This thesis presents an incremental proof search procedure based on a variable splitting sequent calculus for first-order logic without equality. By means of an index system for formulae, the calculus generates variable sharing derivations in which gamma-inferences for different occurrences of the same formula introduce identical variables. Formula indices are utilized to keep track of how variables are split into different branches of a derivation. This allows closing substitutions to instantiate occurrences of the same variable differently under certain conditions. We represent closing substitutions as syntactic constraints and define an incremental method for calculating these constraints alongside derivation expansion.