Beyond Beta-reduction in Church's lambda-arrow

In this paper, we shall writ.e A __ using a notation, item notation, which enables one to make more redexes visible, and sha11 extend j3-reduction to all visible redexes. We will prove that A_ written in item notation and accommodated with extended reduction, satisfies a1l its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if I translates classical terms to our notation, then I(t,t2) '= (I(t,)6)I(t!) and I(>'v".t) '= (p>'v)I(t). For example, t == ((..\'X7:X4 .(AX6:X3 '''\L'~:X I-X:;! .X5X4)X3)X2)Xl J call be written in our item notation as I(t) '= (x,6)(x26)(X.>'x,)(X36)(X3>'x,)((X, -4 X,)>'x,)(X.6)X5 where the visible redexes are based on al1 the matching oA-couples. So here, the redexes are based on (x 26)(X.>'.,), (X36)(X3>'x,) and (",6)((X, -4 X 2»..,). In classical notation however, only the redexes based 011 (A;t:7:X4 , -)X2 and (AX6:X3' -)X3 are immediately visible. The third redex, (.\.!:~:(Xr ..... X:d' -)X2' only becomes visible when the first two red exes have been cont,racted. We extend .B-reduction so that we can contract newly visible redexes even before other redexes have been contracted. So in our example above, (x,6)((X, -4 X 2)>'x,) can be collt,ract.ed before (x,6)(X.>',,) or (X36)(X3>'x,). This refinement (which cannot be done ill classical notat,ion) is achieved by generalising the axiom (3 from (t,6)(p>.,,)t2 -+~ t,[11 := td t.o (t,6)8(p>'v)t, ~~ 8(t,[V := td) for 8 consisting of matching 6>.-couples only. Hence, as (x 26)(X.>'x, )(X36)(X3>'x,) consists of matching 6>'couples, we get. t.hat I(t) ~~ (a:28)(X4>'x,)(X36)(X3>'x,)(((X46)X5)[X5 := xd). Furthermore, wit.h our item notat.ion, it. is possible to refine reduction by rewriting (or reshufJling) terms so that matching b'A-couples occur adjacent t.o each ot.her. For example, we can rewrite I(t) above as ("'28)(X.>',,)(X36)(X3>'x,)(x,6)((X, -+ X 2)>'x,)(x.6)X5. We shall formalise term reshuffling and shall show that. it is correct and preserves both .B-reduction and t,yping.