Minimal from classical proofs

Minimal from Classical Proofs

Let A be a formula without implications, and Γ consist of formulas containing disjunction and falsity only negatively and implication only positively. Orevkov (1968) and Nadathur (2000) proved that classical derivability of A from Γ implies intuitionistic derivability, by a transformation of derivations in sequent calculi. We give a new proof of this result (for minimal rather than intuitionist...

Simplifying Programs Extracted from Classical Proofs

Curry-Howard isomorphism extended to classical logic by associating the rule of double-negation elimination with the control operator C allows to view a natural deduction derivation of a Π2 formula as a program. However, programs produced by this method are often hard to understand, in part because of the presence of control operators. In this paper, we show how to simplify such programs and ma...

Re ned Program Extraction from Classical Proofs

It is well known that it is undecidable in general whether a given programmeets its speci cation In contrast it can be checked easily by a machine whether a formal proof is correct and from a constructive proof one can automatically extract a corresponding program which by its very construction is correct as well This at least in principle opens a way to produce correct software e g for safety ...

Classical Proofs and Programs

Getting results from programs extracted from classical proofs

We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y]∧S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a non trivial example: a proof of Dickson’s lemma.