Obfuscation of Abstract Data-Types - Rowan

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APPENDIX A. LIST ASSERTION 147 For this proof, we declare (A.1) as a lemma and so L(A.2) = {(A.1)}. The tree for this proof is: (A.2) defs | | For this proof, | | (A.1) (A.1) | | Case 1 + defs defs C (A.2) = 16 + C (A.1) = 27 and H (A.2) = 6 + max(3, H (A.1)) = 13 | | | | Case 2 defs + | | (A.1) (A.1) A.2.2 Version 2 As an alternative, 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) For this operation, we want to prove |cat asp xsp ysp| asp = |xsp| asp + |ysp| asp Before we prove (A.3), first we prove So, |cons asp a〈l,r〉 asp | asp = 1 + |〈l,r〉 asp | asp (A.3) (A.4) |cons asp a 〈l,r〉 asp | asp = {definition of cons asp } |〈a : r,l〉 asp | asp = {definition of | | asp } |a : r| + |l| = {definition of | |} 1 + |r| + |l| = {definition of | | asp } 1 + |〈l,r〉 asp | asp The tree for this proof is: | | asp | | (A.4) | | asp cons asp
APPENDIX A. LIST ASSERTION 148 For this proof, C (A.4) = 4 and H (A.4) = 4. We are now ready to prove Equation (A.3) by induction on xsp. Base Case Suppose that xsp = 〈[ ], [ ]〉 asp and so |xsp| asp = 0. Then |cat asp xsp ysp| asp = {definition of xsp} |cat asp 〈[ ], [ ]〉 asp ysp| asp = {definition of cat asp } |ysp| asp = {arithmetic} 0 + |ysp| asp = {definition of | | asp } |xsp| asp + |ysp| asp = {definition of xsp} |xsp| asp + |ysp| asp Step Case Suppose that xsp = cons asp x 〈l 0 ,r 0 〉 asp , (and then xsp = 〈x : r 0 ,l 0 〉 asp ) and 〈l 0 ,r 0 〉 asp satisfies (A.2). Then |cat asp xsp ysp | asp = {definition of xsp} |cat asp 〈x : r 0 ,l 0 〉 asp ysp| asp = {definition of cat asp } |cons asp x (cat asp 〈l 0 ,r 0 〉 asp ysp)| asp = {Equation (A.4)} 1 + |cat asp 〈l 0 ,r 0 〉 asp ysp | asp = {induction hypothesis} 1 + |〈l 0 ,r 0 〉 asp | asp + |ysp| asp = {Equation (A.4)} |cons x 〈l 0 ,r 0 〉 asp | asp + |ysp| asp = {definition of xsp} |xsp| asp + |ysp| asp
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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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Page 141 and 142: BIBLIOGRAPHY 140 [13] Thomas H. Cor
Page 143 and 144: BIBLIOGRAPHY 142 [43] Simon Peyton
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Obfuscation of Abstract Data-Types - Rowan

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