James Brotherston and Alex Simpson worked on non-well-founded proofs, see

J. Brotherston and A. Simpson: Sequent calculi for induction and infinite descent. J Logic Computation (2010)

as well as the talk "On Proof by Innite Descent" by Alex Simpson at "Algebra & Coalgebra meet Proof Theory" in Bern, April 2012.

James Brotherston and Alex Simpson worked on non-well-founded proofs, see

J. Brotherston and A. Simpson: Sequent calculi for induction and infinite descent. J Logic Computation (2010)

as well as the talk "On Proof by Infinite Descent" by Alex Simpson at "Algebra & Coalgebra meet Proof Theory" in Bern, April 2012.