)
DOI : 10.4204/EPTCS.88.1

URL : https://hal.archives-ouvertes.fr/hal-00924926

A. Assaf and S. Perdrix, Completeness of algebraic CPS simulations, Proceedings of the 7th International Workshop on Developments of Computational Methods, 2011.
DOI : 10.4204/EPTCS.88.2

URL : https://hal.archives-ouvertes.fr/hal-00932770

H. P. Barendregt, Lambda calculi with types, volume II of Handbook of logic in computer science, 1992.

G. Boudol, Lambda-Calculi for (Strict) Parallel Functions, Information and Computation, vol.108, issue.1, pp.51-127, 1994.
DOI : 10.1006/inco.1994.1003

URL : https://hal.archives-ouvertes.fr/inria-00075174

O. Bournez and M. Hoyrup, Rewriting Logic and Probabilities, Proceedings of RTA-2003, pp.61-75, 2003.
DOI : 10.1007/3-540-44881-0_6

URL : https://hal.archives-ouvertes.fr/inria-00099620

P. Buiras, A. Díaz-caro, and M. Jaskelioff, Confluence via strong normalisation in an algebraic ?calculus with rewriting, Proceedings of the 6th Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2011), 2011.
URL : https://hal.archives-ouvertes.fr/hal-00924938

A. Church, An Unsolvable Problem of Elementary Number Theory, American Journal of Mathematics, vol.58, issue.2, pp.345-363, 1936.
DOI : 10.2307/2371045

A. Church, A formulation of the simple theory of types, The Journal of Symbolic Logic, vol.1, issue.02, pp.56-68, 1940.
DOI : 10.2307/2371199

C. Dev and . Team, The Coq proof assistant reference manual. INRIA, 8.2 edition, 2009.

U. De-'liguoro and A. Piperno, Non deterministic extensions of untyped ?-calculus, Information and Computation, vol.122, issue.2, pp.149-177, 1995.

D. Cosmo, Review of Isomorphisms of Types:, ACM SIGACT News, vol.28, issue.4, 1995.
DOI : 10.1145/270563.571468

A. D. Pierro, C. Hankin, and H. Wiklicky, Probabilistic ??-calculus and Quantitative Program Analysis, Journal of Logic and Computation, vol.15, issue.2, pp.159-179, 2005.
DOI : 10.1093/logcom/exi008

A. Díaz-caro, Agregando medición al cálculo de van tonder, 2007.

A. Díaz-caro and B. Petit, Sums in linear algebraic lambda-calculus, 2010.

A. Díaz-caro, S. Perdrix, C. Tasson, and B. Valiron, Equivalence of algebraic ?-calculi, Informal proceedings of HOR-2010, pp.6-11, 2010.

A. Díaz-caro, S. Perdrix, C. Tasson, and B. Valiron, Call by value, call by name and the vectorial behaviour of algebraic ?-calculus, 2011.

D. J. Dougherty, Adding algebraic rewriting to the untyped lambda calculus, Information and Computation, vol.101, issue.2, pp.251-267, 1992.
DOI : 10.1016/0890-5401(92)90064-M

T. Ehrhard, On Köthe sequence spaces and linear logic, Mathematical Structures in Computer Science, vol.12, issue.5, pp.579-623, 2003.

T. Ehrhard, Finiteness spaces, Mathematical Structures in Computer Science, vol.15, issue.4, pp.615-646, 2005.
DOI : 10.1017/S0960129504004645

URL : https://hal.archives-ouvertes.fr/hal-00150276

T. Ehrhard, A Finiteness Structure on Resource Terms, 2010 25th Annual IEEE Symposium on Logic in Computer Science, pp.402-410, 2010.
DOI : 10.1109/LICS.2010.38

URL : https://hal.archives-ouvertes.fr/hal-00448431

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

M. J. Fischer, Lambda calculus schemata, Proceedings of ACM conference on Proving assertions about programs, pp.104-109, 1972.
DOI : 10.1145/942580.807077

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

J. Girard, Interprétation fonctionnelle et élimination des coupures dans l'arith-métique d'ordre supérieure, 1972.

J. Girard, Linear logic, Theoretical Computer Science, vol.50, issue.1, pp.1-102, 1987.
DOI : 10.1016/0304-3975(87)90045-4

URL : https://hal.archives-ouvertes.fr/inria-00075966

J. Girard, Y. Lafont, and P. Taylor, Proofs and Types, volume 7 of Cambridge Tracts in Theoretical Computer Science, 1989.

O. M. Herescu and C. Palamidessi, Probabilistic Asynchronous ??-Calculus, Proceedings of FOSSACS-2000, pp.146-160, 2000.
DOI : 10.1007/3-540-46432-8_10

URL : http://arxiv.org/pdf/cs/0109002v1.pdf

G. Jaeger, Quantum information: An overview, 2007.

J. Jouannaud and H. Kirchner, Completion of a Set of Rules Modulo a Set of Equations, SIAM Journal on Computing, vol.15, issue.4, pp.1155-1194, 1986.
DOI : 10.1137/0215084

J. Krivine, Lambda-calcul: types et modèles. Études et recherches en informatique, 1990.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, 2000.

M. Pagani and S. R. Rocca, Solvability in Resource Lambda-Calculus, Proceedings of FOSSACS-2010, pp.358-373, 2010.
DOI : 10.1007/978-3-642-12032-9_25

M. Pagani and P. Tranquilli, Parallel Reduction in Resource Lambda-Calculus, Proceedings of APLAS-2009, pp.226-242, 2009.
DOI : 10.1007/978-3-642-10672-9_17

URL : https://hal.archives-ouvertes.fr/hal-00699013

B. Petit, A Polymorphic Type System for the Lambda-Calculus with Constructors, Proceedings of TLCA-2009, pp.234-248
DOI : 10.1007/978-3-540-71389-0_23

G. D. Plotkin, Call-by-name, call-by-value and the ??-calculus, Theoretical Computer Science, vol.1, issue.2, pp.125-159, 1975.
DOI : 10.1016/0304-3975(75)90017-1

J. C. Reynolds, Towards a theory of type structure, Programming Symposium: Proceedings of the Colloque sur la Programmation, pp.408-425, 1974.
DOI : 10.1007/3-540-06859-7_148

A. Sabry and P. Wadler, A reflection on call-by-value, ACM Transactions on Programming Languages and Systems, vol.19, issue.6, pp.916-941, 1997.
DOI : 10.1145/267959.269968

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

M. H. Sørensen and P. Urzyczyn, Lectures on the Curry-Howard Isomorphism, of Studies in Logic and the Foundations of Mathematics, 2006.

W. W. Tait, Intensional interpretations of functionals of finite type I. The Journal of Symbolic Logic, pp.198-212, 1967.

C. Tasson, Algebraic Totality, towards Completeness, Proceedings of TLCA-2009, pp.325-340, 2009.
DOI : 10.1007/978-3-540-73449-9_28

URL : https://hal.archives-ouvertes.fr/hal-00440750

B. Valiron, Semantics of a Typed Algebraic Lambda-Calculus, Proceedings DCM-2010 of Electronic Proceedings in Theoretical Computer Science, pp.147-158, 2010.
DOI : 10.4204/EPTCS.26.14

B. Valiron, Coq proof, 2011.

B. Valiron, Coq proof, 2011.

A. Van-tonder, A Lambda Calculus for Quantum Computation, SIAM Journal on Computing, vol.33, issue.5, pp.1109-1135, 2004.
DOI : 10.1137/S0097539703432165

L. Vaux, On Linear Combinations of ??-Terms, Proceedings of RTA-2007, pp.374-388, 2007.
DOI : 10.1007/978-3-540-73449-9_28

URL : https://hal.archives-ouvertes.fr/hal-00383896

L. Vaux, The algebraic lambda calculus, Mathematical Structures in Computer Science, vol.12, issue.05, pp.1029-1059, 2009.
DOI : 10.1016/S0304-3975(03)00392-X

URL : https://hal.archives-ouvertes.fr/hal-00379750

W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature, vol.15, issue.5886, pp.802-803, 1982.
DOI : 10.1038/299802a0