Obfuscation of Abstract Data-Types - Rowan

CHAPTER 5. SETS AND THE SPLITTING 97
not very interesting from an obfuscation point of view because it is very similar
to the unobfuscated operation — except for a reversal in the order of the split
components.
Since filter preserves ordering (i.e. (filter p xs) ✂ xs) then we can use the
definition for minus for unordered lists:
minus xs ys = filter (not · member ys) xs
(note that this definition has quadratic complexity).
Using this, we derive an operation minus asp that satisfies:
minus asp xsp ysp = split asp (minus (unsplit asp xsp) (unsplit asp ysp))
The right-hand of this equation becomes
split asp (filter (not · member (unsplit asp ysp)) (unsplit asp xsp))
Using Equation (5.1), we obtain
minus asp xsp ysp =
split asp (filter (not · member asp ysp) (unsplit asp xsp))
(5.4)
We derive minus asp by induction on xsp.
Base Case Suppose that xsp = 〈[ ], [ ]〉 asp then
split asp (filter (not · member asp ysp) (unsplit asp xsp))
= {definition of xsp}
split asp (filter (not · member asp ysp) (unsplit asp 〈[ ], [ ]〉 asp ))
= {definition of unsplit asp }
split asp (filter (not · member asp ysp) [ ])
= {definition of filter}
split asp [ ]
= {definition of split asp }
〈[ ], [ ]〉 asp
Step Case Let xsp = 〈x : xl,xr〉 asp and for the induction hypothesis, we
suppose that 〈xr,xl〉 asp satisfies Equation (5.4). We have two subcases.
Subcase 1 Suppose that member asp ysp x = False.
split asp (filter (not · member asp ysp) (unsplit asp xsp))
= {definition of xsp}
split asp (filter (not · member asp ysp) (unsplit asp 〈x : xl,xr〉 asp ))
= {definition of unsplit}
split asp (filter (not · member asp ysp) x : (unsplit asp 〈xr,xl〉 asp ))
= {definition of filter with member asp ysp x = False}
split asp (x : (filter (not · member asp ysp) (unsplit asp 〈xr,xl〉 asp )))