Strict Functionals for Termination Proofs

@inproceedings{Pol1995StrictFF,
  title={Strict Functionals for Termination Proofs},
  author={Jaco van de Pol and Helmut Schwichtenberg},
  booktitle={International Conference on Typed Lambda Calculus and Applications},
  year={1995}
}
A semantical method to prove termination of higher order rewrite systems (HRS) is presented. Its main tool is the notion of a strict functional, which is a variant of Gandy's notion of a hereditarily monotonic functional [1]. The main advantage of the method is that it makes it possible to transfer ones intuitions about why an HRS should be terminating into a proof: one has to nd a \strict" interpretation of the constants involved in such a way that the left hand side of any rewrite rule gets a… 

12 : 2 Polymorphic Higher-Order Termination

This work generalises the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism and eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types.

Polymorphic Higher-order Termination

This work generalises the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism and eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types.

Two Different Strong Normalization Proofs?

A proof of ∀t∃nSN(t, n) (term t performs at most n reduction steps) is given, based on strong computability predicates. Using modified realizability, a bound on reduction lengths is extracted from

A domain-theoretic strong normalisation theorem

plus rules for function types. Theorem (Pottinger 1980) For any λ-term M M is typable ⇔ M is strongly normalising The theorem is of limited practial use, because it only applies to pure λ-terms, and

Termination and Reduction Checking in the Logical Framework

The power of the induction component is extended to enable complete induction or so called course-of-value ind and a reduction and termination checker which reasons about orders is presented.

Size-based termination: Semantics and generalizations

A structured approach to proving the correctness of size-based termination and a modification of the classical size-types approach allows us to perform a fine control-flow analysis in a higher-order language.

Czajka and C . Kop 12 : 3 2 Preliminaries

This work generalises the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism and eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types.

The higher-order recursive path ordering

  • J. JouannaudA. Rubio
  • Mathematics
    Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)
  • 1999
This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus

Termination of rewriting in the Calculus of Constructions

The criterion is general enough to accept definitions by rewriting of many well-known higher order functions, for example dependent recursors for inductive types or proof carrying functions, and makes it a very good candidate for inclusion in a proof assistant based on the Curry-Howard isomorphism.

Proof Theory at Work: Program Development in the Minlog System

The old idea that proofs are in some sense functions, has been made precise by the Curry-Howard-correspondence between proofs in natural deduction and terms in typed λ-calculus, and is implemented in Minlog, an interactive proof system designed for generating proof terms and exploring their algorithmic content.
...

References

SHOWING 1-9 OF 9 REFERENCES

Termination Proofs for Higher-order Rewrite Systems

This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [12], which combines the computational aspects of term rewriting and simply typed lambda calculus and produces a proof technique similar to the proof technique for the termination of a well-founded monotone algebra.

Orthogonal Higher-Order Rewrite Systems are Confluent

The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more

Ideas and Results in Proof Theory

Proof Theory and Logical Complexity

Introduction: Elementary Proof Theory. The Fall of Hilbert's Program. Hilbert's Program. Recursive Functions. The First Incompleteness Theorem. The Second Incompleteness Theorem. Exercises. Annex:

Equations and rewrite rules: a survey

Proofs of strong normalization

  • Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism
  • 1980

Confluence for Abstract and Higher-Order Rewriting