Computer theorem proving in math

We give an overview of issues surrounding computerverified theorem proving in the standard pure-mathematical context.

[1]  G. Bancerek Journal of Formalized Mathematics , 1997 .

[2]  Robin Milner,et al.  LCF: A Way of Doing Proofs with a Machine , 1979, MFCS.

[3]  Leslie Lamport,et al.  Should your specification language be typed , 1999, TOPL.

[4]  Kenneth Kunen,et al.  A Ramsey theorem in Boyer-Moore logic , 1995, Journal of Automated Reasoning.

[5]  Sten Agerholm,et al.  Experiments with ZF Set Theory in HOL and Isabelle , 1995, TPHOLs.

[6]  Robert S. Boyer,et al.  Computer Proofs of Limit Theorems , 1971, IJCAI.

[7]  Luís Cruz-Filipe,et al.  A Constructive Formalization of the Fundamental Theorem of Calculus , 2002, TYPES.

[8]  S. Gutzwiller,et al.  Robert S , 2002 .

[9]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[10]  Herman Geuvers,et al.  Proof-Assistants Using Dependent Type Systems , 2001, Handbook of Automated Reasoning.

[11]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[12]  John McCarthy,et al.  Recursive functions of symbolic expressions and their computation by machine, Part I , 1960, Commun. ACM.

[13]  Robert S. Boyer,et al.  Automated Reasoning: Essays in Honor of Woody Bledsoe , 1991, Automated Reasoning.

[14]  John McCarthy,et al.  A basis for a mathematical theory of computation, preliminary report , 1899, IRE-AIEE-ACM '61 (Western).

[15]  Pierre Deligne,et al.  Hodge Cycles, Motives, and Shimura Varieties , 1989 .

[16]  Alan Bundy,et al.  A Survey of Automated Deduction , 1999, Artificial Intelligence Today.

[17]  J. A. Robinson,et al.  Theorem-Proving on the Computer , 1963, JACM.

[18]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[19]  R. Veroff,et al.  A Short Sheffer axiom for Boolean algebra. , 2000 .

[20]  Thierry Coquand,et al.  An Analysis of Girard's Paradox , 1986, LICS.

[21]  Michael J. C. Gordon,et al.  From LCF to HOL: a short history , 2000, Proof, Language, and Interaction.

[22]  Gérard P. Huet,et al.  Constructive category theory , 2000, Proof, Language, and Interaction.

[23]  Laurent Igal Chicli Sur la formalisation des mathématiques dans le calcul des constructions inductives , 2003 .

[24]  Benjamin Werner,et al.  Sets in Types, Types in Sets , 1997, TACS.

[25]  S. Feferman,et al.  Typical Ambiguity: Trying to Have Your Cake and Eat it too , 2003 .

[26]  Benjamin Werner,et al.  Une Théorie des Constructions Inductives , 1994 .

[27]  Vanishing cycles, the generalized Hodge Conjecture and Gr\ , 2003, math/0311180.

[28]  Robert S. Boyer,et al.  Computational Logic , 1990, ESPRIT Basic Research Series.

[29]  J. Harrison Formalized Mathematics , 1996 .

[30]  John McCarthy,et al.  Towards a Mathematical Science of Computation , 1962, IFIP Congress.

[31]  Harvey M. Friedman,et al.  Finite functions and the necessary use of large cardinals , 1998, math/9811187.

[32]  Judicaël Courant,et al.  Explicit Universes for the Calculus of Constructions , 2002, TPHOLs.

[33]  Gérard P. Huet Unification in Typed Lambda Calculus , 1975, Lambda-Calculus and Computer Science Theory.

[34]  Freek Wiedijk,et al.  A Comparison of Mizar and Isar , 2004, Journal of Automated Reasoning.

[35]  Leovigildo Alonso Tarrío,et al.  Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes , 1999 .

[36]  G. Robert Meyerhoff,et al.  Homotopy hyperbolic 3-manifolds are hyperbolic , 1996 .

[37]  van Ls Bert Benthem Jutting,et al.  Checking Landau's “Grundlagen” in the Automath System: Appendices 3 and 4 (The PN-lines; Excerpt for “Satz 27”) , 1994 .

[38]  Clement W. H. Lam,et al.  The nonexistence of ovals in a projective plane of order 10 , 1983, Discret. Math..

[39]  David M. Russinoff A mechanical proof of Quadratic Reciprocity , 1992, Journal of Automated Reasoning.

[40]  Cuihtlauac Alvarado Réflexion pour la réécriture dans le calcul des constructions inductives , 2002 .

[41]  Branden Fitelson,et al.  XCB, the Last of the Shortest Single Axioms for the Classical Equivalential Calculus , 2002, ArXiv.

[42]  J. Paris,et al.  The Type Theoretic Interpretation of Constructive Set Theory , 1978 .

[43]  Giuseppe Rosolini,et al.  Two models of synthetic domain theory , 1997 .

[44]  Thorsten Altenkirch,et al.  Generic Programming within Dependently Typed Programming , 2002, Generic Programming.

[45]  B. Plotkin Algebraic geometry in first-order logic , 2003, math/0312485.

[46]  Zhaohui Luo An extended calculus of constructions , 1990 .

[47]  L. Thiel,et al.  The Non-Existence of Finite Projective Planes of Order 10 , 1989, Canadian Journal of Mathematics.

[48]  Loïc Pottier,et al.  Mathematical Quotients and Quotient Types in Coq , 2002, TYPES.

[49]  A. Neeman A counterexample to a 1961 “theorem” in homological algebra , 2002 .

[50]  Natarajan Shankar,et al.  A mechanical proof of the Church-Rosser theorem , 1988, JACM.