number-theory

154 9 Technical details(m + 1 - i)p (for i = 1, ... , m), and then combining the modified trees as follows:Modifiedtree forModifiedtree for9E2.1 Note The use of (m + 1- i)p in (ii) above has the effect of assigning positionsto Tl,...,Tm backwards, giving position 1 to Tm, 2 to and m to TI. Thismakes it easier to relate positions in the condensed tree of a--*r to those of a andT, though we shall not need this facility in this book.As an example, Fig. 9E2.1a shows the condensed tree for the typeT(a->(b->c))->((a-b)-(a->c))(a-.b-->c)-+(a-*b)--a--*c.9E3 Definition (S-subtypes) The significant subtypes or s-subtypes of a type r arethe types that label the nodes in the condensed tree of T. A proper s-subtype of T isan s-subtype # T.9E3.1 Example The s-subtypes of the type T in Fig. 9E2.1a area, b, c, a-*b-.c, a-*b, T.Note that although b-+c is a subtype of T in the usually accepted sense (9D1) it isnot an s-subtype because it does not correspond to a node on the condensed tree.(a_b-4c)_(a-+b)-> a-* cFig. 9E2.1a.