[. Bibliography, E. Althaus, C. Kruglov, and . Weidenbach, Superposition Modulo Linear Arithmetic SUP(LA), Frontiers of Combining Systems (FroCoS)

G. Armand, B. Faure, C. Grégoire, L. Keller, B. Théry et al., A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses
DOI : 10.1145/1217856.1217859

URL : https://hal.archives-ouvertes.fr/hal-00639130

F. Baader and T. Nipkow, Term Rewriting and All That, 1998.

F. Baader and W. Snyder, Unification Theory In: Handbook of Automated Reasoning, pp.445-532, 2001.

L. Bachmair and H. Ganzinger, Rewrite-based Equational Theorem Proving with Selection and Simplification, Journal of Logic and Computation, vol.4, issue.3, pp.217-247, 1994.
DOI : 10.1093/logcom/4.3.217

[. Bachmair, H. Ganzinger, and U. Waldmann, Refutational theorem proving for hierarchic first-order theories, Applicable Algebra in Engineering, Communication and Computing, vol.1, issue.3-4, pp.193-212, 1994.
DOI : 10.1007/BF01190829

[. Bachmair, A. Tiwari, and L. English, Abstract Congruence Closure, Journal of Automated Reasoning, vol.31, issue.2, pp.129-168, 2003.
DOI : 10.1023/B:JARS.0000009518.26415.49

URL : https://hal.archives-ouvertes.fr/inria-00099511

P. Backeman and P. Rümmer, Efficient Algorithms for Bounded Rigid E-unification, Hans de Nivelle. Lecture Notes in Computer Science, vol.27, issue.1, pp.70-85, 2015.
DOI : 10.1007/10721959_17

P. Backeman and P. Rümmer, Theorem Proving with Bounded Rigid E-Unification, Proc. Conference on Automated Deduction (CADE)
DOI : 10.1007/978-3-319-21401-6_39

[. Bansal, A. Reynolds, T. King, C. Barrett, and T. Wies, Deciding Local Theory Extensions via E-matching, English. In: Computer Aided Verification (CAV). Ed. by Daniel Kroening and Corina S. P?s?reanu
DOI : 10.1007/978-3-319-21668-3_6

URL : http://arxiv.org/abs/1508.06827

H. Barbosa, Efficient Instantiation Techniques in SMT (Work In Progress)
URL : https://hal.archives-ouvertes.fr/hal-01388976

H. Barbosa, J. C. Blanchette, and P. Fontaine, Scalable Fine-Grained Proofs for Formula??Processing, Proc. Conference on Automated Deduction (CADE). Ed. by Leonardo de Moura, pp.398-412, 2017.
DOI : 10.1007/978-3-642-02959-2_10

URL : https://hal.archives-ouvertes.fr/hal-01526841

P. Barbosa, A. Fontaine, and . Reynolds, Congruence Closure with Free Variables, Lecture Notes in Computer Science, vol.205, issue.4, pp.214-230, 2017.
DOI : 10.1007/10721959_17

URL : https://hal.archives-ouvertes.fr/hal-01442691

H. Barendregt and F. Wiedijk, The challenge of computer mathematics, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.42, issue.1835, pp.1835-2351, 2005.
DOI : 10.1098/rsta.2005.1650

[. Barrett, P. Fontaine, and C. Tinelli, The SMT-LIB Standard: Version 2.5, 2015.
DOI : 10.1007/978-3-642-19583-9_2

C. Barrett, R. Sebastiani, S. Seshia, and C. Tinelli, Satisfiability Modulo Theories In: Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol.185, pp.26-825, 2009.

[. Baumgartner, J. Bax, and U. Waldmann, Beagle ??? A Hierarchic Superposition Theorem Prover, Proc. Conference on Automated Deduction (CADE). Ed. by Amy Felty and Aart Middeldorp, 2015.
DOI : 10.1007/978-3-319-21401-6_25

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.698.6807

P. Baumgartner and U. English, Hierarchic Superposition with Weak Abstraction, Proc. Conference on Automated Deduction (CADE). Ed. by MariaPaola Bonacina, pp.39-57, 2013.
DOI : 10.1007/978-3-642-38574-2_3

URL : https://hal.archives-ouvertes.fr/hal-00931919

B. Beckert, Ridig E-Unification Automated Deduction: A Basis for Applications. Foundations: Calculi and Methods

S. Berghofer and T. Nipkow, Proof Terms for Simply Typed Higher Order Logic, Lecture Notes in Computer Science, vol.1869, pp.38-52, 2000.
DOI : 10.1007/3-540-44659-1_3

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.8649

F. Besson, P. Fontaine, and L. Théry, A Flexible Proof Format for SMT: a Proposal, Workshop on Proof eXchange for Theorem Proving (PxTP)
URL : https://hal.archives-ouvertes.fr/hal-00642544

[. Blanchette, S. Böhme, M. Fleury, S. J. Smolka, and A. Steckermeier, Semi-intelligible Isar Proofs from Machine-Generated Proofs, Journal of Automated Reasoning, vol.27, issue.4, pp.155-200, 2016.
DOI : 10.1007/978-3-642-37036-6_12

URL : https://hal.archives-ouvertes.fr/hal-01211748

[. Böhme, C. J. Anthony, T. Fox, T. Sewell, and . Weber, Reconstruction of Z3???s Bit-Vector Proofs in HOL4 and Isabelle/HOL, Certified Programs and Proofs, pp.183-198, 2011.
DOI : 10.1016/j.jal.2007.07.003

S. Böhme and T. Weber, Fast LCF-Style Proof Reconstruction for Z3
DOI : 10.1007/978-3-642-14052-5_14

T. Bouton, D. Caminha, B. De-oliveira, D. Déharbe, and P. Fontaine, veriT: An Open, Trustable and Efficient SMT-Solver, Proc. Conference on Automated Deduction (CADE), pp.151-156, 2009.
DOI : 10.1007/978-3-540-73595-3_38

URL : https://hal.archives-ouvertes.fr/inria-00430634

G. Burel, A Shallow Embedding of Resolution and Superposition Proofs into the ??-Calculus Modulo " . In: Workshop on Proof eXchange for Theorem Proving (PxTP), EPiC Series in Computing. EasyChair, pp.43-57, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01126321

[. Cousineau and G. Dowek, Embedding Pure Type Systems in the

P. Modulo, In: Typed Lambda Calculi and Applications (TLCA), Lecture Notes in Computer Science, vol.4583

M. Davis, G. Logemann, and D. Loveland, A Machine Program for Theorem-proving, Commun. ACM, vol.57, pp.394-397, 1962.
DOI : 10.1145/368273.368557

L. Mendonça-de-moura and N. Bjørner, Artificial Intelligence, and Reasoning (LPAR) Workshops, Proofs and Refutations, and Z3 " . In: Logic for Programming CEUR Workshop Proceedings. CEUR-WS.org, 2008.

L. Mendonça-de-moura and N. Bjørner, Z3: An Efficient SMT Solver

L. Mendonça-de-moura and D. Jovanovic, A Model-Constructing Satisfiability Calculus, Verification, Model Checking, and Abstract Interpretation (VMCAI). 2013, pp.1-12

L. De, M. , and N. Bjørner, Efficient E-Matching for SMT Solvers

L. De, M. , and N. Bjørner, Engineering DPLL(T) + Saturation, International Joint Conference on Automated Reasoning (IJCAR), pp.475-490, 2008.

N. Hans-de, Extraction of Proofs from the Clausal Normal Form Transformation Computer Science Logic (CSL), pp.584-598, 2002.

N. Hans-de, Translation of resolution proofs into short first-order proofs without choice axioms, Information and Computation, vol.1991, pp.24-54, 2005.

A. Degtyarev and A. Voronkov, The undecidability of simultaneous rigid E-unification, Theoretical Computer Science, vol.166, issue.1-2, pp.1-2, 1996.
DOI : 10.1016/0304-3975(96)00092-8

A. Degtyarev and A. Voronkov, What you always wanted to know about rigid E-unification, Journal of Automated Reasoning, vol.201, pp.47-80, 1998.
DOI : 10.1007/3-540-61630-6_4

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.11.4975

A. Degtyarev and A. Voronkov, Equality Reasoning in Sequent-Based Calculi
DOI : 10.1016/B978-044450813-3/50012-6

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.11.5202

D. Déharbe, P. Fontaine, D. L. Berre, and B. Mazure, Computing prime implicants, 2013 Formal Methods in Computer-Aided Design
DOI : 10.1109/FMCAD.2013.6679390

D. Déharbe, P. Fontaine, S. Merz, and B. W. Paleo, Exploiting Symmetry in SMT Problems, Proceedings of the 23rd International Conference on Automated Deduction. Proc. Conference on Automated Deduction (CADE), pp.222-236, 2011.
DOI : 10.1007/s00224-004-1192-0

D. Déharbe, P. Fontaine, and B. W. Paleo, Quantifier Inference Rules for SMT proofs, Workshop on Proof eXchange for Theorem Proving, 2011.

[. Detlefs, G. Nelson, and J. B. Saxe, Simplify: a theorem prover for program checking, Journal of the ACM, vol.52, issue.3, pp.365-473, 2005.
DOI : 10.1145/1066100.1066102

URL : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.1745&rep=rep1&type=pdf

C. Dross, J. Sylvain-conchon, A. Kanig, and . Paskevich, Adding Decision Procedures to SMT Solvers Using Axioms with Triggers, Journal of Automated Reasoning, vol.53, issue.6, 2013.
DOI : 10.1007/978-3-642-18275-4_28

URL : https://hal.archives-ouvertes.fr/hal-00915931

B. Dutertre, Yices??2.2, English. In: Computer Aided Verification (CAV). Ed. by Armin Biere and Roderick Bloem
DOI : 10.1007/978-3-319-08867-9_49

B. Dutertre and L. De-moura, A Fast Linear-Arithmetic Solver for DPLL(T), Lecture Notes in Computer Science, vol.4144, pp.81-94, 2006.
DOI : 10.1007/11817963_11

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.705

S. Zivota, System Description: GAPT 2.0, International Joint Conference on Automated Reasoning (IJCAR)

A. Eggers, E. Kruglov, S. Kupferschmid, K. Scheibler, T. Teige et al., Superposition Modulo Non-linear Arithmetic
DOI : 10.1007/978-3-642-02959-2_10

B. Ekici, G. Katz, C. Keller, A. Mebsout, A. Reynolds et al., Extending SMTCoq, a Certified Checker for SMT (Extended Abstract), Electronic Proceedings in Theoretical Computer Science, vol.210, pp.21-29, 2016.
DOI : 10.4204/EPTCS.210.5

URL : https://hal.archives-ouvertes.fr/hal-01388984

M. Fitting, First-Order Logic and Automated Theorem Proving, 1996.
DOI : 10.1007/978-1-4612-2360-3

P. Fontaine, J. Marion, S. Merz, L. P. Nieto, and A. Tiu, Expressiveness + Automation + Soundness: Towards Combining SMT Solvers and Interactive Proof Assistants, Tools and Algorithms for Construction and Analysis of Systems (TACAS). Ed. by Holger Hermanns and Jens Palsberg, pp.167-181, 2006.
DOI : 10.1007/3-540-45620-1_26

URL : https://hal.archives-ouvertes.fr/inria-00001088

Y. Ge, C. Barrett, and C. English, Solving Quantified Verification Conditions Using Satisfiability Modulo Theories, Proc. Conference on Automated Deduction (CADE). Ed. by Frank Pfenning, pp.167-182, 2007.
DOI : 10.1007/978-3-540-73595-3_12

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.6628

[. Ge and L. De-moura, Complete Instantiation for Quantified Formulas in Satisfiabiliby Modulo Theories, Computer Aided Verification (CAV)
DOI : 10.1007/978-3-540-78800-3_19

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.150.833

M. Giese, Incremental Closure of Free Variable Tableaux, English. In: International Joint Conference on Automated Reasoning (IJCAR), pp.545-560, 2001.
DOI : 10.1007/3-540-45744-5_46

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.126.544

M. Giese and E. , A Model Generation Style Completeness Proof for Constraint Tableaux with Superposition
DOI : 10.1007/3-540-45616-3_10

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.16.3781

T. [. Gordon and . Melham, Introduction to HOL: A Theorem Proving Environment for Higher Order Logic, 1993.

J. C. Michael, R. Gordon, C. P. Milner, and . Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation, LNCS, vol.78, 1979.

J. Goubault, A rule-based algorithm for rigid E-unification, pp.202-210, 1993.
DOI : 10.1007/BFb0022569

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.5053

[. Graham-lengrand, Psyche: A Proof-Search Engine Based on Sequent Calculus with an LCF-Style Architecture " . In: Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX)
DOI : 10.1007/978-3-642-40537-2_14

URL : https://hal.archives-ouvertes.fr/hal-00906789

L. Hadarean, C. W. Barrett, A. Reynolds, C. Tinelli, and M. Deters, Fine Grained SMT Proofs for the Theory of??Fixed-Width Bit-Vectors
DOI : 10.1007/978-3-662-48899-7_24

R. Hähnle, Tableaux and Related Methods In: Handbook of Automated Reasoning, pp.1853-1964, 2001.

[. Harper, F. Honsell, and G. D. Plotkin, A framework for defining logics, Journal of the ACM, vol.40, issue.1, pp.194-204, 1987.
DOI : 10.1145/138027.138060

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.5854

S. Hetzl, T. Libal, M. Riener, and M. Rukhaia, Understanding Resolution Proofs through Herbrand???s Theorem, Didier Galmiche and Dominique Larchey-Wendling
DOI : 10.1007/978-3-642-40537-2_15

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.640.7106

[. Jovanovi? and L. De-moura, Solving Non-linear Arithmetic, English . In: International Joint Conference on Automated Reasoning (IJCAR), pp.339-354, 2012.
DOI : 10.1007/978-3-642-31365-3_27

[. Katz, C. W. Barrett, C. Tinelli, A. Reynolds, and L. Hadarean, Lazy proofs for DPLL(T)-based SMT solvers, 2016 Formal Methods in Computer-Aided Design (FMCAD), pp.93-100, 2016.
DOI : 10.1109/FMCAD.2016.7886666

L. Kovács and A. Voronkov, First-Order Theorem Proving and Vampire
DOI : 10.1007/978-3-642-39799-8_1

. English, Computer Aided Verification (CAV) Ed. by Natasha Sharygina and Helmut Veith, Lecture Notes in Computer Science, vol.8044, pp.1-35, 2013.

M. [. Rustan, M. Leino, X. Musuvathi, and . Ou, A Two-Tier Technique for Supporting Quantifiers in a Lazily Proof-Explicating Theorem Prover " . In: Tools and Algorithms for Construction and Analysis of Systems (TACAS), Lecture Notes in Computer Science, vol.3440

M. [. Rustan, C. Leino, and . Pit, Trigger Selection Strategies to Stabilize Program Verifiers, Computer Aided Verification (CAV). Ed. by Swarat Chaudhuri and Azadeh Farzan

[. Mclaughlin, C. Barrett, and Y. Ge, Cooperating Theorem Provers: A Case Study Combining HOL-Light and CVC Lite, Electronic Notes in Theoretical Computer Science, vol.144, issue.2
DOI : 10.1016/j.entcs.2005.12.005

A. Meier, System Description: Tramp: Transformation of Machine-Found Proofs into Natural Deduction Proofs at the Assertion Level, Proc. Conference on Automated Deduction (CADE). Ed. by David McAllester, pp.460-464, 2000.
DOI : 10.1007/10721959_37

M. Moskal, Rocket-Fast Proof Checking for SMT Solvers Tools and Algorithms for Construction and Analysis of Systems (TACAS), pp.486-500, 2008.

M. Moskal, J. ?opusza?ski-moskal, J. ?opusza?ski, and J. R. Kiniry, E-matching for Fun and Profit, Electronic Notes in Theoretical Computer Science, vol.198, issue.2, pp.19-35, 2006.
DOI : 10.1016/j.entcs.2008.04.078

URL : http://doi.org/10.1016/j.entcs.2008.04.078

G. Nelson and D. C. Oppen, Simplification by Cooperating Decision Procedures, ACM Transactions on Programming Languages and Systems, vol.1, issue.2, pp.245-257, 1979.
DOI : 10.1145/357073.357079

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.104.2828

G. Nelson and D. C. Oppen, Fast Decision Procedures Based on Congruence Closure, Journal of the ACM, vol.27, issue.2, pp.356-364, 1980.
DOI : 10.1145/322186.322198

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.125.3226

R. Nieuwenhuis and A. Oliveras, Fast congruence closure and extensions, Information and Computation, vol.205, issue.4
DOI : 10.1016/j.ic.2006.08.009

URL : http://doi.org/10.1016/j.ic.2006.08.009

[. Nieuwenhuis, A. Oliveras, and C. Tinelli, Solving SAT and SAT Modulo Theories, Journal of the ACM, vol.53, issue.6, pp.937-977, 2006.
DOI : 10.1145/1217856.1217859

R. Nieuwenhuis and A. Rubio, Paramodulation-Based Theorem Proving
DOI : 10.1016/B978-044450813-3/50009-6

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.6344

[. Nipkow, L. C. Paulson, and M. Wenzel, Isabelle/HOL: A Proof Assistant for Higher-Order Logic, LNCS, vol.2283, 2002.
DOI : 10.1007/3-540-45949-9

A. Nonnengart, C. Weidenbach, and S. , Computing Small Clause Normal Forms, pp.335-367, 2001.
DOI : 10.1016/B978-044450813-3/50008-4

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.222

C. Lawrence and . Paulson, A Higher-Order Implementation of Rewriting

C. Lawrence, K. Paulson, and . Susanto, Source-Level Proof Reconstruction for Interactive Theorem Proving In: Theorem Proving in Higher Order Logics (TPHOLs), pp.232-245, 2007.

[. Piskac, T. Wies, and D. Zufferey, GRASShopper, Lecture Notes in Computer Science, vol.8413, pp.124-139, 2014.
DOI : 10.1007/978-3-642-54862-8_9

W. Pugh, The Omega test: a fast and practical integer programming algorithm for dependence analysis, Proceedings of the 1991 ACM/IEEE conference on Supercomputing , Supercomputing '91, pp.4-13, 1992.
DOI : 10.1145/125826.125848

[. Reger, N. Bjorner, M. Suda, and A. Voronkov, AVATAR Modulo Theories, Global Conference on Artificial Intelligence (GCAI) EPiC Series in Computing. EasyChair, pp.39-52, 2016.

A. Reynolds, Conflicts, Models and Heuristics for Quantifier Instantiation in SMT, EPiC Series in Computing. EasyChair, pp.1-15, 2016.
DOI : 10.1007/978-3-319-21668-3_12

A. Reynolds, C. Tinelli, and L. Mendonça-de-moura, Finding conflicting instances of quantified formulas in SMT, 2014 Formal Methods in Computer-Aided Design (FMCAD), pp.195-202, 2014.
DOI : 10.1109/FMCAD.2014.6987613

A. Reynolds, C. Tinelli, A. Goel, and S. English, Finite Model Finding in SMT, Computer Aided Verification (CAV). Ed. by Natasha Sharygina and Helmut Veith, pp.640-655, 2013.
DOI : 10.1007/978-3-642-39799-8_42

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.676.9522

C. Barrett, Quantifier Instantiation Techniques for Finite Model Finding in SMT, Proc. Conference on Automated Deduction (CADE). Ed. by Maria Paola Bonacina, pp.377-391, 2013.

P. Rümmer, A Constraint Sequent Calculus for First-Order Logic with Linear Integer Arithmetic, Logic for Programming, Artificial Intelligence, and Reasoning (LPAR). Ed. by Iliano Cervesato, Helmut Veith, and Andrei Voronkov, pp.274-289, 2008.
DOI : 10.1007/978-3-540-89439-1_20

P. Rümmer, E-Matching with Free Variables, Logic for Programming, Artificial Intelligence, and Reasoning (LPAR). Ed. by Nikolaj Bjørner and Andrei Voronkov, pp.359-374, 2012.
DOI : 10.1007/978-3-642-28717-6_28

S. Schulz and E. , System Description: E 1 In: Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), Lecture Notes in Computer Science, vol.8312

R. Sebastiani, Lazy Satisability Modulo Theories, pp.3-4, 2007.

R. E. Shostak, Deciding Combinations of Theories, Journal of the ACM, vol.31, issue.1, pp.1-12, 1984.
DOI : 10.1145/2422.322411

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.460.6173

J. Slaney and B. W. Paleo, Conflict Resolution: A First-Order Resolution Calculus with Decision Literals and Conflict-Driven Clause Learning, Journal of Automated Reasoning, vol.43, issue.4
DOI : 10.1016/B978-044450813-3/50029-1

URL : http://arxiv.org/abs/1602.04568

A. Stump, Proof Checking Technology for Satisfiability Modulo Theories, Electronic Notes in Theoretical Computer Science, vol.228, pp.121-133, 2009.
DOI : 10.1016/j.entcs.2008.12.121

URL : http://doi.org/10.1016/j.entcs.2008.12.121

[. Sutcliffe, J. Zimmer, and S. Schulz, TSTP Data-Exchange Formats for Automated Theorem Proving Tools In: Distributed Constraint Problem Solving and Reasoning in Multi-Agent Systems, Frontiers in Artificial Intelligence and Applications, vol.112, pp.201-215, 2004.

A. Tiwari, L. Bachmair, and H. Ruess, Rigid E-Unification Revisited
DOI : 10.1007/10721959_17

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.7046

G. S. Tseitin, On the Complexity of Derivation in Propositional Calculus, Automation of Reasoning: 2: Classical Papers on Computational Logic, 1967.

A. Voronkov, AVATAR: The Architecture for First-Order Theorem Provers
DOI : 10.1007/978-3-319-08867-9_46

[. Weidenbach, D. Dimova, A. Fietzke, R. Kumar, M. Suda et al., SPASS Version 3.5, Proc. Conference on Automated Deduction (CADE), pp.140-145, 2009.
DOI : 10.1007/978-3-540-73595-3_38

[. Zeljic, C. M. Wintersteiger, and P. Rümmer, Deciding Bit-Vector Formulas with mcSAT, Theory and Applications of Satisfiability Testing (SAT), pp.249-266, 2016.
DOI : 10.1007/978-3-642-38088-4_6

J. Zimmer, A. Meier, G. Sutcliffe, and Y. Zhan, Integrated Proof Transformation Services, 2004.