Programming with Invariants through Targeted Synthesis. John Sarracino

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1 Programming with Invariants through Targeted Synthesis John Sarracino 1

2 Programming is Hard

3 Programming is Hard Therac

4 Programming is Hard Ariane

5 Programming is Hard Maersk 2017

6 Programming is Hard Windows 2018

7 Programming is Hard Windows always

8 Computers are a Sadness and I am the cure

9 Programs are a Sadness and I am the cure

10 Software Invariants ubi_assert(e->num!= e1->num); if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); ensure cat_set : has_category(cat) end

11 Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); ensure cat_set : has_category(cat) end

12 Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Java ensure cat_set : has_category(cat) end

13 Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Java ensure cat_set : has_category(cat) end Eiffel

14 Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Language agnostic, Manually maintained. Java ensure cat_set : has_category(cat) end Eiffel

15 Manual Invariant Programming Invariants Algorithm

16 Manual Invariant Programming Invariants Algorithm Code

18 Why are Manual Invariants difficult? Code changes => Programmer checks, maintains invariant Invariant changes => Programmer checks, maintains code

19 The code change problem: Maintaining Invariants Invariant: All urls are in the urlset

22 The code change problem: Maintaining Invariants Tedious and error-prone, particularly with a large codebase Invariant: All urls are in the urlset

23 The invariant change problem: Maintaining Code Invariant: All urls are in the urlset

24 The invariant change problem: Maintaining Code Invariant: All urls are in the urlset The allocset contains the urlset

25 The invariant change problem: Maintaining Code Even more tedious and error-prone Invariant: All urls are in the urlset The allocset contains the urlset

26 My thesis statement Compiling and generating invariants is useful, novel, and broadly applicable.

27 My thesis proposal Compiling and generating invariants is useful, novel, and broadly applicable. I propose a general framework for maintaining invariants

28 Proposed Work Data invariant calculus Extensions Invariants for Security GUI invariants

29 Outline Data invariant calculus Extensions Invariants for Security GUI invariants

31 Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i]

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sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc

40 Budgeting Code for (let index = 0; index!= week.length; ++index) { week[index].oninput = () => { let w = parsefloat(week[index].textcontent); month[index].textcontent = w * 4; year[index].textcontent = w * 12 * 4; refreshcolors(index); month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); year[index].oninput = () => { let y = parsefloat(year[index].textcontent); week[index].textcontent = y / (4 * 12); month[index].textcontent = y / 12; refreshcolors(index); document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

41 Budgeting Code document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].oninput = () => { week[index].style.backgroundcolor = evencolor; let w = parsefloat(week[index].textcontent); month[index].style.backgroundcolor = evencolor; month[index].textcontent = w * 4; year[index].style.backgroundcolor = evencolor; year[index].textcontent = w * 12 * 4; else { refreshcolors(index); week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; Tedious even for this year[index].style.backgroundcolor simple example = oddcolor; month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { year[index].oninput = () => { if (index % 2 == 0) { let y = parsefloat(year[index].textcontent); week[index].style.backgroundcolor = evencolor; week[index].textcontent = y / (4 * 12); month[index].style.backgroundcolor = evencolor; month[index].textcontent = y / 12; year[index].style.backgroundcolor = evencolor; refreshcolors(index); else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

42 Budgeting Code Invariants Algorithm for (let index = 0; index!= week.length; ++index) { week[index].oninput = () => { let w = parsefloat(week[index].textcontent); month[index].textcontent = w * 4; year[index].textcontent = w * 12 * 4; refreshcolors(index); month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); year[index].oninput = () => { let y = parsefloat(year[index].textcontent); week[index].textcontent = y / (4 * 12); month[index].textcontent = y / 12; refreshcolors(index); document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

43 Budgeting Code Invariants Algorithm for (let index = 0; index!= week.length; ++index) { week[index].oninput = () => { let w = parsefloat(week[index].textcontent); month[index].textcontent = w * 4; year[index].textcontent = w * 12 * 4; refreshcolors(index); month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); year[index].oninput = () => { let y = parsefloat(year[index].textcontent); week[index].textcontent = y / (4 * 12); month[index].textcontent = y / 12; refreshcolors(index); document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

44 Budgeting Code Invariants Algorithm document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].oninput = () => { week[index].style.backgroundcolor = evencolor; let w = parsefloat(week[index].textcontent); month[index].style.backgroundcolor = evencolor; month[index].textcontent = w * 4; year[index].style.backgroundcolor = evencolor; year[index].textcontent = w * 12 * 4; refreshcolors(index); Vast majority else of { code week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; month[index].oninput manually = () => { maintains invariant let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { year[index].oninput = () => { if (index % 2 == 0) { let y = parsefloat(year[index].textcontent); week[index].style.backgroundcolor = evencolor; week[index].textcontent = y / (4 * 12); month[index].style.backgroundcolor = evencolor; month[index].textcontent = y / 12; year[index].style.backgroundcolor = evencolor; refreshcolors(index); else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

46 Invariant Synthesis Invariants Algorithm Code

47 Language Design Goals Programmer thinks about invariants once Compiler automatically handles invariants

48 Compiling Invariants Program Synthesis compiles invariants* Inputs: 1) program with holes e.g. x :=??(y, x) 2) precondition e.g. y = 15 3) requirement e.g. x = y

49 Compiling Invariants Program Synthesis compiles invariants* Inputs: expression 1) program using with holes x and e.g. y x :=??(y, x) 2) precondition e.g. y = 15 3) requirement e.g. x = y

51 Naive Algorithm Synthesize at the end of every function. Use invariants as preconditions, requirements.

52 Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] =??(value, i, weeks[i]); years[i] =??(value, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

53 Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] = 4*weeks[i]; years[i] = 4*12*weeks[i]; 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

54 Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Does not scale to realistic programs Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] =??(value, i, weeks[i]); years[i] =??(value, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

55 Problem: Loops don t scale function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

56 Problem: Loops don t scale function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Timeout even for this simple example Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

57 Key insight: move hole inside loop function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

58 Key insight: move hole inside loop function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

59 Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

60 Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

61 Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

62 Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Targeted Synthesis! Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

63 Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] = months[i] * scale; years[i] = years[i] * scale; 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

64 Spyder Separate invariant, imperative programs Compiler combines with targeted synthesis

65 Spyder Automatic Invariant maintenance Separate invariant, imperative programs Compiler combines with targeted synthesis

66 Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val;

67 Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val; Loops and Arrays

68 Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val; In-out iterators

69 Imperative AST

70 Invariant Language foreach (week, month) in (weeks, months) { week = 4 * month

71 Invariant Language foreach (week, month) in (weeks, months) { week = 4 * month Elementwise Quantification

72 Invariant AST

73 Compilation Algorithm

75 Compilation Algorithm 1) bring necessary variables into scope 2) specialize invariants to loops 3)??? (find template) 4) profit (synthesize)

77 Format Invariants Code Explanatory Caption Problem/Result

78 Simple Example x > 2 * y y = y + 1;

79 Simple Example x > 2 * y y = y + 1; X ~ Y

80 Simple Example x > 2 * y y = y + 1; X ~ Y Y edited

81 Simple Example x > 2 * y y = y + 1; X ~ Y Y edited x =??(x, y);

82 Simple Example x > 2 * y y = y + 1; X, Y in scope x =??(x, y);

83 Simple Example x > 2 * y y = y + 1; x =??(x, y);

84 Simple Example x > 2 * y y = y + 1; x = x + 2;

87 Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05;

88 Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05; weeks ~ months

89 Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05; weeks ~ months need months

90 Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05; weeks ~ months need months

91 Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; weeks ~ months need months

94 Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05;

95 Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant

96 Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant m =??(m, w);

97 Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant m = m * 1.05;

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104 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue;

105 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; bkg ~ bkgcolors bkg edited

106 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; bkg ~ bkgcolors bkg edited for (color: int) in (bkgcolors){ color =??(bkg, blue);

107 Loops foreach (color) in (bkgcolors) { color = bkg That s the skeleton bkg = blue; What about the invariant? for (color: int) in (bkgcolors){ color =??(bkg, blue);

108 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color =??(bkg, blue);

109 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; Use foreach spec for body for (color: int) in (bkgcolors){ color =??(bkg, blue);

110 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color =??(bkg, blue);

111 Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color = blue;

114 Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red;

115 Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; Implications as conditionals

116 Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; Implications as conditionals if (idx % 2 == 0) { color =??(even, red, idx);

117 Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; if (idx % 2 == 0) { color =??(even, red, idx);

118 Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; if (idx % 2 == 0) { color = even;

119 Case studies* Game of Life (1d, 2d) Budgeting calculator Split expenses spreadsheet *

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sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc Code Interactive cells Inflation Backgrounds *

122 <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> Split expenses Invariants 8i.2 mid[i] =me[i]+you[i] 8i.debt[i] =mid[i] you[i] 8i.debt[i] < 0 =) color[i] =black 8i.debt[i] >= 0 =) color[i] =red Code Interactive cells Backgrounds *

125 Formalisms

126 Formalisms

127 Formalisms Interesting Translation Rule

128 Formalisms What we need Formal problem definition Full translation semantics Proof of correctness

129 Formal Correctness Problem: Correctness:

130 Formal Correctness Problem: Given a program with arrays and invariants Correctness:

131 Formal Correctness Problem: Given a program with arrays and invariants Correctness: We find a program with augmented semantics

132 Formal Correctness Problem: Given a program with arrays and invariants Thank you, Reviewer A!!! Correctness: We find a program with augmented semantics

133 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Consider Sortedness

134 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j]

135 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Limited to maplike properties

136 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Cannot express :(

137 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Key insight: change of perspective!! Cannot express :(

138 <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j]

139 <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> 8i, j. i < j =) sorted[i] <sorted[i + 1] 8i. sorted[i] <sorted[i + 1]

140 <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> 8i, j. i < j =) Hard sorted[i] <sorted[i + 1] Easy 8i. sorted[i] <sorted[i + 1]

141 <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> Hard 8i, j. P (i, j) Easy 8i.P (i, i + 1)

142 <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> Hard 8i, j. P (i, j) Thanks Nadia!! Easy 8i.P (i, i + 1)

143 Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi

144 Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem

145 Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem sorted = foldl adder ([], false) sorted adder (es, d) next =...

146 Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem But in imperative language sorted = foldl adder ([], false) sorted adder (es, d) next =...

147 Implications Not clear need to adjust at least 1) loop translation 2) synthesis Formal correctness would help!!

150 Security is really hard