number-theory

9Technical detailsTo avoid interrupting the main lines of thought in the earlier chapters some conceptswere defined there only in outline and their main properties were stated withoutproof. This chapter gives the full definitions and proofs. It should be read onlyas required to follow the arguments in the other chapters. Its sections are largelyindependent of each other.9A The structure of a termFrom the viewpoint of logical order this section is best read between 1A3 and 1A4.As remarked in 1A4, a subterm of a term may have more than one occurrence.The present section introduces a precise notation to distinguish such different occurrences;it is rather clumsy and the reader should avoid using it whenever possible,but in some proofs its precision will be vital. The first step is to define a set of expressionscalled positions that can be assigned to different occurrences of a subtermto show where they occur.9A1 Definition (Positions) A position p = ii ... i is any finite (perhaps empty) stringof symbols such that i1,..., irri_1 are integers and in, is either an integer or an asterisk,*. Its length is m, and if m = 0 we say p = 0.If m 1 and n, = 1 we call p a function position;if m 1 and i,,, = 2 we call p an argument position;if m > 1 and ,n = 0 we call p a body position; andif m 1 and n, = * we call p an abstractor position.(Positions containing integers >_ 3 will be used later but not in the present section.)The concatenation pq of positions p = i1 ... i, and q = j1... j is defined thus:p0 = p, Oq = q, and if m, n > 1 and in * definepq = i1 ... imjl ... jn.(pq is undefined if m,n >_ 1 and in, _ *.)A refinement of p is any position with form pq; it is proper if q * 0.Two positions p = i1 ... in and q = j1 ... j, are said to diverge if neither is arefinement of the other, i.e. if there exists h such thatih # jh,1 < h < Min {m, n}.140