Obfuscation of Abstract Data-Types - Rowan

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APPENDIX B. SET OPERATIONS 163 |ly| = |ry| Proof of (i) (⇐) If member xs a then head zs = a and ⇒ ⇒ |ly| = |ry| {Property (B.1)} unsplit 〈lz,rz〉 asp = zs {applying head to both sides (which are finite non-empty lists)} head (unsplit 〈lz,rz〉 asp ) = head zs ⇒ {member xs a} head (unsplit 〈lz,rz〉 asp ) = a ⇒ {property of head asp } head asp 〈lz,rz〉 asp = a ⇒ {definition of head asp } ⇒ head lz = a {property of member} member l a Proof of (ii) We can prove this result in the same way as we proved (i) or we can note that: ¬(member l a) ⇔ ¬(|ly| = |ry|) Since member xs a then ¬(member l a) ≡ member r a and so (ii) holds. The tree for (i) (⇒) is inv (5.3) arith | | asp (B.1) (B.3⇒) span asp xs Since (B.1) is only used once for this proof we do not declare it as a lemma (note that if we wanted the tree for the entire proof then (B.1) would be declared as a lemma). Thus C (B.3 ⇒) = 6 + C (B.1) = 18 H (B.3 ⇒) = 3 + max(2, H (B.1)) = 12
APPENDIX B. SET OPERATIONS 164 B.2 Insertion for the alternating split To prove the correctness of insert asp , we need to show that Equation (3.6) holds for insert asp . Therefore, we are required to prove: unsplit asp (insert asp a (split asp xs)) = insert a xs We will need the properties from Sections 4.3.2 and B.1 and use the definition of + asp . The definition of insert asp is insert asp a 〈l,r〉 asp = 〈ly,ry〉 asp + asp (if member lz a then 〈lz,rz〉 asp else if |ly| == |ry| then 〈a : rz,lz〉 asp else if member rz a then 〈rz,lz〉 asp else 〈a : lz,rz〉 asp ) where (ly,lz) = span (< a) l (ry,rz) = span (< a) r Suppose that xs ❀ 〈l,r〉 asp and consider insert asp a 〈l,r〉 asp . Let (ly,lz) = span (< a) l, (ry,rz) = span (< a) r and (ys,zs) = span (< a) xs. We have four cases corresponding to the conditions in the definition of insert asp . Case (i) Suppose that member a lz and head zs = a. Since member a l then by Equation (B.3), |ly| = |ry|. So, unsplit asp (insert asp a (split asp xs)) = {xs ❀ 〈l,r〉 asp } unsplit asp (insert asp a 〈l,r〉 asp ) = {definition of insert asp } unsplit asp (〈ly,ry〉 asp + asp 〈lz,rz〉 asp ) = {property of + asp } (unsplit asp 〈ly,ry〉 asp ) + (unsplit asp 〈lz,rz〉 asp ) = {Property (B.1)} ys + zs = {definition of insert with head zs = a} insert a xs Case (ii) Now suppose that ¬(member a lz) and |ly| = |ry|. By Equation (B.3), ¬(member a xs). Then, unsplit asp (insert asp a (split asp xs)) = {xs ❀ 〈l,r〉 asp } unsplit asp (insert asp a 〈l,r〉 asp ) = {definition of insert asp } unsplit asp (〈ly,ry〉 asp + asp 〈a : rz,lz〉 asp ) = {property of + asp } (unsplit asp 〈ly,ry〉 asp ) + (unsplit asp 〈a : rz,lz〉 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 5. SETS AND THE SPLITTING 9
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Obfuscation of Abstract Data-Types - Rowan

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