. Mu1,

. Mu3, =Piecewise[{{x(1-rn),nm<x<=nM},{x-as,-nm<=x<=nm},{x(1+rn)

. Nu3, =Piecewise[{{x(1+rn),nm<x<=nM},{x+as,-nm<=x<=nm},{x(1-rn),-nM<=x<-nm}}] assum, pp.0-0

, We use Mathematica to assist us. Definitions in Appendice A are assumed

C. Alias, Multi-dimensional Rankings, Program Termination and Complexity Bounds of Flowchart Programs, Proc. of the 17th International Static Analysis Symposium (SAS'10). Ed. by Radhia Cousot and Matthieu Martel, vol.6337, pp.117-133, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00523298

R. Bagnara, A New Look at the Automatic Synthesis of Linear Ranking Functions, Information and Computation, vol.215, pp.47-67, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01186145

M. Amir and . Ben-amram, Size-Change Termination, Monotonicity Constraints and Ranking Functions, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, pp.109-123, 2009.

M. Amir and . Ben-amram, Monotonicity Constraints for Termination in the Integer Domain, Logical Methods in Computer Science, vol.7, issue.3, 2011.

M. Amir and . Ben-amram, Ranking Functions for Linear-Constraint Loops, Proc. of the 1st International Workshop on Verification and Program Transformation (VPT'13), vol.16, pp.1-8, 2013.

M. Amir and . Ben-amram, Mortality of Iterated Piecewise Affine Functions over the Integers: Decidability and complexity, pp.19-56

M. Amir, S. Ben-amram, and . Genaim, Ranking Functions for LinearConstraint Loops, Journal of the ACM, vol.61, p.55, 2014.

M. Amir, S. Ben-amram, and . Genaim, Complexity of Bradley-MannaSipma Lexicographic Ranking Functions, Proc. of the 27th International Conference on Computer Aided Verification (CAV'15), vol.9207, pp.304-321, 2015.

M. Amir, S. Ben-amram, and . Genaim, On Multiphase-Linear Ranking Functions, Proc. of the 29th International Conference on Computer Aided Verification (CAV'17), Part II, vol.10427, pp.601-620, 2017.

M. Amir, S. Ben-amram, A. N. Genaim, and . Masud, On the Termination of Integer Loops, ACM Transactions on Programming Languages and Systems, vol.34, p.24, 2012.

M. Bozga, R. Iosif, and F. Konecný, Deciding Conditional Termination, Logical Methods in Computer Science, vol.10, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01418906

, Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.3-540, 2006.

C. Baier and J. Katoen, Principles of Model Checking, 2008.

M. Amir, C. S. Ben-amram, and . Lee, Program Termination Analysis in Polynomial Time, ACM Transactions on Programming Languages and Systems, vol.29, p.37, 2007.

, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, 2009.

R. Bagnara and F. Mesnard, Eventual Linear Ranking Functions, Proc. of the 15th International Symposium on Principles and Practice of Declarative Programming (PPDP'13), pp.229-238, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01451694

M. S. Belaid, C. Michel, and M. Rueher, Boosting Local Consistency Algorithms over Floating-Point Numbers, Proc. of the 18th International Conference on Principles and Practice of Constraint Programming (CP'12). Ed. by Michela Milano, vol.7514, pp.127-140, 2012.

A. R. Bradley, Z. Manna, and H. B. Sipma, Linear Ranking with Reachability, Proc. of the 17th International Conference on Computer Aided Verification (CAV'05), vol.3576, pp.491-504, 2005.

A. R. Bradley, Z. Manna, and H. B. Sipma, Termination of Polynomial Programs, Proc. of the 6th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI'05). Ed. by Radhia Cousot, vol.3385, pp.113-129, 2005.

S. Boldo and T. M. Nguyen, Hardware-Independent Proofs of Numerical Programs, Proc. of the 2nd NASA Formal Methods Symposium (NFM'10). Ed. by César A. Muñoz. Vol. NASA/CP-2010-216215. NASA Conference Proceedings, pp.14-23, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00534410

S. Boldo and T. M. Nguyen, Proofs of Numerical Programs when the Compiler Optimizes, Innovations in Systems and Software Engineering, vol.7, pp.151-160, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00777639

S. Boldo, Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number, In: Electronic Notes in Theoretical Computer Science, vol.317, pp.27-32, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01148409

M. Braverman, Termination of Integer Linear Programs, Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.372-385, 2006.
DOI : 10.1007/11817963_34

URL : http://www.cs.toronto.edu/~mbraverm/CAV2006.pdf

8. ,

S. Berardi and S. Steila, An Intuitionistic Version of Ramsey's Theorem and its use in Program Termination, Annals of Pure and Applied Logic, vol.166, pp.1382-1406, 2015.

P. Cousot and R. Cousot, A Gentle Introduction to Formal Verification of Computer Systems by Abstract Interpretation, NATO Science for Peace and Security Series-D: Information and Communication Security, vol.25, pp.1-29, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00543886

P. Cousot and R. Cousot, An Abstract Interpretation Framework for Termination, Proc. of the 39th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp.245-258, 2012.

P. Cousot and R. Cousot, Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints, Proc. of the 4th ACM Symposium on Principles of Programming Languages (POPL'77), pp.238-252, 1977.

P. E. Ceruzzi, The Early Computers of Konrad Zuse, IEEE Annals of the History of Computing, vol.3, pp.241-262, 1935.

,

H. Chen, S. Flur, and S. Mukhopadhyay, Termination Proofs for Linear Simple Loops, Software Tools for Technology Transfer, vol.17, pp.47-57, 2015.

A. Church, A Note on the Entscheidungsproblem, The Journal of Symbolic Logic, vol.1, pp.40-41, 1936.

M. Codish, P. J. Vitaly-lagoon, and . Stuckey, Testing for Termination with Monotonicity Constraints, Proc. of the 21st International Conference on Logic Programming (ICLP'05), vol.3668, pp.326-340, 2005.

, Proc. of the 17th International Static Analysis Symposium (SAS'10), vol.6337, 2010.

B. Cook, Proving Stabilization of Biological Systems, Proc. of the 12th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI'11), pp.134-149, 2011.

/. Dx,

B. Cook, Ranking Function Synthesis for Bit-vector Relations, Formal Methods in System Design, vol.43, pp.93-120, 2013.

B. Cook, A. Podelski, and A. Rybalchenko, Terminator: Beyond Safety". In: Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.415-418, 2006.

B. Cook, A. Podelski, and A. Rybalchenko, Proving Program Termination, Communications of the ACM, vol.54, pp.88-98, 2011.

B. Cook, A. See, and F. Zuleger, Ramsey vs. Lexicographic Termination Proving, Proc. of the 19th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'13)

, Lecture Notes in Computer Science, vol.7795, pp.47-61, 2013.

R. Cytron, Efficiently Computing Static Single Assignment Form and the Control Dependence Graph, ACM Transactions on Programming Languages and Systems, vol.13, pp.451-490, 1991.

C. Delhommé, Decomposition of Tree-Automatic Structures, 2016.

C. David, D. Kroening, and M. Lewis, Unrestricted Termination and Non-termination Arguments for Bit-Vector Programs, Proc. of the 24th European Symposium on Programming (ESOP'15). Ed. by Jan Vitek, vol.9032, pp.183-204, 2015.
DOI : 10.1007/978-3-662-46669-8_8

URL : http://arxiv.org/pdf/1410.5089

L. Fousse, MPFR: A Multiple-precision Binary Floating-point Library with Correct Rounding, ACM Transactions on Mathematical Software, vol.33, p.13, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00070266

K. Ghorbal, E. Goubault, and S. Putot, The Zonotope Abstract Domain Taylor1+, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, pp.627-633, 2009.
DOI : 10.1007/978-3-642-02658-4_47

J. Giesl, Automated Termination Proofs with AProVE, Proc. of the 15th International Conference on Rewriting Techniques and Applications (RTA'04). Ed. by Vincent van Oostrom, vol.3091, pp.210-220, 2004.
DOI : 10.1007/978-3-540-25979-4_15

URL : http://www-i2.informatik.rwth-aachen.de/giesl/papers/RTA04-distribute.pdf

J. Giesl, Termination Competition (TermComp'15), Proc. of the 25th International Conference on Automated Deduction (CADE'15)
DOI : 10.1007/978-3-319-21401-6_6

URL : http://upcommons.upc.edu/bitstream/2117/85179/1/termCompCADE2015.pdf

, Lecture Notes in Computer Science, vol.9195, pp.105-108, 2015.

J. Giesl, Analyzing Program Termination and Complexity Automatically with AProVE, Journal of Automated Reasoning, vol.58, pp.3-31, 2017.
DOI : 10.1007/s10817-016-9388-y

URL : http://eprints.bbk.ac.uk/16551/1/JAR-AProVE.pdf

S. Graillat, V. Lefèvre, and J. Muller, On the Maximum Relative Error when Computing Integer Powers by Iterated Multiplications in Floating-point Arithmetic, Numerical Algorithms, vol.70, pp.1572-9265, 2015.
URL : https://hal.archives-ouvertes.fr/ensl-00945033

L. Gonnord, D. Monniaux, and G. Radanne, Synthesis of Ranking Functions using Extremal Counterexamples, Proc. of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI'15), pp.608-618, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01144622

D. Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys, vol.23, pp.5-48, 1991.

J. R. Hauser, Handling Floating-Point Exceptions in Numeric Programs, ACM Transactions on Programming Languages and Systems, vol.18, pp.139-174, 1996.

M. Heizmann, N. D. Jones, and A. Podelski, Size-Change Termination and Transition Invariants, Proc. of the 17th International Static Analysis Symposium (SAS'10). Ed. by Radhia Cousot and Matthieu Martel, vol.6337, pp.22-50, 2010.

, IEEE Standard for Binary Floating-Point Arithmetic, 1985.

, IEEE Standard for Floating-Point Arithmetic, IEEE Standard, pp.1-70, 2008.

N. D. Jones, Computability and Complexity-From a Programming Perspective. Foundations of computing series, pp.978-978, 1997.

C. , P. Jeannerod, and S. M. Rump, On Relative Errors of Floating-point Operations: Optimal Bounds and Applications, Mathematics of Computation, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00934443

G. L. Steele and J. L. White, How to Print Floating-point Numbers Accurately (with Retrospective), A Selection from 20 Years of the ACM SIGPLAN Conference on Programming Language Design and Implementation 1979-1999, pp.1-58113, 2004.

W. Kahan, Idempotent Binary->Decimal->Binary Conversion, 2002.

W. Kahan, Why do we Need a Floating-point Arithmetic Standard, 1981.

L. G. Khachiyan, Polynomial Algorithms in Linear Programming", In: USSR Computational Mathematics and Mathematical Physics, vol.20, pp.53-72, 1980.

D. Kroening and M. Lewis, Second-Order SAT Solving using Program Synthesis, Computing Research Repository, 2014.

J. C. Lagarias, The Ultimate Challenge: the 3x + 1 Problem, pp.978-0821849408, 2011.

X. Leroy, Formal Verification of a Realistic Compiler, Communications of the ACM 52, vol.7, pp.107-115, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00415861

J. Leike and M. Heizmann, Ranking Templates for Linear Loops, Logical Methods in Computer Science, vol.11, issue.1, 2015.

C. S. Lee, N. D. Jones, and A. M. Ben-amram, The Size-change Principle for Program Termination, Proc. of the 28th ACM SIGPLANSIGACT Symposium on Principles of Programming Languages

, , pp.81-92, 2001.

S. Lucas and J. Meseguer, Termination of Just/Fair Computations in Term Rewriting, Information and Computation, vol.206, pp.652-675, 2008.

N. Lindenstrauss, Y. Sagiv, and A. Serebrenik, TermiLog: A System for Checking Termination of Queries to Logic Programs, Proc. of the 9th International Conference on Computer Aided Verification (CAV'97)

, Lecture Notes in Computer Science, vol.1254

, , vol.isbn, pp.3-540, 1997.

S. Lucas, Polynomials Over the Reals in Proofs of Termination: From Theory to Practice, Theoretical Informatics and Applications, vol.39, pp.547-586, 2005.

Y. Li, G. Zhu, and Y. Feng, The L-Depth Eventual Linear Ranking Functions for Single-Path Linear Constraint Loops, Proc. of the 10th International Symposium on Theoretical Aspects of Software Engineering (TASE'16), pp.30-37, 2016.

D. W. Matula, In-and-out Conversions, Communications of the ACM, vol.11, issue.1, pp.47-50, 1968.

A. Miné, Relational Abstract Domains for the Detection of FloatingPoint Run-Time Errors, Computing Research Repository, 2007.

J. Misra, A Proof of Infinite Ramsey Theorem

F. Maurica, F. Mesnard, and É. Payet, On the Linear Ranking Problem for Simple Floating-Point Loops, Proc. of the 23rd International Static Analysis Symposium (SAS'16). Ed. by Xavier Rival, vol.9837, pp.300-316, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01451688

F. Maurica, F. Mesnard, and É. Payet, Termination Analysis of Floating-Point Programs using Parameterizable Rational Approximations, Proc. of the 31st ACM Symposium on Applied Computing (SAC'16). Ed. by Sascha Ossowski, pp.1674-1679, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01451687

F. Maurica, F. Mesnard, and É. Payet, Optimal Approximation for Efficient Termination Analysis of Floating-Point Loops, Proc. of the 1st IEEE Conference on on Next Generation Computing Applications (NextComp'17, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01579791

J. Muller, J. Nicolas, and X. Roblot, Number of Solutions to (A 2 + B 2 = C 2 + C) in a Binade, 2003.
URL : https://hal.archives-ouvertes.fr/inria-00071634

D. Monniaux, The Pitfalls of Verifying Floating-Point Computations, ACM Transactions on Programming Languages and Systems, vol.30, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00128124

F. Mesnard and S. Ruggieri, On Proving Left Termination of Constraint Logic Programs, ACM Transactions on Computational Logic, vol.4, pp.207-259, 2003.

J. Muller, Handbook of Floating-Point Arithmetic. Birk-häuser, 2010.
URL : https://hal.archives-ouvertes.fr/ensl-00379167

J. Muller, On the Definition of ulp(x), p.16, 2005.
URL : https://hal.archives-ouvertes.fr/inria-00070503

É. Payet, Loop Detection in Term Rewriting using the Eliminating Unfoldings, Theoretical Computer Science, vol.403, pp.307-327, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01186183

É. Payet and F. Mesnard, Nontermination Inference of Logic Programs, ACM Transactions on Programming Languages and Systems, vol.28, issue.2, pp.256-289, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00129526

A. Podelski and A. Rybalchenko, A Complete Method for the Synthesis of Linear Ranking Functions, Proc. of the 5th International Conference on Verification, Model Checking, and Abstract Interpretation, vol.2937, pp.239-251, 2004.

A. Podelski and A. Rybalchenko, Transition Invariants, Proc. of the 19th IEEE Symposium on Logic in Computer Science (LICS' 04), pp.32-41, 2004.
DOI : 10.1109/lics.2004.1319598

A. Podelski and A. Rybalchenko, Transition Predicate Abstraction and Fair Termination, ACM Transactions on Programming Languages and Systems, vol.29, 2007.

A. Podelski and A. Rybalchenko, Transition Invariants and Transition Predicate Abstraction for Program Termination, Proc. of 17th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'11), vol.6605, pp.3-10, 2011.

É. Payet and F. Spoto, Experiments with Non-Termination Analysis for Java Bytecode, In: Electronic Notes in Theoretical Computer Science, vol.253, pp.83-96, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01188696

P. Frank and . Ramsey, On a Problem of Formal Logic, Proc. of the London Mathematical Society, pp.264-286, 1930.

X. Rival, Static Analysis-23rd International Symposium, vol.9837, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01388205

, , pp.978-981

D. De-schreye and S. Decorte, Termination of Logic Programs: The Never-Ending Story, Journal of Logic Programming, vol.19, pp.90027-90029, 1994.

M. Sipser, Introduction to the Theory of Computation, pp.978-978, 1997.

P. Schrammel and D. Kroening, 2LS for Program Analysis-(Competition Contribution)". In: Proc. of the 22nd International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Ed. by Marsha Chechik and Jean-François Raskin, vol.9636, pp.905-907, 2016.
DOI : 10.1007/978-3-662-49674-9_56

F. Spoto, F. Mesnard, and É. Payet, A Termination Analyzer for Java Bytecode Based on Path-length, ACM Transactions on Programming Languages and Systems, vol.32, issue.3, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01186167

F. Spoto, Proc. of the 23rd International Static Analysis Symposium (SAS'16). Ed. by Xavier Rival, vol.9837, pp.39-57, 2016.

A. Serebrenik and D. D. Schreye, Termination of Floating-Point Computations, Journal of Automated Reasoning, vol.34, pp.141-177, 2005.

P. H. Sterbenz, Floating-Point Computation. Prentice-Hall series in automatic computation, 1974.

C. Strachey, An Impossible Program, The Computer Journal, vol.7, p.313, 1965.

A. Tiwari, Termination of Linear Programs, Proc. of the 16th International Conference on Computer Aided Verification (CAV'04), vol.3114, pp.70-82, 2004.

A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical society, vol.2, pp.230-265, 1937.

C. Urban, Static Analysis by Abstract Interpretation of Functional Temporal Properties of Programs, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01176641

G. Zhu, Y. Li, and W. Wu, Eventual Linear Ranking Functions for Multi-path Linear Loops, Proc. of the 2016 IEEE Conference on Information Technology, Networking, Electronic and Automation Control (ITNEC'16, pp.331-337, 2016.