Obfuscation of Abstract Data-Types - Rowan

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APPENDIX A. LIST ASSERTION 149 The tree for this proof is: (A.3) Base Step xsp xsp cat asp cat asp | | asp arith IH (A.4) xsp (A.4) We declare (A.4) as a lemma and so L(A.3) = {(A.4)}. So, for this proof: xsp C (A.3) = 11 + C (A.4) = 15 and H (A.3) = 6 + max(1, H (A.4)) = 10 A.3 Block Split for Lists We want to prove that |xsp + b(k) ysp| b(k) = |xsp| b(k) + |ysp| b(k) (A.5) The definition of + b(k) is 〈[ ], [ ]〉 b(k) + b(k) ysp = ysp 〈a : l, [ ]〉 b(k) + b(k) ysp = cons b(k) a (〈l, [ ]〉 b(k) + b(k) ysp) 〈a : l,r〉 b(k) + b(k) ysp = cons b(k) a (〈l + [head r], tail r〉 b(k) + b(k) ysp) We first need to prove that |cons b(k) a〈l,r〉 b(k) | b(k) = 1 + |〈l,r〉 b(k) | b(k) (A.6) We have two cases: Case 1 If |l| < k then |cons b(k) a 〈l,r〉 b(k) | b(k) = {definition of cons b(k) with |l| < k} |〈a : l,r〉 bk | b(k) = {definition of | | b(k) } |a : l| + |r| = {definition of | |} 1 + |l| + |r| = {definition of | | b(k) } 1 + |〈l,r〉 b(k) | b(k)
APPENDIX A. LIST ASSERTION 150 Case 2 If |l| ≥ k then |cons b(k) a 〈l,r〉 b(k) | b(k) = {definition of cons b(k) with |l| ≥ k} |〈a : (init l), (last l) : r〉 b(k) | b(k) = {definition of | | b(k) } |a : (init l)| + |(last l) : r| = {definition of | |} 1 + |init l| + |last l| + |r| = {init l + [last l] = l and (A.1)} 1 + |l| + |r| = {definition of | | b(k) } 1 + |〈l,r〉 b(k) | b(k) The tree for this proof is: (A.6) Case 1 | | b(k) cons Case 2 | | b(k) cons | | b(k) | | | | b(k) + (A.1) | | Since (A.1) is only used once, we do not declare it as a lemma. So, for this proof, C (A.6) = 9 + C (A.1) = 20 and H (A.6) = 5 + max(1, H (A.1)) = 12 Now, we prove Equation (A.5) by induction on xsp. Base Case Suppose that xsp = 〈[ ], [ ]〉 b(k) and so |xsp| b(k) = 0. Then |xsp + b(k) ysp| b(k) = {definition of xsp} |〈[ ], [ ]〉 b(k) + b(k) ysp| b(k) = {definition of + b(k) } |ysp| b(k) = {arithmetic} 0 + |ysp| b(k) = {definition of xsp} |xsp| b(k) + |ysp| b(k)
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Obfuscation of Abstract Data-Types
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Contents I The Current View of Obfu
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5.4.3 Third Definition . . . . . .
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List of Figures 1 An example of obf
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8 public class c { d t1 = new d();
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10 • Obfuscations can be proved c
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Chapter 1 Obfuscation Scully: “No
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CHAPTER 1. OBFUSCATION 14 These thr
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CHAPTER 1. OBFUSCATION 16 by changi
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CHAPTER 1. OBFUSCATION 18 We could
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CHAPTER 1. OBFUSCATION 20 B 1 [0] =
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Chapter 2 Obfuscations for Intermed
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CHAPTER 2. OBFUSCATIONS FOR INTERME
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Chapter 3 Techniques for Obfuscatio
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CHAPTER 3. TECHNIQUES FOR OBFUSCATI
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Part III Data-Type Case Studies 62
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CHAPTER 4. SPLITTING HEADACHES 64 (
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CHAPTER 4. SPLITTING HEADACHES 66 a
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CHAPTER 4. SPLITTING HEADACHES 70 L
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CHAPTER 4. SPLITTING HEADACHES 74 C
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CHAPTER 4. SPLITTING HEADACHES 84 E
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CHAPTER 4. SPLITTING HEADACHES 86 b
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Page 143 and 144: BIBLIOGRAPHY 142 [43] Simon Peyton
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Obfuscation of Abstract Data-Types - Rowan

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