By Robert S. Boyer, J. Strother Moore
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Additional resources for A Computational Logic (ACM monograph series)
However, one apparently mild generalization of the above scheme results in our being able to define functions that are considerably more interesting. This generalization allows the use of f as a function symbol in the body of its own definition.
2. 11. We present there an informal proof that LESSP is well-founded. 3. Whenever we have two previously assumed well-founded relations, we assume that the lexicographic relation induced by them is well-founded. 10 we define “induced” and present an informal proof of the wellfoundedness of induced lexicographic relations. The fact that a function has been assumed to be a wellfounded relation is used only in our principles of induction and definition and in the formation of induced lexicographic relations.
For example, the natural numbers can be thought of as shells: a number is either a blue 0 or a blue 1-tuple containing another blue object (namely, the predecessor of the tuple). Ordered pairs can be red 2-tuples containing arbitrary objects. The type consisting of lists of numbers can be either the green, empty list of numbers or else green 2-tuples, x,y , such that x is a number (blue) and y is a list of numbers 1 One way to make sure that T is not a number or to escape from asking what is the successor of T is to employ a typed syntax.