• Corpus ID: 53090076

Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms

@article{Bentkamp2018FormalizationOT,
  title={Formalization of the Embedding Path Order for Lambda-Free Higher-Order Terms},
  author={Alexander Bentkamp},
  journal={FLAP},
  year={2018},
  volume={8},
  pages={2447-2470}
}
The embedding path order, introduced in this article, is a variant of the recursive path order (RPO) for untyped λ-free higher-order terms (also called applicative first-order terms). Unlike other higher-order variants of RPO, it is a groundtotal and well-founded simplification order, making it more suitable for the superposition calculus. I formally proved the order’s theoretical properties in Isabelle/HOL and evaluated the order in a prototype based on the superposition prover Zipperposition. 
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References

SHOWING 1-10 OF 57 REFERENCES
A Lambda-Free Higher-Order Recursive Path Order
TLDR
This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work and appears promising as the basis of a higher-order superposition calculus.
A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms
TLDR
The Knuth–Bendix order is generalized to higher-order terms without \(\lambda \)-abstraction and appears promising as the basis of a higher- order superposition calculus.
Superposition for Lambda-Free Higher-Order Logic
TLDR
Refutationally complete superposition calculi for intentional and extensional \(\lambda \)-free higher-order logic, two formalisms that allow partial application and applied variables, appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher- order logic.
A Higher-Order Iterative Path Ordering
TLDR
An iterative version of HORPO is presented by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and Klop, and well-foundedness of the iterative definition is studied.
The recursive path and polynomial ordering for first-order and higher-order terms
TLDR
A simple ordering is presented that combines both RPO and POLO and defines a family of orderings that includes both and is extended to higher-order terms, providing a new fully automatable use of polynomial interpretations in combination with beta-reduction.
Polymorphic higher-order recursive path orderings
TLDR
A family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols is defined, which can be generated from two given well-founded orderings, on the function symbols and on the type constructors.
A static higher-order dependency pair framework
TLDR
A new rewrite formalism designed for general applicability in termination proving of higher-order rewriting, Algebraic Functional Systems with Meta-variables is introduced and a modular dependency pair framework for this higher- order setting is proposed.
The computability path ordering
TLDR
The computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols, is introduced and the well-foundedness proof shows that core CPO captures the essence of computability arguments ^#224, la Tait and Girard, therefore explaining its name.
Well-Founded Recursive Relations
We give a short constructive proof of the fact that certain binary relations > are well-founded, given a lifting ≫ a la Ferreira-Zantema and a well-founded relation ▹. This construction generalizes
Extending a Brainiac Prover to Lambda-Free Higher-Order Logic
TLDR
This work proposes to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features, explaining how to extend the prover’s data structures, algorithms, and heuristics to higher- order logic, a formalism that supports partial application and applied variables.
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