We address the problem of proving the satisfiability of Constrained Horn Clauses (CHCs) with Algebraic Data Types (ADTs), such as lists and trees. We propose a new technique for transforming CHCs with ADTs into CHCs where predicates are defined over basic types, such as integers and booleans, only. Thus, our technique avoids the explicit use of inductive proof rules during satisfiability proofs. The main extension over previous techniques for ADT removal is a new transformation rule, called

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... erential replacement, which allows us to introduce auxiliary predicates corresponding to the lemmas used when making inductive proofs. We present an algorithm that applies the new rule, together with the traditional folding/unfolding rules, for the automatic removal of ADTs. We prove that if the set of the transformed clauses is satisfiable, then so is the set of the original clauses. By an experimental evaluation, we show that the use of the new rule significantly improves the effectiveness of ADT removal, and that our approach is competitive with respect to a state-of-the-art tool that extends the CVC4 solver with induction.