The proof-checker ÆtnaNova, aka Ref, processes proof scenarios to establish whether or not they are formally correct. A scenario, typically written by a working mathematician or computer scientist, consists of definitions, theorem statements and proofs of the theorems. There is a construct enabling one to package definitions and theorems into reusable proofware components. The deductive system underlying Ref mainly first-order, but with an important second-order feature: the packaging construct just mentioned is a variant of the Zermelo-Fraenkel set theory, ZFC, with axioms of regularity and global choice. This is apparent from the very syntax of the language, borrowing from the set-theoretic tradition many constructs, e.g. abstraction terms. Much of Ref’s naturalness, comprehensiveness, and readability, stems from this foundation; much of its effectiveness, from the fifteen or so built-in mechanisms, tailored on ZFC, which constitute its inferential armory. Rather peculiar aspects of Ref, in comparison to other proof-assistants (Mizar to mention one), are that Ref relies only marginally on predicate calculus and that types play no significant role, in it, as a foundation. This talk illustrates the state-of-the-art of proof-verification technology based on set theory, by reporting on ‘proof-pearl’ scenarios currently under development and by examining some small-scale, yet significant, examples of use of Ref. The choice of examples will reflect today’s tendency to bring Ref’s scenarios closer to algorithm-correctness verification, mainly referred to graphs. The infinity axiom rarely plays a role in applications to algorithms; nevertheless the availability of all resources of ZFC is important in general: for example, relatively unsophisticated argumentations enter into the proof that the Davis-Putnam-Logemann-Loveland satisfiability test is correct, but in order to prove the compactness of propositional logic or Stone’s representation theorem for Boolean algebras one can fruitfully resort to Zorn’s lemma.

Proof verification within set theory

OMODEO, EUGENIO
2013

Abstract

The proof-checker ÆtnaNova, aka Ref, processes proof scenarios to establish whether or not they are formally correct. A scenario, typically written by a working mathematician or computer scientist, consists of definitions, theorem statements and proofs of the theorems. There is a construct enabling one to package definitions and theorems into reusable proofware components. The deductive system underlying Ref mainly first-order, but with an important second-order feature: the packaging construct just mentioned is a variant of the Zermelo-Fraenkel set theory, ZFC, with axioms of regularity and global choice. This is apparent from the very syntax of the language, borrowing from the set-theoretic tradition many constructs, e.g. abstraction terms. Much of Ref’s naturalness, comprehensiveness, and readability, stems from this foundation; much of its effectiveness, from the fifteen or so built-in mechanisms, tailored on ZFC, which constitute its inferential armory. Rather peculiar aspects of Ref, in comparison to other proof-assistants (Mizar to mention one), are that Ref relies only marginally on predicate calculus and that types play no significant role, in it, as a foundation. This talk illustrates the state-of-the-art of proof-verification technology based on set theory, by reporting on ‘proof-pearl’ scenarios currently under development and by examining some small-scale, yet significant, examples of use of Ref. The choice of examples will reflect today’s tendency to bring Ref’s scenarios closer to algorithm-correctness verification, mainly referred to graphs. The infinity axiom rarely plays a role in applications to algorithms; nevertheless the availability of all resources of ZFC is important in general: for example, relatively unsophisticated argumentations enter into the proof that the Davis-Putnam-Logemann-Loveland satisfiability test is correct, but in order to prove the compactness of propositional logic or Stone’s representation theorem for Boolean algebras one can fruitfully resort to Zorn’s lemma.
http://ceur-ws.org/Vol-1068/
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11368/2769413
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