Some of the logics admit Gentzen-style sequent calculus. Are there universal algorithms that allow to find proof (derivation of the proof) in sequent caculus for every hypthetical theorem?
Some of the proof assistants support formalization of general sequent calculus that can allow to define sequent calculus for the concrete object logics and find proofs for these logics (i.e. this is the use case how proof assistant ar adapted for use as theorem provers):
Isabelle/HOL has formalization https://www.cl.cam.ac.uk/research/hvg/Isabelle/dist/library/Sequents/Sequents/Sequents.html https://www.cl.cam.ac.uk/research/hvg/Isabelle/dist/library/Sequents/Sequents/ILL.html
Coq has formalization like http://www.cs.nuim.ie/~jpower/Research/LinearLogic/
So - current theorem provers have more or less universal backward reasoning algorithms for theorem proving. Where can I find the reference for these algorithms? I guess I can refactor the mentioned theorem provers but I hope that theoretical descriptions of these algorithms can be found.
Especially I am interested in quantified linear logic but algorithms for other logics are interesting for me.
I am aware that the original goal for the proof assistants was proof checking and not the derivation of the proof but current state of the art is that the proof assistants can automatically derive large parts of the proof and human intervention is necessary only for the hardest cases. Human intervention can take the two forms: 1) developer can program new tactics for the automatic derivation of the proof; 2) mathematician can choose the specific tactic for the proving the current subgoal. The second task can be left to Artificial Intelligence.
So - I am seeking for the reference - where the proof algorithms for sequent calculus are described. It is quite possible that full algorithms does not exists (similarly - proof assistants require new tactics) but still - are there good reference about such automatiazion?
