Classical Logic with Mendler Induction - A Dual Calculus and Its Strong Normalization

We investigate (co-)induction in Classical Logic under the propositions-as-types paradigm, considering propositional, second-order, and (co-)inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-)iterator that remains strongly normalizing under head reduction. We prove this using a non-constructive realizability argument.

[1]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[2]  Philip Wadler,et al.  Call-by-value is dual to call-by-name , 2003, ACM SIGPLAN International Conference on Functional Programming.

[3]  Stefano Berardi,et al.  A Symmetric Lambda Calculus for Classical Program Extraction , 1994, Inf. Comput..

[4]  Daniel J. Dougherty,et al.  Strong Normalization of the Dual Classical Sequent Calculus , 2005, LPAR.

[5]  Anuj Dawar,et al.  Fixed point logics , 2002, Bull. Symb. Log..

[6]  Daisuke Kimura,et al.  Dual Calculus with Inductive and Coinductive Types , 2009, RTA.

[7]  Stefano Berardi,et al.  "Classical" Programming-with-Proofs in lambdaPASym: An Analysis of Non-confluence , 1997, TACS.

[8]  Tristan Crolard.  A Formulae-as-Types Interpretation of Subtractive Logic , 2004, J. Log. Comput..

[9]  Chung-Kil Hur,et al.  The power of parameterization in coinductive proof , 2013, POPL.

[10]  Ralph Matthes,et al.  Iteration and coiteration schemes for higher-order and nested datatypes , 2005, Theor. Comput. Sci..

[11]  Tim Sheard,et al.  A hierarchy of mendler style recursion combinators: taming inductive datatypes with negative occurrences , 2011, ICFP '11.

[12]  Gerhard Gentzen,et al.  Investigations into Logical Deduction , 1970 .

[13]  HerbelinHugo,et al.  The duality of computation , 2000 .

[14]  Michel Parigot.  Strong Normalization of Second Order Symmetric lambda-Calculus , 2000, FSTTCS.

[15]  Tarmo Uustalu,et al.  Mendler-Style Inductive Types, Categorically , 1999, Nord. J. Comput..

[16]  WadlerPhilip.  Call-by-value is dual to call-by-name , 2003 .

[17]  Ralph Matthes,et al.  Extensions of system F by iteration and primitive recursion on monotone inductive types , 1998 .

[18]  Nikos Tzevelekos.  Investigations on the Dual Calculus , 2006, Theor. Comput. Sci..

[19]  N. P. Mendler,et al.  Inductive Types and Type Constraints in the Second-Order lambda Calculus , 1991, Ann. Pure Appl. Log..

[20]  Hugo Herbelin,et al.  The duality of computation , 2000, ICFP '00.