number-theory

9A The structure of a term 1419A2 Definition (Occurs, occurrence) The phrase "P occurs in M at position p" (or"M contains P at position p") is defined by induction on the length of p, thus:(i) M occurs in M at position 0;(ii) if Pi P2 occurs in M at p, then P, occurs in M at pi for i = 1,2;(iii) if occurs in M at p, then Q occurs in M at p0 and x occurs in M at p*.An occurrence of P in M is a triple (P, p, M) such that P occurs in M at p. IfP - x and the last symbol of p is * we call (x, p, M) a binding occurrence of x or anoccurrence of ).x.An abstractor is an occurrence of Ax for some x.A subterm of M is any term P that occurs in M.9A2.1 Note It is easy to prove that at most one term can occur in M at one position.(We assume of course that a term cannot be simultaneously an application and anabstraction, or a composite term and an atom; also thatP1P2 Q1Q2 P1 Qi & P2 Q2,x = y & P = Q.)9A2.2 Notation An occurrence (P, p, M) may be called simply "P" when no confusionis likely. A binding occurence of x may be called either of9A3 Definition (Components) A component of a term M is any occurrence in Mother than a binding occurrence of a variable, i.e. any occurrence (P, p, M) suchthat the last symbol in p is not *.A component is called proper if it is not (M, 0, M).9A3.1 Note The reason that binding occurrences of variables are denied the name"components" in the above definition is that they play a very different role fromother occurrences. In a term such as we shall think of x and y as being thematerial from which the term is built, but think of z as being part of one of theoperations which do the building.9A4 Definition (The construction-tree of a term) We can display the structure of aterm as a tree with each node carrying two labels, a position and a subterm thatoccurs at that position. This tree is defined for an arbitrary term M as follows.(i) If M = x, its tree is a single node labelled with x and the empty position, thus:x.0.(ii) If M = PQ, its tree is obtained by first concatenating "1" onto the left end ofeach position-label in the tree for P, then concatenating "2" onto the left of each