Abstract. We present an automatic approach for instantiating existentially quantified variables in inductive specifications proofs. Our approach uses first-order meta-variables in place of existentially quantified variables and combines logical proof search with rippling techniques. We avoid the non-termination problems which usually occur in the presence of existentially quantified variables. Moreover, we are able to synthesize conditional substitutions for the meta-variables. We illustrate our approach by discussing the specification of the integer square root.