Superposition with Lambdas

@article{Bentkamp2019SuperpositionWL,
  title={Superposition with Lambdas},
  author={Alexander Bentkamp and Jasmin Christian Blanchette and Sophie Tourret and Petar Vukmirovi{\'c} and Uwe Waldmann},
  journal={ArXiv},
  year={2019},
  volume={abs/2102.00453}
}
We designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on $$\beta \eta $$ β η -equivalence classes of $$\lambda $$ λ -terms and rely on higher-order unification to achieve refutational completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a… 
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  • 2020
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References

SHOWING 1-10 OF 96 REFERENCES
Superposition for Lambda-Free Higher-Order Logic ( Technical Report )
TLDR
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.
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 Unification Algorithm for Typed lambda-Calculus
  • G. Huet
  • Computer Science
    Theor. Comput. Sci.
  • 1975
Encoding Monomorphic and Polymorphic Types
TLDR
This work extends the approach to rank-1 polymorphism and presents alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of "cover", and finds them vastly superior to previous schemes.
A Comprehensive Framework for Saturation Theorem Proving
TLDR
A framework for formal refutational completeness proofs of abstract provers that implement saturation calculi, such as ordered resolution or superposition, based on modular extensions of lifted redundancy criteria is presented.
A Focused Sequent Calculus for Higher-Order Logic
TLDR
A focused intuitionistic sequent calculus for higher-order logic that has primitive support for equality and mixes λ-term conversion with equality reasoning and is proved sound with respect to Church's simple type theory.
Boolean Reasoning in a Higher-Order Superposition Prover
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
HOT: A Concurrent Automated Theorem Prover Based on Higher-Order Tableaux
  • K. Konrad
  • Computer Science, Mathematics
    TPHOLs
  • 1998
TLDR
An improved variant of the calculus which closely corresponds to the proof procedure implemented in Hot is introduced and the design of Hot's design is discussed that can be characterized as a concurrent blackboard architecture.
A Combinator-Based Superposition Calculus for Higher-Order Logic
TLDR
A refutationally complete superposition calculus for a version of higher-order logic based on the combinatory calculus is presented and a novel method of dealing with extensionality is introduced.
Extensional Higher-Order Resolution
TLDR
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.
...
...