Intuitionistic fixed point logic

The logical system IFP introduced in this paper supports program extraction from proofs, unifying theoretical and practical advantages: Based on first-order logic and powerful strictly positive inductive and coinductive definitions, IFP support abstract axiomatic mathematics with a large amount of classical logic. The Haskell-like target programming language has a denotational and an operational semantics which are linked through a computational adequacy theorem that extends to infinite data. Program extraction is fully verified and highly optimised, thus extracted programs are guaranteed to be correct and free of junk. A case study in exact real number computation underpins IFP's effectiveness

Intuitionistic Fixed Point Logic

We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be sound with respect to a domain-theoretic denotational semantics and a corresponding lazy operational semantics of a functional language for extracted programs. We also show how extracted programs can be translated into Haskell. As an application we extract a program converting the signed digit representation of real numbers to infinite Gray-code from a proof of inclusion of the corresponding coinductive predicates.Comment: 65 page

Monotone Inductive and Coinductive Constructors of Rank 2

A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and nal coalgebras in a functor category and the Curry-Howard-correspondence. The denition of the system as a -calculus requires an appropriate denition of monotonicity to overcome subtle problems, most notably to ensure that the (co-)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes.

Monotone Inductive and Coinductive Constructors of Rank 2

Abstract. A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and final coalgebras in a functor category and the Curry-Howard-correspondence. The definition of the system as a *-calculus requires an appropriate definition of monotonicity to overcome subtle problems, most notably to ensure that the (co-)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes