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By Robert S. Boyer, J. Strother Moore

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6 Natural Numbers We now axiomatize the natural numbers, 0, 1, 2, etc. using the shell principle: Shell Definition. Add the shell ADD1 of one argument with bottom object (ZERO), recognizer NUMBERP, accessor SUB1, type restriction (NUMBERP X1), default value (ZERO), and well-founded relation SUB1P. 42 CHAPTER 3. ” The numbers consist of the new object (ZERO) and all of the objects returned by the new function ADD1. We now informally repeat the axioms added by the shell principle. The numbers in parentheses indicate the corresponding clause of the definition of the shell addition principle.

McCarthy (private communication) has written a recursive function, exhibited as SAMEFRINGE in Appendix A, that is more or less the recursive realization of the usual “coroutining” solution to the problem. Our system can prove the correctness of SAMEFRINGE, and we refer the interested reader to Appendix A. The problem of computing the fringe of a tree relates to computer science as a whole, since similar tree processing underlies such fundamental algorithms as parsers and compilers. 4. FLATTEN formally, explain why they are admitted under our principle of definition, and then work through the proof of the conjecture above.

LESSP establish that (COUNT X) goes down according to the well-founded relation LESSP in each recursive call. Hence, FLATTEN is accepted under the definition principle. Observe that (LISTP (FLATTEN X)) is a theorem. LESSP establish that (COUNT X) decreases according to the well-founded relation LESSP in each recursive call. FLATTEN is accepted under the definition principle. FLATTEN X ANS)) is a theorem. FLATTEN X ANS) (APPEND (FLATTEN X) ANS)) Name the conjecture *1. Let us appeal to the induction principle.

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