Core Concepts
The authors present a formalization of higher-order logic in the Isabelle proof assistant, building directly on the foundational Isabelle/Pure framework. The goal is to provide a gentle introduction to higher-order logic and proof assistants, without the complexity of Isabelle/HOL.
Abstract
The paper introduces a formalization of higher-order logic in the Isabelle proof assistant, called HOL_Pure, which is built directly on the Isabelle/Pure framework. This is presented as an alternative to the more complex Isabelle/HOL system, with the aim of providing a gentle introduction to higher-order logic and proof assistants.
The authors first discuss their approach, which involves a graceful extension from first-order to higher-order logic, and present their main contributions. They then discuss related work on using proof assistants for teaching formal reasoning.
The paper includes sample natural deduction proofs in propositional logic, first-order logic, and higher-order logic (Cantor's theorem) to showcase the Isabelle code and proof style. It also covers the formalization of first-order logic systems, including both implicational axiomatics and natural deduction, with formal proofs of soundness and completeness in Isabelle/HOL.
The authors then build up the intuitionistic and classical higher-order logic theories in Isabelle/Pure, starting from the fundamental concepts of implication and universal quantification. They discuss the differences between the various proof systems and logics presented, and how they are used in the authors' automated reasoning course.
Finally, the paper mentions further developments, including the addition of features like the axiom of choice and set comprehension, and concludes with remarks on the usefulness of the approach for teaching higher-order logic and proof assistants.
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