The role played before by variable-binding is now taken by the infinite arities. From this we may say what it means for M to obey certain first order axioms, or to satisfy some other property, as in Remark 1.6.13 . For this structural induction, it is only necessary to verify the soundness of each proof rule individually. The program may then be simplified to a conjunction of some of its inputs, namely those that need to be evaluated in order to execute the program. This subset may be found mechanically by the compiler, which may then detect which arguments actually need to be evaluated.
Any free theory has an equationally free algebra.
In particular, a nullary operation-symbol can only occur at the end of the list. Otherwise we would not be able to handle the nullary operations, without which the free algebra would be empty.