# A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms

@inproceedings{Becker2017ATK, title={A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms}, author={Heiko Becker and Jasmin Christian Blanchette and Uwe Waldmann and Daniel Wand}, booktitle={CADE}, year={2017} }

We generalize the Knuth–Bendix order (KBO) to higher-order terms without \(\lambda \)-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order…

## 16 Citations

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

- Computer ScienceArch. Formal Proofs
- 2018

The embedding path order is a variant of the recursive path order for untyped λ-free higher-order terms that is a groundtotal and well-founded simplification order, making it more suitable for the superposition calculus.

A Knuth-Bendix-Like Ordering for Orienting Combinator Equations

- MathematicsIJCAR
- 2020

A number of desirable properties about the KBO are proved including it having the subterm property for ground terms, being transitive and being well-founded, and the ordering fails to be a reduction ordering as it lacks compatibility with certain contexts.

Superposition for Lambda-Free Higher-Order Logic

- Computer ScienceIJCAR
- 2018

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.

Superposition for Lambda-Free Higher-Order Logic ( Technical Report )

- Computer Science
- 2018

Refutationally complete superposition calculi for intentional and extensional λ-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.

Implementation of Lambda-Free Higher-Order Superposition

- Computer Science
- 2018

This thesis extends E, a state-of-the-art first-order ATP, to a fragment of HOL that is devoid of lambda abstractions (LFHOL), and devise generalizations of E’s indexing data structures to LFHOL, as well as algorithms like matching and unification to support HOL features in an efficient manner.

Superposition with Lambdas

- Computer ScienceCADE
- 2019

A superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans is designed and implemented in the Zipperposition prover and evaluated on TPTP and Isabelle benchmarks.

Extending a brainiac prover to lambda-free higher-order logic

- Computer ScienceTACAS
- 2019

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.

Foundational (Co)datatypes and (Co)recursion for Higher-Order Logic

- Computer ScienceFroCoS
- 2017

We describe a line of work that started in 2011 towards enriching Isabelle/HOL’s language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive…

Nested Multisets, Hereditary Multisets, and Syntactic Ordinals in Isabelle/HOL

- Mathematics, Computer ScienceFSCD
- 2017

Formal proofs of the main properties of the nested multiset order that are useful in applications are presented: preservation of well-foundedness and preservation of totality (linearity).

Certified ACKBO

- Computer ScienceCPP
- 2019

This work presents its Isabelle/HOL formalization of the ACKBO order, and integrated corresponding check functions into the verified certifier CeTA, which has the more practical consequence of enabling the machine certification of AC-termination proofs generated by automated termination tools.

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