proposition let be the variation of ( caused by single splaying substep ( zigzig-zigor zig-zagfor node in we have the followingd < ( (xr( ) if the substep is zig-zig or zig-zag < ( (xr( )if the substep is zig justificationwe use the fact (see proposition appendix athatif and blog log log ( let us consider the change in ( caused by each type of splaying substep zig-zig(recall figure since the size of each node is one more than the size of its two childrennote that only the ranks of xyand change in zig-zig operationwhere is the parent of and is the parent of alsor (xr( ) ( < ( )and ( < (ythusd (xr (yr (zr(xr(yr(zr (yr (zr(xr( < (xr ( (xnote that (xn (zn ( ( thusr(xr ( ( as per formula that isr ( (xr( this inequality and formula imply < ( ( (xr( ( < ( (xr( ) zig-zag(recall figure againby the definition of size and rankonly the ranks of xyand changewhere denotes the parent of and denotes the parent of alsor(xr(yr(zr (xthusd (xr (yr (zr(xr(yr(zr (yr (zr(xr( < (yr ( ( ( note that (yn (zn ( )hencer (yr ( ( as per formula thusd < ( ( ( (xr( ) < ( (xr( ) zig(recall figure in this caseonly the ranks of and changewhere denotes the parent of alsor (yr(xthusd (yr (xr(yr( < (xr( < ( (xr( )