A. Abate, M. Prandini, J. Lygeros, and S. Sastry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems, Automatica, vol.44, issue.11, pp.2724-2734, 2008.

J. Abrial, The B-book: assigning programs to meanings, 2005.

J. Abrial, W. Su, and H. Zhu, Formalizing hybrid systems with event-b, International Conference on Abstract State Machines, Alloy, B, VDM, and Z, pp.178-193, 2012.

L. Aceto, A. Burgueno, and K. Larsen, Model checking via reachability testing for timed automata, International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp.263-280, 1998.

W. Ahrendt, B. Beckert, D. Bruns, R. Bubel, C. Gladisch et al., The key platform for verification and analysis of java programs, Working Conference on Verified Software: Theories, Tools, and Experiments, pp.55-71, 2014.

E. Altman and V. Gaitsgory, Asymptotic optimization of a nonlinear hybrid system governed by a markov decision process, SIAM Journal on Control and Optimization, vol.35, issue.6, pp.2070-2085, 1997.

R. Alur, C. Courcoubetis, A. Thomas, P. Henzinger, and . Ho, Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems, Hybrid systems, pp.209-229, 1993.

R. Alur, R. Grosu, I. Lee, and O. Sokolsky, Compositional modeling and refinement for hierarchical hybrid systems, The Journal of Logic and Algebraic Programming, vol.68, issue.1-2, pp.105-128, 2006.

R. Alur, A. Thomas, G. Henzinger, G. Lafferriere, and . Pappas, Discrete abstractions of hybrid systems, Proceedings of the IEEE, vol.88, issue.7, pp.971-984, 2000.

J. Karl, B. Aström, and . Wittenmark, Computer-controlled systems: theory and design. Courier Corporation, 2013.

R. Back, L. Petre, and I. Porres, Continuous action systems as a model for hybrid systems, Nord. J. Comput, vol.8, issue.1, pp.2-21, 2001.

R. Back and J. Wright, Refinement calculus: a systematic introduction, 1998.

R. Back and W. Joakim-von, Trace refinement of action systems, 1994.

C. M. Jos and . Baeten, A brief history of process algebra, Theoretical Computer Science, vol.335, issue.2-3, pp.131-146, 2005.

R. Banach, M. Butler, S. Qin, N. Verma, and H. Zhu, Core hybrid event-b i: single hybrid event-b machines, Science of Computer Programming, vol.105, pp.92-123, 2015.

B. Beckert, R. Hähnle, H. Peter, and . Schmitt, Verification of object-oriented software: The KeY approach, 2007.

A. Benveniste, B. Caillaud, D. Nickovic, R. Passerone, J. Raclet et al., Contracts for system design, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00757488

A. Benveniste, P. Caspi, A. Stephen, N. Edwards, P. L. Halbwachs et al., The synchronous languages 12 years later, Proceedings of the IEEE, vol.91, issue.1, pp.64-83, 2003.

A. Benveniste, D. Nickovic, and T. Henzinger, Compositional Contract Abstraction for System Design, INRIA, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00938854

G. Berry and G. Gonthier, The esterel synchronous programming language: Design, semantics, implementation. Science of computer programming, vol.19, pp.87-152, 1992.
URL : https://hal.archives-ouvertes.fr/inria-00075711

Y. Bertot and P. Castéran, Interactive theorem proving and program development: Coq'Art: the calculus of inductive constructions, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00344237

F. Bobot, J. Filliâtre, C. Marché, and A. Paskevich, Why3: Shepherd your herd of provers, Boogie 2011: First International Workshop on Intermediate Verification Languages, pp.53-64, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00790310

B. Bohrer and A. Platzer, A hybrid, dynamic logic for hybrid-dynamic information flow, 2018.

B. Bohrer, V. Rahli, I. Vukotic, M. Völp, and A. Platzer, Formally verified differential dynamic logic, Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, pp.208-221, 2017.

T. Bourke and M. Pouzet, Zélus: A synchronous language with odes, Proceedings of the 16th international conference on Hybrid systems: computation and control, pp.113-118, 2013.

E. Brinksma, T. Krilavi?ciuskrilavi?cius, and Y. Usenko, Process algebraic approach to hybrid systems. IFAC Proceedings Volumes, vol.38, pp.325-330, 2005.

H. Brinksma and T. Krilavicius, Behavioural hybrid process calculus, 2005.

L. Manuela and . Bujorianu, Extended stochastic hybrid systems and their reachability problem, International Workshop on Hybrid Systems: Computation and Control, pp.234-249, 2004.

L. Manuela, J. Bujorianu, M. C. Lygeros, and . Bujorianu, Bisimulation for general stochastic hybrid systems, International Workshop on Hybrid Systems: Computation and Control, pp.198-214, 2005.

C. Zhou-chaochen, A. P. Hoare, and . Ravn, A calculus of durations, Information processing letters, vol.40, issue.5, pp.269-276, 1991.

W. Zhou-chaochen, A. P. Ji, and . Ravn, A formal description of hybrid systems, International Hybrid Systems Workshop, pp.511-530, 1995.

M. Chen, X. Han, T. Tang, S. Wang, M. Yang et al., Mars: A toolchain for modelling, analysis and verification of hybrid systems, Provably Correct Systems, pp.39-58, 2017.

O. Edmund-m-clarke, K. Grumberg, and . Hamaguchi, Another look at ltl model checking. Formal Methods in System Design, vol.10, pp.47-71, 1997.

M. Edmund and . Clarke, Orna Grumberg, and Doron Peled. Model checking, 1999.

T. Coquand and G. Huet, The calculus of constructions. Information and computation, vol.76, pp.95-120, 1988.
URL : https://hal.archives-ouvertes.fr/inria-00076024

P. Cousot and R. Cousot, Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints, Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages, pp.238-252, 1977.

P. Cousot, R. Cousot, J. Feret, L. Mauborgne, A. Miné et al., The astrée analyzer, European Symposium on Programming, pp.21-30, 2005.

L. De-moura and N. Bjørner, Z3: An efficient smt solver, International conference on Tools and Algorithms for the Construction and Analysis of Systems, pp.337-340, 2008.

A. Donzé and O. Maler, Systematic simulation using sensitivity analysis, International Workshop on Hybrid Systems: Computation and Control, pp.174-189

. Springer, , 2007.

I. Dragomir, V. Preoteasa, and S. Tripakis, Compositional semantics and analysis of hierarchical block diagrams, International Symposium on Model Checking Software, pp.38-56, 2016.

A. Fehnker and F. Ivan?i´ivan?i´c, Benchmarks for hybrid systems verification, Hybrid Systems: Computation and Control, pp.326-341, 2004.

J. , C. Filliâtre, and A. Paskevich, Why3-where programs meet provers, European Symposium on Programming, pp.125-128, 2013.

M. Fränzle, E. Moritz-hahn, H. Hermanns, N. Wolovick, and L. Zhang, Measurability and safety verification for stochastic hybrid systems, Proceedings of the 14th international conference on Hybrid systems: computation and control, pp.43-52, 2011.

G. Frehse, Phaver: Algorithmic verification of hybrid systems past hytech, International workshop on hybrid systems: computation and control, pp.258-273

. Springer, , 2005.

N. Fulton, S. Mitsch, B. Bohrer, and A. Platzer, Bellerophon: Tactical theorem proving for hybrid systems, International Conference on Interactive Theorem Proving, pp.207-224, 2017.

N. Fulton, S. Mitsch, J. Quesel, M. Völp, and A. Platzer, Keymaerax: An axiomatic tactical theorem prover for hybrid systems, International Conference on Automated Deduction, pp.527-538, 2015.

N. Fulton and A. Platzer, A logic of proofs for differential dynamic logic, 2016.

S. Gao, J. Avigad, and E. M. Clarke, ?-complete decision procedures for satisfiability over the reals, International Joint Conference on Automated Reasoning, pp.286-300, 2012.

S. Gao, J. Avigad, and E. M. Clarke, Delta-decidability over the reals, Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on, pp.305-314, 2012.

S. Gao, S. Kong, and E. M. Clarke, dreal: An smt solver for nonlinear theories over the reals, International Conference on Automated Deduction, pp.208-214, 2013.

S. Gao, S. Kong, and E. M. Clarke, Satisfiability modulo odes, Formal Methods in Computer-Aided Design (FMCAD), pp.105-112, 2013.

T. Gautier, P. L. Guernic, and L. Besnard, Signal: A declarative language for synchronous programming of real-time systems, Conference on Functional Programming Languages and Computer Architecture, pp.257-277, 1987.
URL : https://hal.archives-ouvertes.fr/inria-00075791

A. Girard, J. George, and . Pappas, Approximate bisimulation: A bridge between computer science and control theory, European Journal of Control, vol.17, issue.5-6, pp.568-578, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00765660

G. Gonthier, A computer-checked proof of the four colour theorem, 2005.

P. Dimitar, S. Guelev, N. Wang, and . Zhan, Compositional hoare-style reasoning about hybrid csp in the duration calculus, International Symposium on Dependable Software Engineering: Theories, Tools, and Applications, pp.110-127, 2017.

A. Ernst-moritz-hahn, H. Hartmanns, J. Hermanns, and . Katoen, A compositional modelling and analysis framework for stochastic hybrid systems, Formal Methods in System Design, vol.43, issue.2, pp.191-232, 2013.

N. Halbwachs, P. Caspi, P. Raymond, and D. Pilaud, The synchronous data flow programming language lustre, Proceedings of the IEEE, vol.79, issue.9, pp.1305-1320, 1991.

N. Halbwachs, Y. Proy, and P. Raymond, Verification of linear hybrid systems by means of convex approximations, International Static Analysis Symposium, pp.223-237, 1994.

Z. Han and B. Krogh, Reachability analysis of large-scale affine systems using low-dimensional polytopes, Hybrid Systems: Computation and Control, pp.287-301, 2006.

D. Harel, D. Kozen, and J. Tiuryn, Dynamic logic. In Handbook of philosophical logic, pp.99-217, 2001.

L. Constance, . Heitmeyer, and . Bg-labaw, A benchmark for comparing different approaches for specifying and verifying real-time systems, 1993.

A. Thomas and . Henzinger, The theory of hybrid automata, Verification of Digital and Hybrid Systems, pp.265-292, 2000.

A. Thomas, P. Henzinger, H. Ho, and . Wong-toi, Hytech: the next generation, Real-Time Systems Symposium, 1995. Proceedings., 16th IEEE, pp.56-65, 1995.

A. Thomas, P. Henzinger, H. Ho, and . Wong-toi, Hytech: A model checker for hybrid systems, International Journal on Software Tools for Technology Transfer, vol.1, issue.1-2, pp.110-122, 1997.

C. Hoare, Communicating sequential processes, Communications of the ACM, vol.21, issue.8, pp.666-677, 1978.

P. Höfner, Algebraic calculi for hybrid systems, BoD-Books on Demand, 2009.

E. John, R. Hopcroft, J. D. Motwani, and . Ullman, Introduction to automata theory, languages, and computation, vol.32, pp.60-65, 2001.

J. Hu, J. Lygeros, and S. Sastry, Towards a theory of stochastic hybrid systems, International Workshop on Hybrid Systems: Computation and Control, pp.160-173, 2000.

A. Jean-raymond, Modelling in event-b. system and software engineering, 2008.

A. Jean-baptiste-jeannin and . Platzer, dtl 2: Differential temporal dynamic logic with nested temporalities for hybrid systems, International Joint Conference on Automated Reasoning, pp.292-306, 2014.

H. E. Jensen, G. Kim, A. Larsen, and . Skou, Modelling and analysis of a collision avoidance protocol using spin and uppaal, BRICS Report Series, vol.3, issue.24, 1996.

H. Jifeng, From csp to hybrid systems, A classical mind, pp.171-189, 1994.

G. Klein, K. Elphinstone, G. Heiser, J. Andronick, D. Cock et al., Formal verification of an os kernel, Proceedings of the ACM SIGOPS 22nd symposium on Operating systems principles, pp.207-220, 2009.

L. Lamport, Hybrid systems in tla+, Hybrid Systems, pp.77-102, 1993.

L. Lamport, The temporal logic of actions, ACM Transactions on Programming Languages and Systems (TOPLAS), vol.16, issue.3, pp.872-923, 1994.

L. Lamport, Specifying systems: the TLA+ language and tools for hardware and software engineers, 2002.

P. Kim-g-larsen, W. Pettersson, and . Yi, Uppaal in a nutshell, International journal on software tools for technology transfer, vol.1, issue.1-2, pp.134-152, 1997.

X. Leroy, The compcert verified compiler. Documentation and user's manual, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01399482

J. Liu, J. Lv, Z. Quan, N. Zhan, H. Zhao et al., A calculus for hybrid csp, Asian Symposium on Programming Languages and Systems, pp.1-15, 2010.

S. M. Loos and A. Platzer, Differential refinement logic, LICS. ACM, 2016.

S. Lunel, B. Boyer, and J. Talpin, Compositional proofs in differential dynamic logic, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01615140

N. Lynch, R. Segala, and F. Vaandrager, Hybrid i/o automata. Information and computation, vol.185, pp.105-157, 2003.

J. Meseguer and R. Sharykin, Specification and analysis of distributed objectbased stochastic hybrid systems, International Workshop on Hybrid Systems: Computation and Control, pp.460-475, 2006.

S. Mitsch and A. Platzer, The keymaera x proof ide-concepts on usability in hybrid systems theorem proving, 2017.

S. Mitsch, J. Quesel, and A. Platzer, Refactoring, refinement, and reasoning, International Symposium on Formal Methods, pp.481-496, 2014.

A. Müller, S. Mitsch, and A. Platzer, Verified traffic networks: component-based verification of cyber-physical flow systems, 2015 IEEE 18th International Conference on Intelligent Transportation Systems, pp.757-764, 2015.

A. Müller, S. Mitsch, W. Retschitzegger, W. Schwinger, and A. Platzer, A component-based approach to hybrid systems safety verification, LNCS, vol.9681, pp.441-456

. Springer, , 2016.

A. Müller, S. Mitsch, W. Retschitzegger, W. Schwinger, and A. Platzer, Change and delay contracts for hybrid system component verification, International Conference on Fundamental Approaches to Software Engineering, pp.134-151, 2017.

T. Nahhal and T. Dang, Test coverage for continuous and hybrid systems, International Conference on Computer Aided Verification, pp.449-462, 2007.

S. Corina and W. Visser, Verification of java programs using symbolic execution and invariant generation, International SPIN Workshop on Model Checking of Software, pp.164-181, 2004.

C. Lawrence and . Paulson, Isabelle: A generic theorem prover, vol.828, 1994.

Y. Peng, S. Wang, N. Zhan, and L. Zhang, Extending hybrid csp with probability and stochasticity, International Symposium on Dependable Software Engineering: Theories, Tools, and Applications, pp.87-102, 2015.

A. Platzer, Cheat sheet of rules in keymaera

A. Platzer, Differential dynamic logic for verifying parametric hybrid systems, International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, pp.216-232, 2007.

A. Platzer, A temporal dynamic logic for verifying hybrid system invariants, International Symposium on Logical Foundations of Computer Science, pp.457-471

. Springer, , 2007.

A. Platzer, Differential dynamic logic for hybrid systems, J. Autom. Reas, vol.41, issue.2, pp.143-189, 2008.

A. Platzer, Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics, 2010.

A. Platzer, Quantified differential dynamic logic for distributed hybrid systems, International Workshop on Computer Science Logic, pp.469-483, 2010.

A. Platzer, Stochastic differential dynamic logic for stochastic hybrid programs, International Conference on Automated Deduction, pp.446-460, 2011.

A. Platzer, The complete proof theory of hybrid systems, LICS, pp.541-550, 2012.

A. Platzer, Logics of dynamical systems, LICS, pp.13-24, 2012.

A. Platzer, A uniform substitution calculus for differential dynamic logic, International Conference on Automated Deduction, pp.467-481, 2015.

A. Platzer and E. M. Clarke, Computing differential invariants of hybrid systems as fixedpoints, International Conference on Computer Aided Verification, pp.176-189, 2008.

A. Platzer and J. Quesel, Keymaera: A hybrid theorem prover for hybrid systems (system description), International Joint Conference on Automated Reasoning, pp.171-178, 2008.

A. Platzer and J. Quesel, European Train Control System: A case study in formal verification, LNCS, vol.5885, pp.246-265, 2009.

A. Pnueli, The temporal logic of programs, Foundations of Computer Science, 1977., 18th Annual Symposium on, pp.46-57, 1977.

R. Vaughan and . Pratt, Dynamic logic, Studies in Logic and the Foundations of Mathematics, vol.104, pp.251-261, 1982.

J. Quesel and A. Platzer, Playing hybrid games with keymaera, International Joint Conference on Automated Reasoning, pp.439-453, 2012.

A. P. Ravn, H. Rischel, and K. M. Hansen, Specifying and verifying requirements of real-time systems, IEEE Transactions on Software Engineering, vol.19, issue.1, pp.41-55, 1993.

D. Ricketts, G. Malecha, M. M. Alvarez, V. Gowda, and S. Lerner, Towards verification of hybrid systems in a foundational proof assistant, Formal Methods and Models for Codesign (MEMOCODE), 2015 ACM/IEEE International Conference on, pp.248-257, 2015.

D. Ricketts, G. Malecha, and S. Lerner, Modular deductive verification of sampled-data systems, Proceedings of the 13th International Conference on Embedded Software, p.17, 2016.

M. Rönkkö and X. Li, Linear hybrid action systems, Nordic Journal of Computing, vol.8, issue.1, pp.159-177, 2001.

M. Rönkkö and . Anders-p-ravn, Action systems with continuous behaviour, International Hybrid Systems Workshop, pp.304-323, 1997.

M. Rönkkö, K. Anders-p-ravn, and . Sere, Hybrid action systems, Theoretical Computer Science, vol.290, issue.1, pp.937-973, 2003.

M. Rönkkö and K. Sere, Refinement and continuous behaviour, International Workshop on Hybrid Systems: Computation and Control, pp.223-237, 1999.

. Ivan-ruchkin, S. Dionisio-de-niz, D. Chaki, and . Garlan, Contract-based integration of cyber-physical analyses, Embedded Software (EMSOFT), 2014 International Conference on, pp.1-10, 2014.

I. Ruchkin, B. Schmerl, and D. Garlan, Architectural abstractions for hybrid programs, Proceedings of the 18th International ACM SIGSOFT Symposium on Component-Based Software Engineering, pp.65-74, 2015.

A. Sangiovanni-vincentelli, W. Damm, and R. Passerone, Taming dr. frankenstein: Contract-based design for cyber-physical systems, European journal of control, vol.18, issue.3, pp.217-238, 2012.

S. Sankaranarayanan, B. Henny, Z. Sipma, and . Manna, Constructing invariants for hybrid systems, International Workshop on Hybrid Systems: Computation and Control, pp.539-554, 2004.

J. Sproston, Decidable model checking of probabilistic hybrid automata, International Symposium on Formal Techniques in Real-Time and Fault-Tolerant Systems, pp.31-45, 2000.

A. Tarski, A decision method for elementary algebra and geometry, Quantifier elimination and cylindrical algebraic decomposition, pp.24-84, 1951.

S. Wang, N. Zhan, and D. Guelev, An assume/guarantee based compositional calculus for hybrid csp, International Conference on Theory and Applications of Models of Computation, pp.72-83, 2012.

S. Wang, N. Zhan, and L. Zhang, A compositional modelling and verification framework for stochastic hybrid systems, Formal Aspects of Computing, vol.29, issue.4, pp.751-775, 2017.

S. Wang, N. Zhan, and L. Zou, An improved hhl prover: an interactive theorem prover for hybrid systems, International Conference on Formal Engineering Methods, pp.382-399, 2015.

S. Wolfram, The mathematica, 1999.

G. Yan, L. Jiao, Y. Li, S. Wang, and N. Zhan, Approximate bisimulation and discretization of hybrid csp, International Symposium on Formal Methods, pp.702-720, 2016.

G. Yan, L. Jiao, S. Wang, and N. Zhan, Synthesizing systemc code from delay hybrid csp, Asian Symposium on Programming Languages and Systems, pp.21-41, 2017.

L. Zhang, Z. She, S. Ratschan, H. Hermanns, and E. Hahn, Safety verification for probabilistic hybrid systems, International Conference on Computer Aided Verification, pp.196-211, 2010.

H. Zhao, M. Yang, N. Zhan, B. Gu, L. Zou et al., Formal verification of a descent guidance control program of a lunar lander, International Symposium on Formal Methods, pp.733-748, 2014.

L. Zou, N. Zhan, S. Wang, and M. Fränzle, Formal verification of simulink/stateflow diagrams, International Symposium on Automated Technology for Verification and Analysis, pp.464-481, 2015.

L. Zou, N. Zhan, S. Wang, M. Fränzle, and S. Qin, Verifying simulink diagrams via a hybrid hoare logic prover, Proceedings of the Eleventh ACM International Conference on Embedded Software, 2013.

P. Zuliani, A. Platzer, and E. M. Clarke, Bayesian statistical model checking with application to simulink/stateflow verification, Proceedings of the 13th ACM international conference on Hybrid systems: computation and control, pp.243-252, 2010.

, /* Guarantees */ ((wl1 >= 3 & 7 >= wl1) /* the water-level stay in the desired range */ & wl1=(fin-fout1)*(t-tctrl1)+wlm1)

, Assumptions from the second water-level controller */ /* value of fout2 according to the measured water-level *

, /* Guarantees */ ((wl2 >= 3 & 7 >= wl2) /* the water-level stay in the desired range */ & wl2=(fout1-fout2)*(t-tctrl2)+wlm2)

, /* Initial value */ & wlm1 >= 3 & 7 >= wlm1 /* initial assumption on the measured water-level */ /* initial assumption on the relation between the measured water-level and inlet flow *

. }*-],

, /* Initial values */ & wlm2 >= 3 & 7 >= wlm2 /* initial assumption on the measured water-level */ /* initial assumption on the relation between the measured water-level and inlet flow

. }*-],

, /* assumption that the measured water-level is in the range, vol.3

, / */ & wl1=(fin-fout1)*(t-tctrl1)+wlm1 -> [ /* Behavior, vol.3

, } /* behavior of the water-level */ ++ {wl1' = fin -fout1

, /* behavior of the second water-level controller */ {?tctrl2 + delta2 >= t, vol.2, p.2

, } /* behavior of the second water-level */ ++ {wl2' = fout1 -fout2

, /* assumption that the measured water-level is in the range, vol.3

, /* the first water-level is in the range, vol.3

, } /* behavior of the second water-level controller */ ++ {?tctrl2 + delta2 >= t, vol.2, p.2

, } /* parallel composition of two water-levels */ ++ {wl1' = fin -fout1, wl2' = fout1 -fout2

. End,

, 5->fin=0)&(3.5>=wlm1->fin=1)&(wlm1>3.5&6.5>wlm1->fin=0|fin= master, closeId, andL(-1) ; andL(-5) ; andL(-4) ; andL(-5) ; andL(-6)

. <(-boxand,

. <(-boxand,

, <( master, boxAnd(1) ; andR(1)