# Extensional Higher-Order Paramodulation and RUE-Resolution

@inproceedings{Benzmller1999ExtensionalHP, title={Extensional Higher-Order Paramodulation and RUE-Resolution}, author={Christoph Benzm{\"u}ller}, booktitle={CADE}, year={1999} }

This paper presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69], and a calculus ERUE adapting FO RUE-Resolution [Dig79] to classical type theory, i.e., HO logic based on Church's simply typed λ-calculus. EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both…

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## References

SHOWING 1-10 OF 24 REFERENCES

### Equality and extensionality in automated higher order theorem proving

- Computer Science
- 1999

The three new calculi ER, ERUE, EP and ERUE which improve the mechanisation of defined and primitvie equality in classical type theory and these calculi reach Henkin completeness without requiring additional extensionality axioms are introduced.

### System Description: LEO - A Higher-Order Theorem Prover

- Computer ScienceCADE
- 1998

Leo uses a higher-order Logic based upon Church's simply typed λ-calculus, so that the comprehension axioms are implicitly handled by αβη-equality, and extensionality principles are build in into Leo’s unification, and hence do not have to be axiomatized in order to achieve Henkin completeness.

### Extensional Higher-Order Resolution

- MathematicsCADE
- 1998

An extensional higher-order resolution calculus that is complete relative to Henkin model semantics is presented and the long-standing conjecture, that it is sufficient to restrict the order of primitive substitutions to the orders of input formulae is proved.

### Resolution in type theory

- PhilosophyJournal of Symbolic Logic
- 1971

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very…

### The Clausal Theory of Types

- Computer Science
- 1990

This book introduces just such a theory, based on a lambda-calculus formulation of a clausal logic with equality, known as the Clausal Theory of Types, which is a concise form of logic programming that incorporates functional programming.

### Proofs in Higher-Order Logic

- Computer Science
- 1983

This work resolves the open question of what is a sound definition of skolemization in higher-order logic but also provides a direct, syntactic proof of its correctness.

### Higher-Order Unification Revisited: Complete Sets of Transformations

- Computer ScienceJ. Symb. Comput.
- 1989

### Resolution by Unification and Equality

- Mathematics
- 1983

In resolution by unification and equality, we recast the theory of binary resolution on the basis of the properties of the equality relationship as stated by the equality axioms.
In standard binary…

### Model Existence for Higher Order Logic

- Mathematics, Computer Science
- 1997

A semantical meta-theory that will support the development of higher-order calculi for automated theorem proving like the corresponding methodology has in first-order logic and establish classes of models that adequately characterize the existing theorem-proving calculi.

### An introduction to mathematical logic and type theory - to truth through proof

- PhilosophyComputer science and applied mathematics
- 1986

This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.