Obfuscation of Abstract Data-Types - Rowan

CHAPTER 4. SPLITTING HEADACHES 76
Let us now consider an operation cat asp which corresponds to +, i.e. ( +) ❀
cat asp . By writing + as an uncurried operation then we can use Equation (3.9)
so that cat asp satisfies
cat asp (xsp,ysp) = split asp ( + (unsplit asp xsp, unsplit asp ysp))
This is equivalent to:
cat asp xsp ysp = split asp ((unsplit asp xsp) + (unsplit asp ysp)) (4.17)
We derive a definition for cat asp by structural induction on xsp.
Base Case Suppose that xsp = 〈[ ], [ ]〉 asp and so unsplit asp xsp = [ ].
split asp ((unsplit asp xsp) + (unsplit asp ysp))
= {definitions of xsp and unsplit asp }
split asp ([ ] + (unsplit asp ysp))
= {definition of +}
split asp (unsplit asp ysp)
= {Equation (4.13)}
ysp
Step Case Suppose that xsp = 〈x : r,l〉 asp and for the induction hypothesis,
we assume that 〈l,r〉 asp satisfies Equation (4.17).
split asp ((unsplit asp xsp) + (unsplit asp ysp))
= {definition of xsp}
split asp ((unsplit asp (〈x : r,l〉 asp )) + (unsplit asp ysp))
= {definition of unsplit asp }
split asp ((x : (unsplit asp 〈l,r〉 asp )) + (unsplit asp ysp))
= {definition of +}
split asp (x : ((unsplit asp 〈l,r〉 asp ) + (unsplit asp ysp)))
= {Property (4.16)}
cons asp x (split asp ((unsplit asp 〈l,r〉 asp ) + (unsplit asp ysp)))
= {induction hypothesis}
cons asp x (cat asp 〈l,r〉 asp ysp)
Thus, we can define
cat asp 〈[ ], [ ]〉 asp ysp = ysp
cat asp 〈x : r 0 ,l 0 〉 asp ysp = cons asp x (cat asp 〈l 0 ,r 0 〉 asp ysp)
As an alternative, we could define:
{
〈l0 + l
〈l 0 ,r 0 〉 asp + asp 〈l 1 ,r 1 〉 asp = 1 ,r 0 + r 1 〉 asp if |l 0 | = |r 0 |
〈l 0 + r 1 ,r 0 + l 1 〉 asp otherwise
In Appendix A, we discuss which of these two definitions produces the better
obfuscation with respect to one of the assertions. We find that + asp gives rise