Some Axioms for Mathematics

12 The λΠ-calculus modulo theory is a logical framework in which many logical systems can be expressed 13 as theories. In this paper, we present such a theory, the theory U , where the proofs of several 14 logical systems can be expressed: Minimal, Constructive, and Ecumenical predicate logic; Minimal, 15 Constructive, and Ecumenical simple type theory; Simple type theory with predicate subtyping, 16 prenex predicative polymorphism, or both; the Calculus of constructions, and the Calculus of 17 constructions with prenex predicative polymorphism. Moreover, we identify a sub-theory of the 18 theory U corresponding to each of these systems, and prove that, when a proof in U uses only 19 symbols of a sub-theory, then it is a proof in that sub-theory. This result is a consequence of a 20 general theorem on the dependency, in the λΠ-calculus modulo theory, of typing with respect to the 21 constant declarations and rewriting rules constituting the theory. 22 2012 ACM Subject Classification Theory of computation → Logic; Theory of computation → Type 23 theory; Theory of computation → Equational logic and rewriting 24

[1]  F. Blanqui,et al.  Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory , 2021, TYPES.

[2]  Frédéric Blanqui,et al.  Type Safety of Rewrite Rules in Dependent Types , 2020, FSCD.

[3]  Gabriel Hondet,et al.  The New Rewriting Engine of Dedukti (System Description) , 2020, FSCD.

[4]  F. Thiré.  Interoperability between proof systems using the logical framework Dedukti. (Interopérabilité entre systèmes de preuve en utilisant le cadre logique Dedukti) , 2020 .

[5]  G. Genestier.  Dependently-Typed Termination and Embedding of Extensional Universe-Polymorphic Type Theory using Rewriting. (Terminaison en présence de types dépendants et encodage par réécriture d'une théorie des types extensionelle avec polymorphisme d'univers) , 2020 .

[6]  François Thiré.  Sharing a Library between Proof Assistants: Reaching out to the HOL Family , 2018, LFMTP@FSCD.

[7]  Frédéric Gilbert.  Extending higher-order logic with predicate subtyping: Application to PVS. (Extension de la logique d'ordre supérieur avec le sous-typage par prédicats) , 2018 .

[8]  Luiz Carlos Pereira,et al.  Normalization, Soundness and Completeness for the Propositional Fragment of Prawitz’ Ecumenical System , 2017 .

[9]  Gilles Dowek.  On the definition of the classical connectives and quantifiers , 2016, ArXiv.

[10]  Ali Assaf,et al.  A framework for defining computational higher-order logics. (Un cadre de définition de logiques calculatoires d'ordre supérieur) , 2015 .

[11]  D. Prawitz.  Classical versus intuitionistic logic , 2015 .

[12]  Olivier Hermant,et al.  Semantic A-translations and Super-Consistency Entail Classical Cut Elimination , 2013, LPAR.

[13]  Denis Cousineau,et al.  Embedding Pure Type Systems in the Lambda-Pi-Calculus Modulo , 2007, TLCA.

[14]  Frédéric Blanqui,et al.  Type theory and rewriting , 2006, ArXiv.

[15]  Gilles Dowek,et al.  Arithmetic as a Theory Modulo , 2005, RTA.

[16]  Claude Kirchner,et al.  Theorem Proving Modulo , 2003, Journal of Automated Reasoning.

[17]  Gilles Dowek,et al.  Proof normalization modulo , 1998, Journal of Symbolic Logic.

[18]  Enno Ohlebusch,et al.  Term Rewriting Systems , 2002 .

[19]  Shankar Natarajan,et al.  The Formal Semantics of PVS , 1999 .

[20]  Antonius J. C. Hurkens.  A Simplification of Girard's Paradox , 1995, TLCA.

[21]  J. H. Geuvers,et al.  The calculus of constructions and higher order logic , 1995 .

[22]  Lawrence C. Paulson,et al.  Isabelle: The Next 700 Theorem Provers , 2000, ArXiv.

[23]  Vincent van Oostrom,et al.  Combinatory Reduction Systems: Introduction and Survey , 1993, Theor. Comput. Sci..

[24]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

[25]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[26]  Wolfgang Thomas,et al.  Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics , 1990 .

[27]  Thierry Coquand,et al.  The Calculus of Constructions , 1988, Inf. Comput..

[28]  Gopalan Nadathur,et al.  An Overview of Lambda-PROLOG , 1988, ICLP/SLP.

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

[30]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[31]  J. Y. Girard,et al.  Interpretation fonctionelle et elimination des coupures dans l'aritmetique d'ordre superieur , 1972 .

[32]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[33]  L. M.-T..  Grundzüge der theoretischen Logik , 1929, Nature.