# Translating Higher-Order Clauses to First-Order Clauses

@article{Meng2007TranslatingHC, title={Translating Higher-Order Clauses to First-Order Clauses}, author={Jia Meng and Lawrence Charles Paulson}, journal={Journal of Automated Reasoning}, year={2007}, volume={40}, pages={35-60} }

Interactive provers typically use higher-order logic, while automatic provers typically use first-order logic. To integrate interactive provers with automatic ones, one must translate higher-order formulas to first-order form. The translation should ideally be both sound and practical. We have investigated several methods of translating function applications, types, and λ-abstractions. Omitting some type information improves the success rate but can be unsound, so the interactive prover must…

## 116 Citations

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.

Set of Support for Higher-Order Reasoning

- Computer SciencePAAR@FLoC
- 2018

Limiting how axioms introduced during translation can improve proof search with higher-order problems is shown and heuristics based on the set-of-support strategy for minimising the effects are introduced.

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.

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- 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.

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- Mathematics
- 2012

Nitpick is the development of Nitpick, a counterexample generator that builds on a first-order relational model finder that heuristically selects facts relevant to the conjecture to prove and delegates the problem to first- order resolution provers and SMT solvers.

Proving Theorems of Higher-Order Logic with SMT Solvers

- Computer Science
- 2012

It is found that many theorems can now be proved automatically and almost instantaneously for which time-consuming user ingenuity was previously required.

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- Computer ScienceIWIL@LPAR
- 2017

Modifications to the higher-order automated theorem prover Leo-III are presented for turning it into a reasoning system for rank-1 polymorphic HOL and a suitable paramodulation-based calculus are sketched.

TFF1: The TPTP Typed First-Order Form with Rank-1 Polymorphism

- Computer ScienceCADE
- 2013

The TFF1 format is introduced, an extension of TFF0 with rank-1 polymorphism, designed to be easy to process by existing reasoning tools that support ML-style polymorphism.

A Polymorphic Intermediate Verification Language: Design and Logical Encoding

- Computer ScienceTACAS
- 2010

The paper gives two encodings of types and formulae in simply typed logic such that SMT solvers and other theorem provers can be used to discharge verification conditions.

Boolean Reasoning in a Higher-Order Superposition Prover

- Computer SciencePAAR+SC²@IJCAI
- 2020

We present a pragmatic approach to extending a Boolean-free higher-order superposition calculus to support Boolean reasoning. Our approach extends inference rules that have been used only in a…

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