Extracting ω's programs from proofs in the calculus of constructions

We define in this paper a notion of realizability for the Calculus of Constructions. The extracted programs are terms of the Calculus that do not contain dependent types. We introduce a distinction between informative and non-informative propositions. This distinction allows the removal of the “logical” part in the development of a program. We show also how to use our notion of realizability in order to interpret various axioms like the axiom of choice or the induction on integers. A practical example of development of program is given in the appendix.

[1]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[2]  A. Troelstra.  Metamathematical investigation of intuitionistic arithmetic and analysis , 1973 .

[3]  J. Leblanc.  THÈSE DE 3ÈME CYCLE , 1978 .

[4]  Bengt Nordström,et al.  Types and Specifications , 1983, IFIP Congress.

[5]  Thierry Coquand,et al.  Constructions: A Higher Order Proof System for Mechanizing Mathematics , 1985, European Conference on Computer Algebra.

[6]  Thierry Coquand,et al.  Concepts mathématiques et informatiques formalisés dans le calcul des constructions , 1985, Logic Colloquium.

[7]  Peter Dybjer,et al.  Program Verification in a Logical Theory of Constructions , 1985, FPCA.

[8]  M. Beeson.  Foundations of Constructive Mathematics: Metamathematical Studies , 1985 .

[9]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[10]  Christine Mohring,et al.  Algorithm Development in the Calculus of Constructions , 1986, Logic in Computer Science.

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

[12]  中野 裕,et al.  PX, a computational logic , 1988 .

[13]  Michel Parigot,et al.  Programming with Proofs: A Second Order Type Theory , 1988, ESOP.

[14]  T. Coquand,et al.  Metamathematical investigations of a calculus of constructions , 1989 .

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

[16]  Michel Parigot,et al.  Programming with Proofs , 1990, J. Inf. Process. Cybern..