Principles of Superdeduction

In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focusing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system "superdeduction ". We introduce a proof-term language and a cut-elimination procedure both based on Christian Urban\´s work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently sketch how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants like lemuridae, our prototypal implementation of superdeduction modulo.