Obfuscation of Abstract Data-Types - Rowan

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APPENDIX C. MATRICES 167 ([v] + ws) : (((zipWith ( +)).(map wrap)) vs wss) = {property of : and +} (v : ws) : (((zipWith ( +)).(map wrap)) vs wss) = {definition of f } (v : ws) : (f 1 vs wss) Now if we let f 2 = zipWith (:) then f 2 (v : vs) (ws : wss) = (v : ws) : (f 2 vs wss) which is the same pattern as above. Note that f 2 [ ] yss = [ ] and f 2 (v : vs) [[ ]] = [[v]]. Thus (zipWith ( +)) · (map wrap) = zipWith (:) and so we can write transpose = foldr (zipWith (:)) blanks C.2 Using fold fusion We need to prove Equation (C.1) and, by the results above, we need to prove that transpose (foldr (zipWith (:)) blanks mss) = mss Note that this result only holds if valid mss is true, i.e. when mss represents a matrix. We prove this result by using the Fold Fusion Theorem (Property 1) taking f = transpose, g = zipWith (:) and a = blanks. So, f is strict and f a = [ ] and so we take b = [ ]. Now we need a function h that f (g x y) = h x (f y). Since (C.1) is only true for matrices we need to ensure that |x | = |y|. Before we prove that transpose is self-inverse we need a property involving f and g. Property 12. With the definitions of f and g above: f (x : xs) = g x (f xs). Proof. f (x : xs) = {f = transpose} transpose (x : xs) = {foldr definition of transpose} foldr (zipWith (:)) blanks (x : xs) = {definition of foldr} (zipWith (:)) x (foldr (zipWith (:)) blanks xs) = {definitions of f and g} g x (f xs) For the fold fusion, we propose that h = (:) (note that foldr (:) [ ] = id) and we prove this by induction, i.e. that if |xs| = |yss| then f (g xs yss) = xs : (f yss) Proof. We prove Equation (C.2) by induction on |xs|. (C.2)
APPENDIX C. MATRICES 168 Base Case Suppose that xs = [x] and let yss = [ys] so that |xs| = 1 = |yss| f (g xs yss) = {definitions of xs, yss and g} f [x : ys] = {foldr1 definition of f } foldr1 (zipWith ( +)) (map (map wrap) [x : ys]) = {definitions of map and wrap} foldr1 (zipWith ( +)) [[x] : (map wrap ys)] = {definition of foldr1} [x] : (map wrap ys) = {[definition of foldr1} [x] : (foldr1 (zipWith ( +)) [map wrap ys]) = {definition of map} [x] : (foldr1 (zipWith ( +)) (map (map wrap) [ys])) = {foldr1 definition of f } [x] : (f [ys]) = {definitions} xs : (f yss) Step Case Suppose that xs = v : vs and yss = ws : wss where |vs| = |wss|. For the induction hypothesis, we suppose that vs and wss satisfy Equation (C.2). f (g xs yss) = {definitions} f (g (v : vs) (ws : wss)) = {definition of g (g is an instance of zipWith)} f ((v : ws) : (g vs wss)) = {Property 12} g (v : ws) (f (g vs wss)) = {induction hypothesis} g (v : ws) (vs : (f wss)) = {g is an instance of zipWith} (v : vs) : (g ws (f wss))) = {Property 12} (v : vs) : (f (ws : wss)) = {definitions} xs : (f yss)
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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 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 OBFUSCATI
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