Short proofs of tautologies using the schema of equivalence

2007

It is well-known that there are "hard" and "simple" tautolo- gies, but in the capacity of the logical functions they all are equal to each other. In our opinion this thesis is not entirely correct. We suggest a new conception of equality of tautologies, with the help of the notion of j-determinative conjunct, which was defined in (1) for every tautology j.

2017

In proof theory the notion of canonical proof is rather basic, and it is usually taken for granted that a canonical proof of a sentence must be unique up to certain minor syntactical details (such as, e.g., change of bound variables). When setting up a proof theory for equality one is faced with a rather unexpected situation where there may not be a unique canonical proof of an equality statement. Indeed, in a (1994–5) proposal for the formalisation of proofs of propositional equality in the Curry–Howard style [41], we have already uncovered such a peculiarity. Totally independently, and in a different setting, Hofmann & Streicher (1994) [14] have shown how to build a model of Martin-Löf’s Type Theory in which uniqueness of canonical proofs of identity types does not hold. The intention here is to show that, by considering as sequences of rewrites and substitution, it comes a rather natural fact that two (or more) distinct proofs may be yet canonical and are none to be preferred ove...

Theoretical Computer Science, 2006

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1998

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Lecture Notes in Computer Science, 2008

2009