Linear Quantifier Elimination

This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection).

[1]  Amine Chaieb,et al.  Verifying Mixed Real-Integer Quantifier Elimination , 2006, IJCAR.

[2]  Tobias Nipkow,et al.  Verifying and Reflecting Quantifier Elimination for Presburger Arithmetic , 2005, LPAR.

[3]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[4]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[5]  C. H. Langford.  Some Theorems on Deducibility , 1926 .

[6]  John Harrison,et al.  A Proof-Producing Decision Procedure for Real Arithmetic , 2005, CADE.

[7]  Markus Wenzel,et al.  Constructive Type Classes in Isabelle , 2006, TYPES.

[8]  Robert S. Boyer,et al.  Metafunctions: Proving Them Correct and Using Them Efficiently as New Proof Procedures. , 1979 .

[9]  Steven Obua,et al.  Proving Bounds for Real Linear Programs in Isabelle/HOL , 2005, TPHOLs.

[10]  T. Nipkow,et al.  Reflecting Quantifier Elimination for Linear Arithmetic , 2008 .

[11]  Jeanne Ferrante,et al.  A Decision Procedure for the First Order Theory of Real Addition with Order , 1975, SIAM J. Comput..

[12]  Assia Mahboubi.  Contributions à la certification des calculs dans R : théorie, preuves, programmation. (Contributions to the certification of computations in R : theory, proofs, implementation) , 2006 .

[13]  Clemens Ballarin,et al.  Interpretation of Locales in Isabelle: Theories and Proof Contexts , 2006, MKM.

[14]  John Harrison,et al.  Handbook of Practical Logic and Automated Reasoning , 2009 .

[15]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

[16]  J. Harrison.  Introduction to Logic and Automated Theorem Proving , 2007 .

[17]  John Harrison.  Complex quantifier elimination in HOL , 2001 .

[18]  Rüdiger Loos,et al.  Applying Linear Quantifier Elimination , 1993, Comput. J..

[19]  Georges Gonthier.  A computer-checked proof of the Four Colour Theorem , 2005 .

[20]  J. Farkas.  Theorie der einfachen Ungleichungen. , 1902 .

[21]  Th. Motzkin.  Beiträge zur Theorie der linearen Ungleichungen , 1936 .

[22]  Michael Norrish.  Complete Integer Decision Procedures as Derived Rules in HOL , 2003, TPHOLs.

[23]  David Monniaux.  A Quantifier Elimination Algorithm for Linear Real Arithmetic , 2008, LPAR.

[24]  Tobias Nipkow,et al.  A Proof Assistant for Higher-Order Logic , 2002 .

[25]  Tobias Nipkow,et al.  Linear Quantifier Elimination , 2008, Journal of Automated Reasoning.