Obfuscation of Abstract Data-Types - Rowan

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APPENDIX A. LIST ASSERTION 145 = {definition of xs} |(t : ts) + ys| = {definition of +} |t : (ts + ys)| = {definition of | |} 1 + |ts + ys| = {induction hypothesis} 1 + |ts| + |ys| = {definition of | |} |t : ts| + |ys| = {definition of xs} |xs| + |ys| The tree is this proof is: (A.1) Base Step xs xs + + arith | | | | IH xs | | For this, C (A.1) = 11 and H (A.1) = 7. xs A.2 Alternating Split for Lists We want to prove that |xsp + asp ysp| asp = |xsp| asp + |ysp| asp (A.2) For the alternating split, we have two operations equivalent to +. A.2.1 Version 1 We first define: 〈l 0 ,r 0 〉 asp + asp 〈l 1 ,r 1 〉 asp = { 〈l0 + l 1 ,r 0 + r 1 〉 asp if |l 0 | = |r 0 | 〈l 0 + r 1 ,r 0 + l 1 〉 asp otherwise To prove (A.2), let xsp = 〈l 0 ,r 0 〉 asp and ysp = 〈l 1 ,r 1 〉 asp and we have two cases:
APPENDIX A. LIST ASSERTION 146 Case 1 Suppose that |l 0 | = |r 0 |. Then |xsp + asp ysp| asp = {definitions of xsp and ysp} |〈l 0 ,r 0 〉 asp + asp 〈l 1 ,r 1 〉 asp | asp = {definition of + asp } |〈l 0 + l 1 ,r 0 + r 1 〉 asp | asp = {definition of | | asp } |l 0 + l 1 | + |r 0 + r 1 | = {Equation (A.1)} |l 0 | + |l 1 | + |r 0 + r 1 | = {Equation (A.1)} |l 0 | + |l 1 | + |r 0 | + |r 1 | = {definition of | | asp } |〈l 0 ,r 0 〉 asp | asp + |l 1 | + |r 1 | = {definition of | | asp } |〈l 0 ,r 0 〉 asp | asp + |〈l 1 ,r 1 〉 asp | asp = {definitions of xsp and ysp} |xsp| asp + |ysp| asp Case 2 Suppose that |l 0 | ≠ |r 0 |. Then |xsp + asp ysp| asp = {definitions of xsp and ysp} |〈l 0 ,r 0 〉 asp + asp 〈l 1 ,r 1 〉 asp | asp = {definition of + asp } |〈l 0 + r 1 ,r 0 + l 1 〉 asp | asp = {definition of | | asp } |l 0 + r 1 | + |r 0 + l 1 | = {Equation (A.1)} |l 0 | + |r 1 | + |r 0 + l 1 | = {Equation (A.1)} |l 0 | + |r 1 | + |r 0 | + |l 1 | = {definition of | | asp } |〈l 0 ,r 0 〉 asp | asp + |l 1 | + |r 1 | = {definition for | | asp } |〈l 0 ,r 0 〉 asp | asp + |〈l 1 ,r 1 〉 asp | asp = {definitions of xsp and ysp} |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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CHAPTER 4. SPLITTING HEADACHES 68 T
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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 80 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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CHAPTER 4. SPLITTING HEADACHES 88 W
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CHAPTER 4. SPLITTING HEADACHES 90 T
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Chapter 5 Sets and the Splitting Bu
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Page 141 and 142: BIBLIOGRAPHY 140 [13] Thomas H. Cor
Page 143 and 144: BIBLIOGRAPHY 142 [43] Simon Peyton
Page 145: Appendix A List assertion We want t
Page 149 and 150: APPENDIX A. LIST ASSERTION 148 For
Page 151 and 152: APPENDIX A. LIST ASSERTION 150 Case
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Page 167 and 168: Appendix C Matrices We prove the as
Page 169 and 170: APPENDIX C. MATRICES 168 Base Case
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Page 187: APPENDIX E. OBFUSCATION EXAMPLE 186
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Obfuscation of Abstract Data-Types - Rowan

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