number-theory

152 9 Technical detailsthe set of these variables is called Vars(T). The set of all variables occurring in afinite sequence (ti,. .. , will be calledVars (T 1, ... , T,,).9D2 Definition (Construction-tree of a type) The construction-tree of a type T is alabelled tree showing how T is built up (cf. the tree of a term in 9A4). Each nodehas two labels, a type and a position, and the tree is defined thus:(i) If T is an atom, say T = e, its tree has just one node:e.0.(ii) If T =_ p-.a, its tree is built from the trees for p and a by first putting "1" onthe left end of each position-label in the tree for p, next putting "2" on the left endof each position-label in the tree for a, and then placing an extra node beneath thetwo modified trees, thus:Modified tree....*for p. Modified tree....-'for ap-*a 09D2.1 Example Let T - (a-(b--+c))-+((a-+b)-*(a-*c)); its tree is as shown inFig. 9D2.1a.0(a-. (b-+ c))- ((a-. b)-* (ate c))Fig. 9D2.1 a.9D3 Definition (Positive and negative occurrences) Let a - (a, p, T) be an occurrenceof a. We call a positive (and say that a occurs positively in r) if p = 0 or the numberof 1's in p is even. We call a negative and say that a occurs negatively in T if thenumber of l's in p is odd.