I've read this paper http://www21.in.tum.de/~berghofe/papers/TYPES2002_slides.pdf .
It contains calculus for Isabelle type system on page 7. In this calculus "term has type" and "proof has term".
1) Does exist more detailed paper on this calculus?
2) The only two rules which I do not understand are those which relate to constants.
2.1) What does "c_{[\vec{\tau}_n / \vec{\alpha}_n]}" mean? It is a proofs which are constants. We have, for example, constant "impI" on page 9. How these tau and alpha relate to "term of impI".
2.2) Could you please clarify the rule $\Sigma(c) [\vec{\tau}_n / \vec{\alpha}_n] $. ?
At first glance it may be translated as "any set of terms which depend on proof(!) constant with any amount of types instead of atomic types", but I don't see any sense in it.
3)I am also interested in papers on semantics of this calculus and/or correctness theorem of it.
I appreciate any related information. Thanks!
