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[(Compound)-240.9(abstract)-228.9(semantic)-252.9(functions.)]TJ
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<006d>Tj
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12.5 -4.0625 TD
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-10.602 -3.875 TD
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[(ed)-325.2(some)-337.3(of)-337.3(the)-337.3(e)12(xpressions)-313.2(by)-325.2(using)-337.3(rules)-337.3(from)-313.2(Figure)-337.3(4.)-578.2(The)-325.2(abstract)-337.3(states)-349.3(in)-337.3(the)]TJ
-8.2143 -1.2045 TD
[(equations)-216.8(refer)-228.9(to)-240.9(the)-240.9(corresponding)-204.8(states)-240.9(in)-240.9(Figure)-240.9(5,)-228.9(which)-240.9(sho)24.1(ws)-240.9(the)-228.9(control)]TJ
/TT3 1 Tf
31.7975 0 TD
()Tj
/TT2 1 Tf
0.5562 0 TD
[(o)24.1(w)-240.9(graph)-228.9(of)-228.9(the)-240.9(program.)]TJ
/TT9 1 Tf
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0 0 0 1 k
/GS3 gs
(S)Tj
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(1)Tj
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( = )Tj
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/TT2 1 Tf
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( )Tj
/TT9 1 Tf
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(1)Tj
/TT2 1 Tf
0.5 0.5669 TD
( )Tj
/TT5 1 Tf
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(then)Tj
/TT2 1 Tf
2.4004 0 TD
( )Tj
/TT9 1 Tf
0.25 0 TD
(S)Tj
0.5 -0.5669 TD
(3)Tj
/TT2 1 Tf
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( )Tj
/TT5 1 Tf
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(else)Tj
/TT2 1 Tf
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( )Tj
/TT9 1 Tf
0.25 0 TD
(S)Tj
0.5 -0.5669 TD
(4)Tj
-11.9258 -1.4331 TD
(b)Tj
0.5 -0.5669 TD
(1)Tj
0.5 0.5669 TD
( = )Tj
/TT5 1 Tf
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(x = 1)Tj
/TT9 1 Tf
-2.1748 -2 TD
(S)Tj
0.5 -0.5669 TD
(3 )Tj
0.75 0.5669 TD
(= )Tj
/TT5 1 Tf
0.9248 0 TD
(z := 1)Tj
/TT9 1 Tf
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(S)Tj
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(4 )Tj
0.75 0.5669 TD
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/TT5 1 Tf
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(z := 2)Tj
/TT9 1 Tf
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(S = S)Tj
2.1748 -0.5669 TD
(1)Tj
0.5 0.5669 TD
(; S)Tj
1.083 -0.5669 TD
(2)Tj
ET
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1 i
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S
323.484 467.673 m
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f*
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S
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f*
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S
BT
7 0 0 7 331.5653 466.2506 Tm
/GS3 gs
(S)Tj
5.6 0 0 5.6 335.0653 462.2821 Tm
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(S)Tj
5.6 0 0 5.6 363.3672 462.0091 Tm
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ET
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S
383.414 467.728 m
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383.414 466.124 l
383.414 467.728 l
f*
BT
7 0 0 7 302.54 440.4398 Tm
/GS3 gs
(S)Tj
0.5 -0.5669 TD
(1)Tj
8.4 0 0 8.4 309.54 440.4398 Tm
(:)Tj
-0.6682 4.65 TD
(S:)Tj
ET
0.7 w
/GS2 gs
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S
BT
7 0 0 7 351.7459 439.9431 Tm
/GS3 gs
(S)Tj
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(3)Tj
/TT11 1 Tf
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<006d>Tj
/TT9 1 Tf
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ET
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S
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S
BT
7 0 0 7 351.396 407.7431 Tm
/GS3 gs
(S)Tj
5.6 0 0 5.6 354.896 403.7746 Tm
(4)Tj
ET
0.602 w
/GS2 gs
362.992 408.855 m
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S
BT
/TT11 1 Tf
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/GS3 gs
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/TT9 1 Tf
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(6)Tj
ET
/GS2 gs
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391.448 425.126 l
S
391.181 425.126 m
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395.189 425.126 l
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f*
BT
/TT2 1 Tf
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/GS3 gs
(tt)Tj
-0.0269 -3.7647 TD
-0.018 Tc
(ff)Tj
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0 Tc
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 304.0084 462.6515 Tm
(1)Tj
/TT11 1 Tf
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<006d>Tj
/TT9 1 Tf
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-0.074 Tc
(11)Tj
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0 Tc
(S)Tj
0.5 -0.5669 TD
(2)Tj
0.5 0.5669 TD
( = )Tj
/TT5 1 Tf
1.1748 0 TD
(if)Tj
/TT2 1 Tf
1.2002 0 TD
( )Tj
/TT9 1 Tf
0.25 0 TD
(b)Tj
0.5 -0.5669 TD
(2)Tj
/TT2 1 Tf
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( )Tj
/TT5 1 Tf
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(then)Tj
/TT2 1 Tf
2.4004 0 TD
( )Tj
/TT9 1 Tf
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(S)Tj
0.5 -0.5669 TD
(5)Tj
/TT2 1 Tf
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( )Tj
/TT5 1 Tf
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(else)Tj
/TT2 1 Tf
2.4004 0 TD
( )Tj
/TT9 1 Tf
0.25 0 TD
(S)Tj
0.5 -0.5669 TD
(6)Tj
-11.9258 -1.4331 TD
(b)Tj
0.5 -0.5669 TD
(2)Tj
0.5 0.5669 TD
( = )Tj
/TT5 1 Tf
1.1748 0 TD
(y = 1)Tj
/TT9 1 Tf
-2.1748 -2 TD
(S)Tj
0.5 -0.5669 TD
(5 )Tj
0.75 0.5669 TD
(= )Tj
/TT5 1 Tf
0.9248 0 TD
(x := 1)Tj
/TT9 1 Tf
-2.1748 -2 TD
(S)Tj
0.5 -0.5669 TD
(6 )Tj
0.75 0.5669 TD
(= )Tj
/TT5 1 Tf
0.9248 0 TD
(x := 2)Tj
/TT9 1 Tf
14.1851 4.9018 TD
(S)Tj
0.5 -0.5669 TD
(2)Tj
8.4 0 0 8.4 309.0971 378.4899 Tm
(:)Tj
ET
0.7 w
/GS2 gs
346.93 388.117 14.978 -17.294 re
S
BT
7 0 0 7 351.3029 377.9931 Tm
/GS3 gs
(S)Tj
5.6 0 0 5.6 354.8029 374.0246 Tm
(5)Tj
/TT11 1 Tf
7 0 0 7 290.9919 363.3247 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 295.2131 359.3562 Tm
(6)Tj
ET
0.602 w
/GS2 gs
300.44 364.77 m
313.899 364.405 l
S
313.632 364.415 m
313.688 366.017 l
317.638 364.275 l
313.576 362.812 l
313.632 364.415 l
f*
362.199 379.105 m
378.299 379.105 l
S
0.7 w
346.58 355.917 14.978 -17.294 re
S
BT
7 0 0 7 352.9401 345.7932 Tm
/GS3 gs
(S)Tj
5.6 0 0 5.6 354.481 341.8246 Tm
(6)Tj
ET
0.602 w
/GS2 gs
361.849 346.905 m
378.403 346.92 l
S
BT
/TT11 1 Tf
7 0 0 7 398.6501 361.7756 Tm
/GS3 gs
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 402.8713 357.8071 Tm
-0.074 Tc
(11)Tj
ET
0.7 w
/GS2 gs
378.299 379.105 m
378.403 346.92 l
S
0.602 w
378.405 363.176 m
391.005 363.176 l
S
390.738 363.176 m
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390.738 361.572 l
390.738 363.176 l
f*
BT
7 0 0 7 322.6385 362.5579 Tm
/GS3 gs
0 Tc
(b)Tj
5.6 0 0 5.6 326.1385 358.5894 Tm
(2)Tj
ET
0.7 w
/GS2 gs
325.99 372.477 m
326.165 378.683 l
325.919 355.769 m
325.964 347.018 l
S
BT
/TT2 1 Tf
5.6 0 0 5.6 330.1635 372.253 Tm
/GS3 gs
(tt)Tj
ET
0.7 w
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325.99 355.663 m
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325.99 372.477 l
334.397 364.07 l
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s
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303.587 426.118 m
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S
315.22 426.118 m
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315.22 424.514 l
315.22 426.118 l
f*
BT
/TT9 1 Tf
7 0 0 7 324.226 424.27 Tm
/GS3 gs
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5.6 0 0 5.6 327.726 420.3015 Tm
(1)Tj
ET
0.7 w
/GS2 gs
327.577 434.189 m
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327.565 417.008 m
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S
0.7 w
327.578 417.375 m
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s
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378.742 441.055 m
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S
BT
/TT11 1 Tf
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/GS3 gs
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/TT9 1 Tf
5.6 0 0 5.6 342.0866 443.0515 Tm
(2)Tj
/TT11 1 Tf
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<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 342.4366 398.864 Tm
(4)Tj
/TT11 1 Tf
7 0 0 7 364.2904 446.6263 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 368.5115 442.6578 Tm
(3)Tj
/TT11 1 Tf
7 0 0 7 364.6404 403.1387 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 368.8616 399.1702 Tm
(5)Tj
/TT11 1 Tf
7 0 0 7 337.4279 384.5231 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 341.6491 380.5546 Tm
(7)Tj
/TT11 1 Tf
7 0 0 7 337.7779 340.3356 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 341.9991 336.3671 Tm
(9)Tj
/TT11 1 Tf
7 0 0 7 363.8529 384.1294 Tm
<006d>Tj
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(8)Tj
/TT11 1 Tf
7 0 0 7 364.2029 339.9418 Tm
<006d>Tj
/TT9 1 Tf
5.6 0 0 5.6 368.4241 335.9733 Tm
(10)Tj
/TT2 1 Tf
-12.1682 23.9048 TD
(~)Tj
16.3801 0.0625 TD
(~)Tj
1.3125 -7.75 TD
(~)Tj
-18.6875 0.2812 TD
(~)Tj
7.8125 3.9062 TD
4.209 Tc
(~~)Tj
0.0625 -7.9375 TD
0 Tc
(~)Tj
4.625 0.0625 TD
(~)Tj
-4.75 -3.3125 TD
(~)Tj
4.6875 -0.0625 TD
(~)Tj
6.1875 -4 TD
(~)Tj
-6.1875 -3.875 TD
(~)Tj
-4.6875 0.0625 TD
(~)Tj
-8.0625 4.1875 TD
(~)Tj
ET
0.602 w
/GS2 gs
327.74 441.245 m
343.84 441.245 l
S
343.573 441.245 m
343.573 442.849 l
347.582 441.245 l
343.573 439.641 l
343.573 441.245 l
f*
328.09 408.695 m
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S
343.923 408.695 m
343.923 410.299 l
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343.923 408.695 l
f*
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343.14 379.295 l
S
342.873 379.295 m
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f*
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S
342.523 346.745 m
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f*
BT
9.9626 0 0 9.9626 205.8 302.4 Tm
0 g
/GS1 gs
[(Figure)-240.9(5:)-301.1(Control)]TJ
/TT3 1 Tf
7.2152 0 TD
()Tj
/TT2 1 Tf
0.5562 0 TD
[(o)24.1(w)-252.9(graphs)-228.9(for)-228.9(the)-252.9(e)12(xample)]TJ
/TT7 1 Tf
-19.8886 -3.6737 TD
[(Solution)-265(of)-265(the)-277(intr)12(oductory)-240.9(example.)]TJ
/TT2 1 Tf
16.9952 0 TD
[(The)-265(Jacobi)-265(iteration)-265(of)-265(the)-265(system)-277(of)]TJ
/TT3 1 Tf
14.9739 0 TD
()Tj
/TT2 1 Tf
0.5562 0 TD
[(o)24.1(w)-265(equations)-252.9(in)-277(Figure)-252.9(6)]TJ
-32.5253 -1.2045 TD
[(will)-252.9(reach)-228.9(a)]TJ
/TT3 1 Tf
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()Tj
/TT2 1 Tf
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[(xpoint)-228.9(after)-240.9(less)-252.9(that)-240.9(10)-240.9(iterations.)-301.1(The)-228.9(abstract)-240.9(states)-265(for)-228.9(tw)12(o)-252.9(interesting)-228.9(program)-216.8(points)-240.9(are:)]TJ
/F4 1 Tf
5.6505 -2.1922 TD
()Tj
/F6 1 Tf
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()Tj
/F3 1 Tf
6.9738 0 0 6.9738 200.88 230.52 Tm
(6)Tj
/F4 1 Tf
9.9626 0 0 9.9626 208.08 231.96 Tm
0.2822 Tc
(=[)Tj
/F5 1 Tf
1.337 0 TD
0 Tc
(x)Tj
/F1 1 Tf
0.807 0 TD
()Tj
T*
()Tj
/F7 1 Tf
1.2768 0 TD
-0.0001 Tc
(ID)Tj
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1.2286 0 TD
0 Tc
(,)Tj
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(y)Tj
/F1 1 Tf
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()Tj
T*
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0.0017 Tc
(NID)Tj
/F6 1 Tf
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0 Tc
(,)Tj
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(z)Tj
/F1 1 Tf
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()Tj
T*
()Tj
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-0.0001 Tc
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0 Tc
(,)Tj
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0.005 Tc
(flow)Tj
/F1 1 Tf
2.3849 0 TD
0 Tc
()Tj
T*
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0 Tc
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()Tj
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(11)Tj
/F4 1 Tf
9.9626 0 0 9.9626 206.4 210 Tm
0.2822 Tc
(=[)Tj
/F5 1 Tf
1.337 0 TD
0 Tc
(x)Tj
/F1 1 Tf
0.795 0 TD
()Tj
T*
()Tj
/F7 1 Tf
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0.0017 Tc
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1.9754 0 TD
0 Tc
(,)Tj
/F5 1 Tf
0.4457 0 TD
(y)Tj
/F1 1 Tf
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()Tj
T*
()Tj
/F7 1 Tf
1.2768 0 TD
0.0017 Tc
(NID)Tj
/F6 1 Tf
1.9874 0 TD
0 Tc
(,)Tj
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0.4457 0 TD
(z)Tj
/F1 1 Tf
0.795 0 TD
()Tj
T*
()Tj
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1.2768 0 TD
-0.0001 Tc
(ID)Tj
/F6 1 Tf
1.2407 0 TD
0 Tc
(,)Tj
/F5 1 Tf
0.4457 0 TD
0.005 Tc
(flow)Tj
/F1 1 Tf
2.3729 0 TD
0 Tc
()Tj
T*
()Tj
/F7 1 Tf
1.2768 0 TD
0.0017 Tc
(NID)Tj
/F4 1 Tf
1.9754 0 TD
0 Tc
(])Tj
/TT2 1 Tf
-31.9074 -1.7947 TD
[(The)-204.8(result)-204.8(of)-204.8(the)-216.8(analysis)-204.8(of)-204.8(the)]TJ
/TT3 1 Tf
12.5649 0 TD
()Tj
/TT2 1 Tf
0.5562 0 TD
[(rst)-216.8(if-statement)-204.8(\()]TJ
/F4 1 Tf
6.5786 0 TD
()Tj
/F6 1 Tf
-0.0482 0 TD
()Tj
/F3 1 Tf
6.9738 0 0 6.9738 287.28 190.68 Tm
(6)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 291.72 192.12 Tm
[(\))-216.8(i)0(s)-216.8(that)]TJ
/TT5 1 Tf
3.1499 0 TD
(z)Tj
/TT2 1 Tf
0.8049 0 TD
[(is)-228.9(set)-216.8(to)]TJ
/TT9 1 Tf
3.2181 0 TD
(ID)Tj
/TT2 1 Tf
1.0552 0 TD
[(.)-204.8(The)-204.8(reason)-204.8(for)-192.7(this)-216.8(is)-228.9(that)-204.8(the)]TJ
/TT3 1 Tf
12.0878 0 TD
()Tj
/TT2 1 Tf
0.5562 0 TD
(rst)Tj
-41.6136 -1.2045 TD
(condition)Tj
/TT5 1 Tf
4.0062 0 TD
[(if)-602.3(x)-590.2(=)-602.3(1)]TJ
/TT2 1 Tf
5.0602 0 TD
[(is)-265(input)-240.9(dependent)-228.9(\(since)]TJ
/TT5 1 Tf
10.2195 0 TD
(x)Tj
/TT2 1 Tf
0.853 0 TD
[(is\).)-349.3(Therefore,)]TJ
/TT5 1 Tf
6.0196 0 TD
(z)Tj
/TT2 1 Tf
0.8651 0 TD
[(is)-265(re)12(garded)-228.9(as)-265(input)-240.9(dependent)-228.9(\(e)24.1(v)12(e)0(n)-240.9(i)0(f)]TJ
/TT5 1 Tf
-27.0235 -1.2045 TD
(z)Tj
/TT2 1 Tf
0.841 0 TD
[(is)-252.9(assigned)-240.9(constants)-240.9(in)-240.9(both)-240.9(edges!\).)]TJ
0.6526 -1.1925 TD
[(The)-240.9(result)-252.9(of)-240.9(the)-252.9(analysis)-240.9(of)-240.9(the)-252.9(entire)-240.9(e)12(xample)-228.9(\()]TJ
/F4 1 Tf
19.3323 0 TD
()Tj
/F6 1 Tf
-0.0482 0 TD
()Tj
/F3 1 Tf
6.9738 0 0 6.9738 298.68 154.8 Tm
-0.0016 Tc
(11)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 307.08 156.24 Tm
0 Tc
[(\))-252.9(i)0(s)-265(that)]TJ
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3.2583 0 TD
(z)Tj
/TT2 1 Tf
0.853 0 TD
[(is)-252.9(set)-265(to)]TJ
/TT9 1 Tf
3.3145 0 TD
(ID)Tj
/TT2 1 Tf
1.3081 0 TD
[(\(as)-252.9(described)-228.9(abo)12(v)12(e)0(\))-228.9(and)-228.9(that)]TJ
/TT5 1 Tf
-31.0173 -1.2045 TD
(x)Tj
/TT2 1 Tf
0.9253 0 TD
[(is)-325.2(set)-337.3(to)]TJ
/TT9 1 Tf
3.5434 0 TD
(NID)Tj
/TT2 1 Tf
2.0474 0 TD
[(due)-313.2(to)-325.2(the)-325.2(second)-301.1(condition)]TJ
/TT5 1 Tf
11.5634 0 TD
[(if)-602.3(y)-590.2(=)-602.2(1)]TJ
/TT2 1 Tf
5.1204 0 TD
[(is)-337.3(not)-313.2(input)-313.2(dependent)-289.1(\(since)]TJ
/TT5 1 Tf
12.0755 0 TD
(y)Tj
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[(b)24.1(u)0(t)-265(not)-252.9(both.)-361.4(The)-265(second)-252.9(statement)-265(in)-265(2)-265(also)-265(follo)24.1(ws)-265(quite)-265(directly)60.2(,)-252.9(from)]TJ
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[(,)-337.3(which)-301.1(pro)12(v)12(e)0(s)-301.1(the)]TJ
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[(In)-493.8(Case)-505.9(5,)-542(we)-505.9(ha)24.1(v)12(e)-481.8(the)-493.8(cases)]TJ
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[(In)-240.9(an)12(y)-228.9(case)-252.9(holds)-228.9(that)]TJ
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(.)Tj
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[(Finally)60.2(,)-361.3(i)0(n)-337.3(Case)-349.3(6,)-361.3(we)-349.3(use)]TJ
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[(x)12(e)0(d)-325.2(point)-337.3(induction)-313.2(in)-349.3(the)-337.3(same)-337.3(manner)-325.2(as)-349.3(in)-337.3(the)-337.3(proof)-325.2(of)-337.3(Theorem)-313.2(9.1)]TJ
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[(\(e)12(xcept)-277(that)-301.1(this)-313.2(proof)-277(concerns)-289.1(the)-301.1(standard)-289.1(semantics,)-313.2(not)-301.1(the)-301.1(trace)-301.1(semantics\).)-469.8(Denote)-289.1(the)-301.1(partial)-301.1(state)]TJ
0 -1.1925 TD
[(transition)-349.3(functions)-337.3(successi)24.1(v)12(ely)-361.4(approximating)]TJ
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[(.W)746.8(e)289.1(h)662.5(a)686.6(v)674.5(e)]TJ
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[()24.2(.)]TJ
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[(ned)-204.8(for)-216.8(an)12(y)-228.9(state\),)-228.9(and)]TJ
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[()24.2(.)]TJ
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[(.)-301.1(Assume)-240.9(that)]TJ
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[(kil)-49.5(l)]TJ
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[(Denote)-240.9(the)-240.9(complement)-216.8(of)-240.9(the)-240.9(set)]TJ
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[(Lemma)-240.9(9.1)]TJ
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[(F)108.4(o)0(r)-433.6(all)-433.6(pr)48.2(o)12(g)0(r)12(ams)]TJ
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[(,)-481.8(and)-421.6(states)]TJ
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[(ned,)-457.7(holds)-421.6(that)]TJ
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[(ki)13.1(l)-45.6(l)]TJ
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[(ki)13.1(l)-45.7(l)]TJ
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[(Pr)48.2(oof)12(.)]TJ
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[(By)-252.9(induction)-216.8(o)12(v)12(e)0(r)-228.9(program)-216.8(structure.)]TJ
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(1)Tj
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0.2281 Tc
[(]:)228.8(=)]TJ
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(2)Tj
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[(:)-337.3(immediate.)-337.3(\(In)-252.9(the)-252.9(tw)12(o)-265(latter)-265(cases,)-265(since)-265(no)-252.9(aliasing)]TJ
-18.4168 -1.1925 TD
[(of)-240.9(v)24.1(ariables)-240.9(can)-240.9(occur)36.1(,)-216.8(n)0(o)-240.9(other)-228.9(v)24.1(ariable)-228.9(than)]TJ
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[(can)-240.9(be)-240.9(touched)-228.9(by)-240.9(the)-240.9(respecti)24.1(v)12(e)-228.9(assignment.\))]TJ
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(;)Tj
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[(:)-277(assume)-192.7(true)-180.7(for)]TJ
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[(.)-289.1(W)84.3(e)-192.7(then)-180.7(ha)24.1(v)12(e)]TJ
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[(ki)13.1(l)-45.6(l)]TJ
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[(ki)13.1(l)-45.6(l)]TJ
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[(,)-204.8(which)-180.7(im-)]TJ
-35.7014 -1.2045 TD
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[(ki)13.1(l)-45.7(l)]TJ
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[(ki)13.1(l)-45.6(l)]TJ
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[(ki)13.1(l)-45.7(l)]TJ
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[(.)-566.1(W)84.3(e)-349.3(also)-325.2(ha)24.1(v)12(e)]TJ
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[(ki)13.1(l)-45.7(l)]TJ
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[(]])-168.7(\()]TJ
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(]])Tj
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(\)\))Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(\)\))Tj
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[(,)-2360.8(which)-1927.2(implies)]TJ
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(\)\))Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(2)Tj
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(\)\))Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(1)Tj
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(\)\))Tj
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[(.)-1132.2(Since)]TJ
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(C)Tj
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(\()Tj
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(\))Tj
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()Tj
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(\()Tj
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(Y)Tj
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(\)=)Tj
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(X)Tj
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()Tj
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(\))Tj
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[(it)-530(follo)24.1(ws)-505.9(that)]TJ
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()Tj
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(|)Tj
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(C)Tj
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(\()Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(1)Tj
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(\))Tj
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[(ki)13.1(l)-45.7(l)]TJ
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4.9813 0 0 4.9813 180 343.8 Tm
(2)Tj
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0.001 Tc
(\)\))Tj
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9.9626 0 0 9.9626 193.32 346.56 Tm
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(=\()Tj
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(S)Tj
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([[)Tj
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(2)Tj
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-0.1694 Tc
[(]])-168.7(\()]TJ
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(S)Tj
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(1)Tj
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-0.1694 Tc
(]])Tj
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()Tj
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(\)\))Tj
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(|)Tj
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(C)Tj
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(\()Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(\()Tj
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(1)Tj
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(\))Tj
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[(ki)13.1(l)-45.7(l)]TJ
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(2)Tj
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(\)\))Tj
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()Tj
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(then)Tj
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(1)Tj
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(else)Tj
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(2)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 285.12 327 Tm
1.8068 Tc
[(:I)1806.8(f)]TJ
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3.7503 0 TD
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(B)Tj
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([[)Tj
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(b)Tj
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(]])Tj
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()Tj
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(=)Tj
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0.005 Tc
(tt)Tj
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[(,)-1180.4(then)]TJ
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()Tj
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(|)Tj
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6.9738 0 0 6.9738 446.16 325.2 Tm
(C)Tj
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(\()Tj
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0.0062 Tc
[(ki)13.1(l)-45.6(l)]TJ
/F3 1 Tf
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(\()Tj
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(S)Tj
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(1)Tj
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(\)\))Tj
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(=)Tj
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(\()Tj
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(S)Tj
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([[)Tj
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(if)Tj
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(b)Tj
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(then)Tj
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(1)Tj
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(else)Tj
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(2)Tj
/F4 1 Tf
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-0.1694 Tc
(]])Tj
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()Tj
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(\))Tj
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(|)Tj
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(C)Tj
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(\()Tj
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0.0062 Tc
[(ki)13.1(l)-45.6(l)]TJ
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(\()Tj
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(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 288.48 312.24 Tm
(1)Tj
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(\)\))Tj
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9.9626 0 0 9.9626 299.04 315 Tm
1.9995 Tc
[(.I)1999.5(f)]TJ
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(B)Tj
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([[)Tj
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(b)Tj
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(]])Tj
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()Tj
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(=)Tj
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()Tj
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0.8311 0 TD
[(,)-951.6(then)]TJ
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()Tj
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(|)Tj
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(C)Tj
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(\()Tj
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0.0062 Tc
[(ki)13.1(l)-45.7(l)]TJ
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(\()Tj
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(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 478.68 312.24 Tm
(1)Tj
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0.001 Tc
(\)\))Tj
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9.9626 0 0 9.9626 502.44 315 Tm
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(=)Tj
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(\()Tj
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(S)Tj
/F4 1 Tf
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([[)Tj
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(if)Tj
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(b)Tj
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(then)Tj
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(S)Tj
/F3 1 Tf
6.9738 0 0 6.9738 179.52 301.68 Tm
(1)Tj
/F5 1 Tf
9.9626 0 0 9.9626 188.4 303.12 Tm
0.005 Tc
(else)Tj
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(S)Tj
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6.9738 0 0 6.9738 219.84 301.68 Tm
(2)Tj
/F4 1 Tf
9.9626 0 0 9.9626 224.28 303.12 Tm
-0.1694 Tc
(]])Tj
/F6 1 Tf
0.3854 0 TD
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()Tj
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(\))Tj
/F1 1 Tf
0.3854 0 TD
(|)Tj
/F16 1 Tf
6.9738 0 0 6.9738 240.72 301.32 Tm
(C)Tj
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(\()Tj
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0.0062 Tc
[(ki)13.1(l)-45.7(l)]TJ
/F3 1 Tf
1.7036 0 TD
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(\()Tj
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(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 270 300.36 Tm
(2)Tj
/F3 1 Tf
6.9738 0 0 6.9738 273.84 301.32 Tm
0.001 Tc
(\)\))Tj
/TT2 1 Tf
9.9626 0 0 9.9626 280.56 303.12 Tm
0 Tc
[(.)-879.3(I)0(n)-433.6(a)0(n)12(y)-433.6(case,)-481.8(it)-445.7(holds)-433.6(that)]TJ
/F6 1 Tf
12.2489 0 TD
()Tj
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0.6023 0 TD
(|)Tj
/F16 1 Tf
6.9738 0 0 6.9738 411.84 301.32 Tm
(C)Tj
/F3 1 Tf
0.8948 0 TD
(\()Tj
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0.4474 0 TD
0.0062 Tc
[(ki)13.1(l)-45.7(l)]TJ
/F3 1 Tf
1.6864 0 TD
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(\()Tj
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0.4474 0 TD
(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 441 300.36 Tm
(1)Tj
/F3 1 Tf
6.9738 0 0 6.9738 444.84 301.32 Tm
0.001 Tc
(\)\))Tj
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()Tj
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(C)Tj
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(\()Tj
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0.4474 0 TD
0.0062 Tc
[(ki)13.1(l)-45.6(l)]TJ
/F3 1 Tf
1.6692 0 TD
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(\()Tj
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0.4474 0 TD
(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 485.52 300.36 Tm
(2)Tj
/F3 1 Tf
6.9738 0 0 6.9738 489.48 301.32 Tm
0.001 Tc
(\)\))Tj
/F4 1 Tf
9.9626 0 0 9.9626 502.44 303.12 Tm
0 Tc
(=)Tj
-39.3992 -1.2045 TD
(\()Tj
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(S)Tj
/F4 1 Tf
0.6745 0 TD
-0.1694 Tc
([[)Tj
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0.005 Tc
(if)Tj
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(b)Tj
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0.005 Tc
(then)Tj
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3.2281 0 TD
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(S)Tj
/F3 1 Tf
6.9738 0 0 6.9738 199.92 289.68 Tm
(1)Tj
/F5 1 Tf
9.9626 0 0 9.9626 215.52 291.12 Tm
0.005 Tc
(else)Tj
/F6 1 Tf
3.2281 0 TD
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(S)Tj
/F3 1 Tf
6.9738 0 0 6.9738 253.8 289.68 Tm
(2)Tj
/F4 1 Tf
9.9626 0 0 9.9626 258.24 291.12 Tm
-0.1694 Tc
(]])Tj
/F6 1 Tf
0.3854 0 TD
0 Tc
()Tj
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0.6023 0 TD
(\))Tj
/F1 1 Tf
0.3854 0 TD
(|)Tj
/F16 1 Tf
6.9738 0 0 6.9738 274.68 289.32 Tm
(C)Tj
/F3 1 Tf
0.8948 0 TD
(\()Tj
/F15 1 Tf
0.4474 0 TD
0.0062 Tc
[(ki)13.1(l)-45.7(l)]TJ
/F3 1 Tf
1.7036 0 TD
0 Tc
(\()Tj
/F16 1 Tf
0.4474 0 TD
(S)Tj
/F23 1 Tf
4.9813 0 0 4.9813 303.96 288.36 Tm
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[(X<)69.7(S)25.6(>)69.7(X)]TJ
/F1 1 Tf
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[(kil)-49.5(l)]TJ
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[(Pr)48.2(oof)12(.)]TJ
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[(.)-433.6(Similarly)60.2(,)]TJ
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[(ki)13.1(l)-45.7(l)]TJ
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[(=\()357.9(\()]TJ
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()Tj
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(X)Tj
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[(ki)13.1(l)-45.6(l)]TJ
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0 Tc
[(.)-445.7(When)]TJ
/F6 1 Tf
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()Tj
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(|)Tj
/F16 1 Tf
6.9738 0 0 6.9738 93.84 646.92 Tm
(X)Tj
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(=)Tj
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()Tj
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(X)Tj
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[(we)-337.3(obtain)]TJ
/F4 1 Tf
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([[)Tj
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[(ki)13.1(l)-45.7(l)]TJ
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[(=\()442.2(\()]TJ
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()Tj
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[(ki)13.1(l)-45.7(l)]TJ
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[(,)-361.4(that)-325.2(is:)]TJ
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(X)Tj
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[(ki)13.1(l)-45.7(l)]TJ
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()Tj
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[(ki)13.1(l)-45.7(l)]TJ
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/TT2 1 Tf
9.9626 0 0 9.9626 160.56 636.48 Tm
(.)Tj
ET
q
7.08 0 0 -7.08 503.268 643.548 cm
/Im1 Do
Q
BT
9.9626 0 0 9.9626 99.96 612.48 Tm
[(Let)-349.3(us)-349.3(no)24.1(w)-337.3(turn)-337.3(to)-349.3(the)-349.3(abstract)-337.3(semantics)-349.3(for)-337.3(the)-349.3(While)-349.3(language.)-590.2(The)-337.3(abstract)-349.3(states)-349.3(map)-349.3(program)]TJ
-1.4936 -1.1925 TD
[(v)24.1(ariables)-421.6(to)]TJ
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[(.)-867.2(F)12(or)-421.6(an)12(y)-421.6(abstract)-421.6(state)]TJ
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()Tj
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[(we)-433.6(de)]TJ
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[(ne)-433.6(the)-421.6(set)-445.7(of)-421.6(data-independent)-385.4(v)24.1(ariables)-421.6(as)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(.)-337.3(Abstract)-252.9(states)-265(can)-240.9(be)-252.9(represented)-228.9(by)-252.9(such)-252.9(sets)-265(of)-240.9(v)24.1(ariables,)-240.9(and)-252.9(all)-252.9(oper)24.1(-)]TJ
-12.8883 -1.2045 TD
[(ations)-228.9(on)-228.9(abstract)-228.9(states)-240.9(carry)-228.9(o)12(v)12(e)0(r)-204.8(t)0(o)-240.9(set)-240.9(operations.)-277(F)12(o)0(r)-228.9(instance,)]TJ
/F7 1 Tf
26.3115 0 TD
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[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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()Tj
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6.9738 0 0 6.9738 378.36 575.16 Tm
(1)Tj
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(2)Tj
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(\)=)Tj
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[(nv)6.2(a)9.1(r)]TJ
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()Tj
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(1)Tj
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(\))Tj
/F1 1 Tf
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()Tj
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[(nv)6.2(a)9.1(r)]TJ
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(\()Tj
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()Tj
/F3 1 Tf
6.9738 0 0 6.9738 499.32 575.16 Tm
(2)Tj
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9.9626 0 0 9.9626 503.88 576.6 Tm
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-42.4225 -1.1925 TD
[(Also,)-313.2(rather)-289.1(than)-301.1(computing)-277(a)-301.1(least)]TJ
/TT3 1 Tf
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()Tj
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[(x)12(e)0(d)-289.1(point)-301.1(for)-289.1(abstract)-301.1(states,)-325.2(one)-289.1(can)-301.1(compute)-289.1(a)-301.1(greatest)]TJ
/TT3 1 Tf
23.0498 0 TD
()Tj
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[(x)12(e)0(d)-289.1(point)]TJ
-38.7029 -1.2045 TD
[(\(w)60.2(.r)60.2(.t.)-228.9(set)-252.9(inclusion\))-228.9(for)-228.9(the)-252.9(v)24.1(ariable)-228.9(sets.)]TJ
/TT7 1 Tf
0 -2.2042 TD
[(Lemma)-240.9(9.2)]TJ
/TT9 1 Tf
5.2616 0 TD
[(F)108.4(o)0(r)-192.7(all)-204.8(pr)48.2(o)12(g)0(r)12(ams)]TJ
/F6 1 Tf
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(S)Tj
/TT9 1 Tf
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[(that)-192.7(terminate)-192.7(for)-192.7(all)-204.8(inputs,)-192.7(and)-192.7(abstr)12(act)-192.7(states)]TJ
/F4 1 Tf
18.9633 0 TD
()Tj
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()Tj
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[(suc)12(h)-192.7(that)]TJ
/F4 1 Tf
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()Tj
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()Tj
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/F7 1 Tf
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0.0024 Tc
[(o)7.6(w)]TJ
/F4 1 Tf
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(\)=)Tj
/F7 1 Tf
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(ID)Tj
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(,)Tj
-42.2978 -1.1925 TD
[(holds)-228.9(that)]TJ
/F4 1 Tf
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0.6497 0.2529 TD
()Tj
/F1 1 Tf
-0.2409 -0.2529 TD
(S)Tj
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([[)Tj
/F6 1 Tf
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(S)Tj
/F4 1 Tf
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[(]])-216.8()]TJ
/F6 1 Tf
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()Tj
/F4 1 Tf
0.6143 0 TD
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(\)\()Tj
/F7 1 Tf
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[(o)7.6(w)]TJ
/F4 1 Tf
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(\)=)Tj
/F7 1 Tf
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(ID)Tj
/TT9 1 Tf
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(and)Tj
/F7 1 Tf
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[(nv)6.2(a)9.1(r)]TJ
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[(]])-216.8()]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(]])-216.8()]TJ
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[(as)-325.2(soon)-313.2(as)]TJ
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[(o)7.6(w)]TJ
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[(.)-517.9(I)0(t)-325.2(i)0(s)-325.2(then)-313.2(easy)-313.2(to)-313.2(see)-325.2(that)-325.2(the)]TJ
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[(]])-168.7(\()]TJ
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[(]])-216.8()]TJ
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[(yields)-240.9(a)-252.9(contradiction.\))]TJ
-20.1153 -1.2045 TD
[(The)-373.4(second)-361.3(statement)-361.4(holds)-361.4(tri)24.1(vially)-373.4(for)]TJ
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(1)Tj
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(2)Tj
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[(,)-421.6(noticing)-361.4(that)-385.4(by)-373.4(induction)]TJ
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[(o)7.6(w)]TJ
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(1)Tj
/F4 1 Tf
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-0.1694 Tc
[(]])-216.8()]TJ
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[(o)7.6(w)]TJ
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[(.)-542(F)12(or)-313.2(if-then-else,)-325.2(the)-325.2(result)-313.2(again)-313.2(follo)24.1(ws)-313.2(by)-313.2(a)-325.2(simple)-325.2(induction)-301.1(on)]TJ
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(1)Tj
/TT2 1 Tf
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(and)Tj
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(S)Tj
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(2)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 507.72 437.16 Tm
(.)Tj
-42.4225 -1.2045 TD
[(F)12(o)0(r)-265(while,)]TJ
/TT3 1 Tf
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()Tj
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[(nally)60.2(,)-277(w)0(e)-277(must)-289.1(ha)24.1(v)12(e)-265(the)-277(case)]TJ
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()Tj
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(B)Tj
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([[)Tj
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/F4 1 Tf
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[(]])-168.7(\()]TJ
0.9997 0.2529 TD
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()Tj
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([[)Tj
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[(]])-216.8()]TJ
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(ID)Tj
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[(:)-373.4(then)]TJ
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()Tj
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([[)Tj
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[(]])-216.8()]TJ
/F6 1 Tf
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()Tj
/TT2 1 Tf
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[(is)-301.1(the)]TJ
-40.4954 -1.1925 TD
[(least)-325.2(solution)-301.1(to)-325.2(the)-313.2(equation)]TJ
/F6 1 Tf
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(w)Tj
/F4 1 Tf
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0.4147 Tc
[(=\()370()]TJ
/F6 1 Tf
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()Tj
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()Tj
/F4 1 Tf
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()Tj
/F1 1 Tf
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([[)Tj
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(]])Tj
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([)Tj
/F7 1 Tf
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[(o)7.6(w)]TJ
/F1 1 Tf
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()Tj
T*
()Tj
/F7 1 Tf
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-0.0001 Tc
(ID)Tj
/F4 1 Tf
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-0.0021 Tc
(]\)[)Tj
/F6 1 Tf
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[(fl)85.9(o)115.6(w)]TJ
/F1 1 Tf
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()Tj
T*
()Tj
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(w)Tj
/F4 1 Tf
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[(o)7.6(w)]TJ
/F4 1 Tf
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(\)])Tj
/TT2 1 Tf
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[(.)-530(This)-325.2(equation)-289.1(can)-325.2(be)]TJ
-33.3648 -1.2045 TD
(simpli)Tj
/TT3 1 Tf
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()Tj
/TT2 1 Tf
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[(ed)-325.2(to)]TJ
/F6 1 Tf
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(w)Tj
/F4 1 Tf
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(=)Tj
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()Tj
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()Tj
/F4 1 Tf
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()Tj
/F1 1 Tf
-0.2289 -0.2529 TD
(S)Tj
/F4 1 Tf
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([[)Tj
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-0.1694 Tc
(]])Tj
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[(.)-590.2(The)-325.2(corresponding)-289.1(equation)-313.2(for)-325.2(v)24.1(ariable)-325.2(sets)-337.3(is)]TJ
/F7 1 Tf
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0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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/F4 1 Tf
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(\)=)Tj
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[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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()Tj
/F7 1 Tf
-41.9012 -1.1925 TD
0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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()Tj
/F1 1 Tf
-0.2289 -0.2529 TD
(S)Tj
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([[)Tj
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(S)Tj
/F4 1 Tf
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-0.1694 Tc
(]])Tj
/F6 1 Tf
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(w)Tj
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(\))Tj
/TT2 1 Tf
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[(:)-469.8(the)-313.2(greatest)]TJ
/TT3 1 Tf
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()Tj
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[(x)12(e)0(d)-301.1(point)-313.2(for)-301.1(this)-325.2(equation)-289.1(corresponds)-289.1(to)-325.2(the)-313.2(least)]TJ
/TT3 1 Tf
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()Tj
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[(x)12(e)0(d)-301.1(point)-301.1(of)-313.2(the)-313.2(origi-)]TJ
-33.5027 -1.2045 TD
[(nal)-313.2(equation.)-505.9(By)-325.2(induction)-289.1(on)]TJ
/F6 1 Tf
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(S)Tj
/TT2 1 Tf
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[(,w)337.3(e)24.1(h)337.3(a)361.4(v)349.3(e)]TJ
/F7 1 Tf
4.2312 0 TD
0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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()Tj
/F1 1 Tf
-0.2409 -0.2529 TD
(S)Tj
/F4 1 Tf
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([[)Tj
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(]])Tj
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/F4 1 Tf
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(\)=)Tj
/F7 1 Tf
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0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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(w)Tj
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(\))Tj
/F1 1 Tf
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(\\)Tj
/F7 1 Tf
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-0.0039 Tc
[(kil)-49.5(l)]TJ
/F4 1 Tf
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(\()Tj
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(S)Tj
/F4 1 Tf
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(\))Tj
/TT2 1 Tf
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[(.)-530(W)84.3(e)-325.2(obtain)-313.2(the)-313.2(equation)]TJ
/F7 1 Tf
-32.4494 -1.2045 TD
0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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(w)Tj
/F4 1 Tf
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0.3579 Tc
(\)=)Tj
/F7 1 Tf
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[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
/F6 1 Tf
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0 Tc
()Tj
/F4 1 Tf
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(\))Tj
/F1 1 Tf
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()Tj
/F4 1 Tf
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(\()Tj
/F7 1 Tf
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0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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(w)Tj
/F4 1 Tf
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(\))Tj
/F1 1 Tf
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(\\)Tj
/F7 1 Tf
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[(kil)-49.5(l)]TJ
/F4 1 Tf
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(S)Tj
/F4 1 Tf
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(\)\))Tj
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[(.)-421.6(I)0(t)-289.1(i)0(s)-289.1(easy)-289.1(to)-277(see)-289.1(that)-277(the)-289.1(greatest)]TJ
/TT3 1 Tf
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()Tj
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[(x)12(e)0(d)-265(point)-277(to)-277(this)-289.1(equation)]TJ
-32.3432 -1.1925 TD
[(indeed)-228.9(is)]TJ
/F7 1 Tf
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0.0039 Tc
[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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/F4 1 Tf
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(\)=)Tj
/F7 1 Tf
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[(nv)6.2(a)9.1(r)]TJ
/F4 1 Tf
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(\()Tj
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()Tj
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(\))Tj
/F1 1 Tf
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(\\)Tj
/F7 1 Tf
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[(kil)-49.5(l)]TJ
/F4 1 Tf
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/F4 1 Tf
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(.)Tj
ET
q
7.08 0 0 -7.08 503.268 360.588 cm
/Im1 Do
Q
BT
9.9626 0 0 9.9626 99.96 329.52 Tm
[(W)84.3(e)-325.2(are)]TJ
/TT3 1 Tf
3.1504 0 TD
()Tj
/TT2 1 Tf
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[(nally)-301.1(in)-301.1(a)-313.2(position)-289.1(to)-301.1(state)-313.2(our)-289.1(main)-301.1(result,)-313.2(which)-301.1(links)-301.1(the)-313.2(abstract)-289.1(v)24.1(alues)-301.1(calculated)-301.1(by)-289.1(our)]TJ
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[(analysis)-301.1(to)-313.2(statements)-313.2(that)-301.1(can)-313.2(be)-301.1(used,)-325.2(through)-277(Theorem)-289.1(9.1,)-313.2(to)-313.2(pro)12(v)12(e)-289.1(input)-301.1(data)-301.1(independence)-277(of)-301.1(state-)]TJ
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[(Theor)12(em)-240.9(9.2)]TJ
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[(F)108.4(o)0(r)-445.7(all)-445.7(abstr)12(act)-433.6(states)]TJ
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[(and)-421.6(terminating)-433.6(pr)48.2(o)12(g)0(r)12(ams)]TJ
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[(,)-493.8(i)0(t)-445.7(holds)-433.6(that)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[()]TJ
/F7 1 Tf
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[(nv)6.2(a)9.1(r)]TJ
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()Tj
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[(]])-216.8()]TJ
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-5.5648 -2.2042 TD
[(Pr)48.2(oof)12(.)]TJ
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[(Note)-252.9(that)-265(by)-252.9(Proposition)-240.9(3)-265(and)-252.9(Lemma)-240.9(9.2,)-252.9(we)-265(ha)24.1(v)12(e)-252.9(already)-240.9(pro)12(v)12(e)0(d)-228.9(the)-265(case)-265(when)]TJ
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[(o)7.6(w)]TJ
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(\)=)Tj
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[(Thus,)-228.9(assume)]TJ
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[(o)7.6(w)]TJ
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[(in)-252.9(the)-240.9(sequel.)]TJ
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[(The)-385.4(proof)-361.3(is)-397.5(by)-385.4(structural)-373.4(induction)-361.3(o)12(v)12(e)0(r)-373.4(programs.)]TJ
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[(is)-397.5(tri)24.1(vial.)-722.7(F)12(o)0(r)-385.4(assignments)]TJ
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[(,i)421.6(f)]TJ
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[(]])-216.8()]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(]])-216.8()]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(.)-493.8(The)-301.1(result)-301.1(then)]TJ
-35.81 -1.1925 TD
[(follo)24.1(ws)-289.1(from)-277(Proposition)-277(2.)-457.7(Otherwise,)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(]])-216.8()]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(,)-313.2(and)-289.1(the)-289.1(result)-301.1(follo)24.1(ws)]TJ
-33.3407 -1.2045 TD
[(from)-228.9(Proposition)-216.8(3.)-301.1(The)-240.9(case)-252.9(for)-240.9(array)-228.9(assignment)-228.9(is)-252.9(similar)-252.9(to)-240.9(the)-240.9(case)-252.9(for)-228.9(ordinary)-216.8(assignments.)]TJ
1.4936 -1.2045 TD
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[(Fo)-12(r)]TJ
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(1)Tj
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(;)Tj
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(2)Tj
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[(,)-566.1(assume)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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([[)Tj
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(1)Tj
/F4 1 Tf
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-0.1694 Tc
[(]])-216.8()]TJ
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(and)Tj
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[(nv)6.2(a)9.1(r)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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([[)Tj
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(2)Tj
/F4 1 Tf
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-0.1694 Tc
[(]])-216.8()]TJ
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[(for)-421.6(all)]TJ
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[(.)-867.2(W)36.1(ith)]TJ
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[(replaced)-409.5(by)]TJ
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(S)Tj
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([[)Tj
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(1)Tj
/F4 1 Tf
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[(]])-216.8()]TJ
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[(in)-421.6(the)-433.6(second)-421.6(statement,)-469.8(we)-433.6(obtain)]TJ
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[(nv)6.2(a)9.1(r)]TJ
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(S)Tj
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([[)Tj
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(S)Tj
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(1)Tj
/F4 1 Tf
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-0.1694 Tc
[(]])-216.8()]TJ
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[(nv)6.2(a)9.1(r)]TJ
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(S)Tj
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([[)Tj
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(2)Tj
/F4 1 Tf
9.9626 0 0 9.9626 130.68 154.2 Tm
-0.1694 Tc
[(]])-168.7(\()]TJ
1.0118 0.2529 TD
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(S)Tj
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([[)Tj
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(1)Tj
/F4 1 Tf
9.9626 0 0 9.9626 159.72 154.2 Tm
-0.1694 Tc
[(]])-216.8()]TJ
/F6 1 Tf
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(\)\))Tj
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[(.)-313.2(Proposition)-228.9(2)-240.9(then)-240.9(yields)-240.9(the)-240.9(result.)]TJ
-7.757 -1.1925 TD
[(Similarly)60.2(,)-638.4(for)]TJ
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(if)Tj
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/F3 1 Tf
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(1)Tj
/F5 1 Tf
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(else)Tj
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6.9738 0 0 6.9738 264.48 140.88 Tm
(2)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 268.92 142.32 Tm
[(,)-650.4(assume)-566.1(the)-566.1(same)-566.1(for)]TJ
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(S)Tj
/F3 1 Tf
6.9738 0 0 6.9738 380.52 140.88 Tm
(1)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 390.6 142.32 Tm
(and)Tj
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(S)Tj
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6.9738 0 0 6.9738 416.88 140.88 Tm
(2)Tj
/TT2 1 Tf
9.9626 0 0 9.9626 421.32 142.32 Tm
[(.)-1276.8(When)]TJ
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(FV)Tj
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(\))Tj
/F1 1 Tf
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[(nv)6.2(a)9.1(r)]TJ
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(\()Tj
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[(we)-469.8(ha)24.1(v)12(e)]TJ
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(B)Tj
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([[)Tj
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[(]])-216.8()]TJ
/F6 1 Tf
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(=)Tj
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(NID)Tj
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()Tj
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(S)Tj
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([[)Tj
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(b)Tj
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(then)Tj
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(S)Tj
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(1)Tj
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(else)Tj
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6.9738 0 0 6.9738 360.96 128.88 Tm
(2)Tj
/F4 1 Tf
9.9626 0 0 9.9626 365.4 130.32 Tm
-0.1694 Tc
[(]])-216.8()]TJ
/F6 1 Tf
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1.6863 0.2529 TD
()Tj
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(S)Tj
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([[)Tj
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(S)Tj
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(1)Tj
/F4 1 Tf
9.9626 0 0 9.9626 417.96 130.32 Tm
-0.1694 Tc
[(]])-216.8()]TJ
/F6 1 Tf
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()Tj
/F4 1 Tf
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()Tj
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(S)Tj
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([[)Tj
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(S)Tj
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6.9738 0 0 6.9738 459 128.88 Tm
(2)Tj
/F4 1 Tf
9.9626 0 0 9.9626 463.56 130.32 Tm
-0.1694 Tc
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