/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 26 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) TransformationProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QReductionProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QDP
        (23) TransformationProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) UsableRulesProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) QReductionProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (30) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] _ = True;
isPrefixOf _ [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isSuffixOf x y = reverse x `isPrefixOf` reverse y;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] wu = True;
isPrefixOf wv [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isSuffixOf x y = reverse x `isPrefixOf` reverse y;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isPrefixOf [] wu = True;
isPrefixOf wv [] = False;
isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;

isSuffixOf :: Eq a => [a]  ->  [a]  ->  Bool;
isSuffixOf x y = reverse x `isPrefixOf` reverse y;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
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490 -> 7[label="",style="solid", color="burlywood", weight=3];
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492 -> 318[label="",style="solid", color="burlywood", weight=3];
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326[label="(:) ww15 ww14",fontsize=16,color="green",shape="box"];327 -> 332[label="",style="dashed", color="red", weight=0];
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337[label="List.isPrefixOf ((:) ww15 ww14) (foldl (flip (:)) ww18 [])",fontsize=16,color="black",shape="box"];337 -> 339[label="",style="solid", color="black", weight=3];
338 -> 332[label="",style="dashed", color="red", weight=0];
338[label="List.isPrefixOf ((:) ww15 ww14) (foldl (flip (:)) (flip (:) ww18 ww1710) ww1711)",fontsize=16,color="magenta"];338 -> 340[label="",style="dashed", color="magenta", weight=3];
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496 -> 342[label="",style="solid", color="burlywood", weight=3];
497[label="ww18/[]",fontsize=10,color="white",style="solid",shape="box"];339 -> 497[label="",style="solid", color="burlywood", weight=9];
497 -> 343[label="",style="solid", color="burlywood", weight=3];
340 -> 324[label="",style="dashed", color="red", weight=0];
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344[label="ww1710",fontsize=16,color="green",shape="box"];345[label="ww18",fontsize=16,color="green",shape="box"];346 -> 348[label="",style="dashed", color="red", weight=0];
346[label="ww15 == ww180 && List.isPrefixOf ww14 ww181",fontsize=16,color="magenta"];346 -> 349[label="",style="dashed", color="magenta", weight=3];
346 -> 350[label="",style="dashed", color="magenta", weight=3];
346 -> 351[label="",style="dashed", color="magenta", weight=3];
347[label="False",fontsize=16,color="green",shape="box"];349[label="ww15 == ww180",fontsize=16,color="blue",shape="box"];498[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 498[label="",style="solid", color="blue", weight=9];
498 -> 352[label="",style="solid", color="blue", weight=3];
499[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 499[label="",style="solid", color="blue", weight=9];
499 -> 353[label="",style="solid", color="blue", weight=3];
500[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 500[label="",style="solid", color="blue", weight=9];
500 -> 354[label="",style="solid", color="blue", weight=3];
501[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 501[label="",style="solid", color="blue", weight=9];
501 -> 355[label="",style="solid", color="blue", weight=3];
502[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 502[label="",style="solid", color="blue", weight=9];
502 -> 356[label="",style="solid", color="blue", weight=3];
503[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 503[label="",style="solid", color="blue", weight=9];
503 -> 357[label="",style="solid", color="blue", weight=3];
504[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 504[label="",style="solid", color="blue", weight=9];
504 -> 358[label="",style="solid", color="blue", weight=3];
505[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 505[label="",style="solid", color="blue", weight=9];
505 -> 359[label="",style="solid", color="blue", weight=3];
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506 -> 360[label="",style="solid", color="blue", weight=3];
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507 -> 361[label="",style="solid", color="blue", weight=3];
508[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 508[label="",style="solid", color="blue", weight=9];
508 -> 362[label="",style="solid", color="blue", weight=3];
509[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 509[label="",style="solid", color="blue", weight=9];
509 -> 363[label="",style="solid", color="blue", weight=3];
510[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 510[label="",style="solid", color="blue", weight=9];
510 -> 364[label="",style="solid", color="blue", weight=3];
511[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];349 -> 511[label="",style="solid", color="blue", weight=9];
511 -> 365[label="",style="solid", color="blue", weight=3];
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512 -> 366[label="",style="solid", color="burlywood", weight=3];
513[label="ww23/True",fontsize=10,color="white",style="solid",shape="box"];348 -> 513[label="",style="solid", color="burlywood", weight=9];
513 -> 367[label="",style="solid", color="burlywood", weight=3];
352[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];352 -> 368[label="",style="solid", color="black", weight=3];
353[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];353 -> 369[label="",style="solid", color="black", weight=3];
354[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];354 -> 370[label="",style="solid", color="black", weight=3];
355[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];355 -> 371[label="",style="solid", color="black", weight=3];
356[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];356 -> 372[label="",style="solid", color="black", weight=3];
357[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];357 -> 373[label="",style="solid", color="black", weight=3];
358[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];358 -> 374[label="",style="solid", color="black", weight=3];
359[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];359 -> 375[label="",style="solid", color="black", weight=3];
360[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];360 -> 376[label="",style="solid", color="black", weight=3];
361[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];361 -> 377[label="",style="solid", color="black", weight=3];
362[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];362 -> 378[label="",style="solid", color="black", weight=3];
363[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];363 -> 379[label="",style="solid", color="black", weight=3];
364[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];364 -> 380[label="",style="solid", color="black", weight=3];
365[label="ww15 == ww180",fontsize=16,color="black",shape="triangle"];365 -> 381[label="",style="solid", color="black", weight=3];
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367[label="True && List.isPrefixOf ww24 ww25",fontsize=16,color="black",shape="box"];367 -> 383[label="",style="solid", color="black", weight=3];
368[label="error []",fontsize=16,color="red",shape="box"];369[label="error []",fontsize=16,color="red",shape="box"];370[label="error []",fontsize=16,color="red",shape="box"];371[label="error []",fontsize=16,color="red",shape="box"];372[label="error []",fontsize=16,color="red",shape="box"];373[label="error []",fontsize=16,color="red",shape="box"];374[label="error []",fontsize=16,color="red",shape="box"];375[label="primEqInt ww15 ww180",fontsize=16,color="burlywood",shape="box"];514[label="ww15/Pos ww150",fontsize=10,color="white",style="solid",shape="box"];375 -> 514[label="",style="solid", color="burlywood", weight=9];
514 -> 384[label="",style="solid", color="burlywood", weight=3];
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515 -> 385[label="",style="solid", color="burlywood", weight=3];
376[label="error []",fontsize=16,color="red",shape="box"];377[label="error []",fontsize=16,color="red",shape="box"];378[label="error []",fontsize=16,color="red",shape="box"];379[label="error []",fontsize=16,color="red",shape="box"];380[label="error []",fontsize=16,color="red",shape="box"];381[label="error []",fontsize=16,color="red",shape="box"];382[label="False",fontsize=16,color="green",shape="box"];383[label="List.isPrefixOf ww24 ww25",fontsize=16,color="burlywood",shape="box"];516[label="ww24/ww240 : ww241",fontsize=10,color="white",style="solid",shape="box"];383 -> 516[label="",style="solid", color="burlywood", weight=9];
516 -> 386[label="",style="solid", color="burlywood", weight=3];
517[label="ww24/[]",fontsize=10,color="white",style="solid",shape="box"];383 -> 517[label="",style="solid", color="burlywood", weight=9];
517 -> 387[label="",style="solid", color="burlywood", weight=3];
384[label="primEqInt (Pos ww150) ww180",fontsize=16,color="burlywood",shape="box"];518[label="ww150/Succ ww1500",fontsize=10,color="white",style="solid",shape="box"];384 -> 518[label="",style="solid", color="burlywood", weight=9];
518 -> 388[label="",style="solid", color="burlywood", weight=3];
519[label="ww150/Zero",fontsize=10,color="white",style="solid",shape="box"];384 -> 519[label="",style="solid", color="burlywood", weight=9];
519 -> 389[label="",style="solid", color="burlywood", weight=3];
385[label="primEqInt (Neg ww150) ww180",fontsize=16,color="burlywood",shape="box"];520[label="ww150/Succ ww1500",fontsize=10,color="white",style="solid",shape="box"];385 -> 520[label="",style="solid", color="burlywood", weight=9];
520 -> 390[label="",style="solid", color="burlywood", weight=3];
521[label="ww150/Zero",fontsize=10,color="white",style="solid",shape="box"];385 -> 521[label="",style="solid", color="burlywood", weight=9];
521 -> 391[label="",style="solid", color="burlywood", weight=3];
386[label="List.isPrefixOf (ww240 : ww241) ww25",fontsize=16,color="burlywood",shape="box"];522[label="ww25/ww250 : ww251",fontsize=10,color="white",style="solid",shape="box"];386 -> 522[label="",style="solid", color="burlywood", weight=9];
522 -> 392[label="",style="solid", color="burlywood", weight=3];
523[label="ww25/[]",fontsize=10,color="white",style="solid",shape="box"];386 -> 523[label="",style="solid", color="burlywood", weight=9];
523 -> 393[label="",style="solid", color="burlywood", weight=3];
387[label="List.isPrefixOf [] ww25",fontsize=16,color="black",shape="box"];387 -> 394[label="",style="solid", color="black", weight=3];
388[label="primEqInt (Pos (Succ ww1500)) ww180",fontsize=16,color="burlywood",shape="box"];524[label="ww180/Pos ww1800",fontsize=10,color="white",style="solid",shape="box"];388 -> 524[label="",style="solid", color="burlywood", weight=9];
524 -> 395[label="",style="solid", color="burlywood", weight=3];
525[label="ww180/Neg ww1800",fontsize=10,color="white",style="solid",shape="box"];388 -> 525[label="",style="solid", color="burlywood", weight=9];
525 -> 396[label="",style="solid", color="burlywood", weight=3];
389[label="primEqInt (Pos Zero) ww180",fontsize=16,color="burlywood",shape="box"];526[label="ww180/Pos ww1800",fontsize=10,color="white",style="solid",shape="box"];389 -> 526[label="",style="solid", color="burlywood", weight=9];
526 -> 397[label="",style="solid", color="burlywood", weight=3];
527[label="ww180/Neg ww1800",fontsize=10,color="white",style="solid",shape="box"];389 -> 527[label="",style="solid", color="burlywood", weight=9];
527 -> 398[label="",style="solid", color="burlywood", weight=3];
390[label="primEqInt (Neg (Succ ww1500)) ww180",fontsize=16,color="burlywood",shape="box"];528[label="ww180/Pos ww1800",fontsize=10,color="white",style="solid",shape="box"];390 -> 528[label="",style="solid", color="burlywood", weight=9];
528 -> 399[label="",style="solid", color="burlywood", weight=3];
529[label="ww180/Neg ww1800",fontsize=10,color="white",style="solid",shape="box"];390 -> 529[label="",style="solid", color="burlywood", weight=9];
529 -> 400[label="",style="solid", color="burlywood", weight=3];
391[label="primEqInt (Neg Zero) ww180",fontsize=16,color="burlywood",shape="box"];530[label="ww180/Pos ww1800",fontsize=10,color="white",style="solid",shape="box"];391 -> 530[label="",style="solid", color="burlywood", weight=9];
530 -> 401[label="",style="solid", color="burlywood", weight=3];
531[label="ww180/Neg ww1800",fontsize=10,color="white",style="solid",shape="box"];391 -> 531[label="",style="solid", color="burlywood", weight=9];
531 -> 402[label="",style="solid", color="burlywood", weight=3];
392[label="List.isPrefixOf (ww240 : ww241) (ww250 : ww251)",fontsize=16,color="black",shape="box"];392 -> 403[label="",style="solid", color="black", weight=3];
393[label="List.isPrefixOf (ww240 : ww241) []",fontsize=16,color="black",shape="box"];393 -> 404[label="",style="solid", color="black", weight=3];
394[label="True",fontsize=16,color="green",shape="box"];395[label="primEqInt (Pos (Succ ww1500)) (Pos ww1800)",fontsize=16,color="burlywood",shape="box"];532[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];395 -> 532[label="",style="solid", color="burlywood", weight=9];
532 -> 405[label="",style="solid", color="burlywood", weight=3];
533[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];395 -> 533[label="",style="solid", color="burlywood", weight=9];
533 -> 406[label="",style="solid", color="burlywood", weight=3];
396[label="primEqInt (Pos (Succ ww1500)) (Neg ww1800)",fontsize=16,color="black",shape="box"];396 -> 407[label="",style="solid", color="black", weight=3];
397[label="primEqInt (Pos Zero) (Pos ww1800)",fontsize=16,color="burlywood",shape="box"];534[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];397 -> 534[label="",style="solid", color="burlywood", weight=9];
534 -> 408[label="",style="solid", color="burlywood", weight=3];
535[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];397 -> 535[label="",style="solid", color="burlywood", weight=9];
535 -> 409[label="",style="solid", color="burlywood", weight=3];
398[label="primEqInt (Pos Zero) (Neg ww1800)",fontsize=16,color="burlywood",shape="box"];536[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];398 -> 536[label="",style="solid", color="burlywood", weight=9];
536 -> 410[label="",style="solid", color="burlywood", weight=3];
537[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];398 -> 537[label="",style="solid", color="burlywood", weight=9];
537 -> 411[label="",style="solid", color="burlywood", weight=3];
399[label="primEqInt (Neg (Succ ww1500)) (Pos ww1800)",fontsize=16,color="black",shape="box"];399 -> 412[label="",style="solid", color="black", weight=3];
400[label="primEqInt (Neg (Succ ww1500)) (Neg ww1800)",fontsize=16,color="burlywood",shape="box"];538[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];400 -> 538[label="",style="solid", color="burlywood", weight=9];
538 -> 413[label="",style="solid", color="burlywood", weight=3];
539[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];400 -> 539[label="",style="solid", color="burlywood", weight=9];
539 -> 414[label="",style="solid", color="burlywood", weight=3];
401[label="primEqInt (Neg Zero) (Pos ww1800)",fontsize=16,color="burlywood",shape="box"];540[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];401 -> 540[label="",style="solid", color="burlywood", weight=9];
540 -> 415[label="",style="solid", color="burlywood", weight=3];
541[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];401 -> 541[label="",style="solid", color="burlywood", weight=9];
541 -> 416[label="",style="solid", color="burlywood", weight=3];
402[label="primEqInt (Neg Zero) (Neg ww1800)",fontsize=16,color="burlywood",shape="box"];542[label="ww1800/Succ ww18000",fontsize=10,color="white",style="solid",shape="box"];402 -> 542[label="",style="solid", color="burlywood", weight=9];
542 -> 417[label="",style="solid", color="burlywood", weight=3];
543[label="ww1800/Zero",fontsize=10,color="white",style="solid",shape="box"];402 -> 543[label="",style="solid", color="burlywood", weight=9];
543 -> 418[label="",style="solid", color="burlywood", weight=3];
403 -> 348[label="",style="dashed", color="red", weight=0];
403[label="ww240 == ww250 && List.isPrefixOf ww241 ww251",fontsize=16,color="magenta"];403 -> 419[label="",style="dashed", color="magenta", weight=3];
403 -> 420[label="",style="dashed", color="magenta", weight=3];
403 -> 421[label="",style="dashed", color="magenta", weight=3];
404[label="False",fontsize=16,color="green",shape="box"];405[label="primEqInt (Pos (Succ ww1500)) (Pos (Succ ww18000))",fontsize=16,color="black",shape="box"];405 -> 422[label="",style="solid", color="black", weight=3];
406[label="primEqInt (Pos (Succ ww1500)) (Pos Zero)",fontsize=16,color="black",shape="box"];406 -> 423[label="",style="solid", color="black", weight=3];
407[label="False",fontsize=16,color="green",shape="box"];408[label="primEqInt (Pos Zero) (Pos (Succ ww18000))",fontsize=16,color="black",shape="box"];408 -> 424[label="",style="solid", color="black", weight=3];
409[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];409 -> 425[label="",style="solid", color="black", weight=3];
410[label="primEqInt (Pos Zero) (Neg (Succ ww18000))",fontsize=16,color="black",shape="box"];410 -> 426[label="",style="solid", color="black", weight=3];
411[label="primEqInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];411 -> 427[label="",style="solid", color="black", weight=3];
412[label="False",fontsize=16,color="green",shape="box"];413[label="primEqInt (Neg (Succ ww1500)) (Neg (Succ ww18000))",fontsize=16,color="black",shape="box"];413 -> 428[label="",style="solid", color="black", weight=3];
414[label="primEqInt (Neg (Succ ww1500)) (Neg Zero)",fontsize=16,color="black",shape="box"];414 -> 429[label="",style="solid", color="black", weight=3];
415[label="primEqInt (Neg Zero) (Pos (Succ ww18000))",fontsize=16,color="black",shape="box"];415 -> 430[label="",style="solid", color="black", weight=3];
416[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];416 -> 431[label="",style="solid", color="black", weight=3];
417[label="primEqInt (Neg Zero) (Neg (Succ ww18000))",fontsize=16,color="black",shape="box"];417 -> 432[label="",style="solid", color="black", weight=3];
418[label="primEqInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];418 -> 433[label="",style="solid", color="black", weight=3];
419[label="ww240 == ww250",fontsize=16,color="blue",shape="box"];544[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 544[label="",style="solid", color="blue", weight=9];
544 -> 434[label="",style="solid", color="blue", weight=3];
545[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 545[label="",style="solid", color="blue", weight=9];
545 -> 435[label="",style="solid", color="blue", weight=3];
546[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 546[label="",style="solid", color="blue", weight=9];
546 -> 436[label="",style="solid", color="blue", weight=3];
547[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 547[label="",style="solid", color="blue", weight=9];
547 -> 437[label="",style="solid", color="blue", weight=3];
548[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 548[label="",style="solid", color="blue", weight=9];
548 -> 438[label="",style="solid", color="blue", weight=3];
549[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 549[label="",style="solid", color="blue", weight=9];
549 -> 439[label="",style="solid", color="blue", weight=3];
550[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 550[label="",style="solid", color="blue", weight=9];
550 -> 440[label="",style="solid", color="blue", weight=3];
551[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 551[label="",style="solid", color="blue", weight=9];
551 -> 441[label="",style="solid", color="blue", weight=3];
552[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 552[label="",style="solid", color="blue", weight=9];
552 -> 442[label="",style="solid", color="blue", weight=3];
553[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 553[label="",style="solid", color="blue", weight=9];
553 -> 443[label="",style="solid", color="blue", weight=3];
554[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 554[label="",style="solid", color="blue", weight=9];
554 -> 444[label="",style="solid", color="blue", weight=3];
555[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 555[label="",style="solid", color="blue", weight=9];
555 -> 445[label="",style="solid", color="blue", weight=3];
556[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 556[label="",style="solid", color="blue", weight=9];
556 -> 446[label="",style="solid", color="blue", weight=3];
557[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];419 -> 557[label="",style="solid", color="blue", weight=9];
557 -> 447[label="",style="solid", color="blue", weight=3];
420[label="ww251",fontsize=16,color="green",shape="box"];421[label="ww241",fontsize=16,color="green",shape="box"];422[label="primEqNat ww1500 ww18000",fontsize=16,color="burlywood",shape="triangle"];558[label="ww1500/Succ ww15000",fontsize=10,color="white",style="solid",shape="box"];422 -> 558[label="",style="solid", color="burlywood", weight=9];
558 -> 448[label="",style="solid", color="burlywood", weight=3];
559[label="ww1500/Zero",fontsize=10,color="white",style="solid",shape="box"];422 -> 559[label="",style="solid", color="burlywood", weight=9];
559 -> 449[label="",style="solid", color="burlywood", weight=3];
423[label="False",fontsize=16,color="green",shape="box"];424[label="False",fontsize=16,color="green",shape="box"];425[label="True",fontsize=16,color="green",shape="box"];426[label="False",fontsize=16,color="green",shape="box"];427[label="True",fontsize=16,color="green",shape="box"];428 -> 422[label="",style="dashed", color="red", weight=0];
428[label="primEqNat ww1500 ww18000",fontsize=16,color="magenta"];428 -> 450[label="",style="dashed", color="magenta", weight=3];
428 -> 451[label="",style="dashed", color="magenta", weight=3];
429[label="False",fontsize=16,color="green",shape="box"];430[label="False",fontsize=16,color="green",shape="box"];431[label="True",fontsize=16,color="green",shape="box"];432[label="False",fontsize=16,color="green",shape="box"];433[label="True",fontsize=16,color="green",shape="box"];434 -> 352[label="",style="dashed", color="red", weight=0];
434[label="ww240 == ww250",fontsize=16,color="magenta"];434 -> 452[label="",style="dashed", color="magenta", weight=3];
434 -> 453[label="",style="dashed", color="magenta", weight=3];
435 -> 353[label="",style="dashed", color="red", weight=0];
435[label="ww240 == ww250",fontsize=16,color="magenta"];435 -> 454[label="",style="dashed", color="magenta", weight=3];
435 -> 455[label="",style="dashed", color="magenta", weight=3];
436 -> 354[label="",style="dashed", color="red", weight=0];
436[label="ww240 == ww250",fontsize=16,color="magenta"];436 -> 456[label="",style="dashed", color="magenta", weight=3];
436 -> 457[label="",style="dashed", color="magenta", weight=3];
437 -> 355[label="",style="dashed", color="red", weight=0];
437[label="ww240 == ww250",fontsize=16,color="magenta"];437 -> 458[label="",style="dashed", color="magenta", weight=3];
437 -> 459[label="",style="dashed", color="magenta", weight=3];
438 -> 356[label="",style="dashed", color="red", weight=0];
438[label="ww240 == ww250",fontsize=16,color="magenta"];438 -> 460[label="",style="dashed", color="magenta", weight=3];
438 -> 461[label="",style="dashed", color="magenta", weight=3];
439 -> 357[label="",style="dashed", color="red", weight=0];
439[label="ww240 == ww250",fontsize=16,color="magenta"];439 -> 462[label="",style="dashed", color="magenta", weight=3];
439 -> 463[label="",style="dashed", color="magenta", weight=3];
440 -> 358[label="",style="dashed", color="red", weight=0];
440[label="ww240 == ww250",fontsize=16,color="magenta"];440 -> 464[label="",style="dashed", color="magenta", weight=3];
440 -> 465[label="",style="dashed", color="magenta", weight=3];
441 -> 359[label="",style="dashed", color="red", weight=0];
441[label="ww240 == ww250",fontsize=16,color="magenta"];441 -> 466[label="",style="dashed", color="magenta", weight=3];
441 -> 467[label="",style="dashed", color="magenta", weight=3];
442 -> 360[label="",style="dashed", color="red", weight=0];
442[label="ww240 == ww250",fontsize=16,color="magenta"];442 -> 468[label="",style="dashed", color="magenta", weight=3];
442 -> 469[label="",style="dashed", color="magenta", weight=3];
443 -> 361[label="",style="dashed", color="red", weight=0];
443[label="ww240 == ww250",fontsize=16,color="magenta"];443 -> 470[label="",style="dashed", color="magenta", weight=3];
443 -> 471[label="",style="dashed", color="magenta", weight=3];
444 -> 362[label="",style="dashed", color="red", weight=0];
444[label="ww240 == ww250",fontsize=16,color="magenta"];444 -> 472[label="",style="dashed", color="magenta", weight=3];
444 -> 473[label="",style="dashed", color="magenta", weight=3];
445 -> 363[label="",style="dashed", color="red", weight=0];
445[label="ww240 == ww250",fontsize=16,color="magenta"];445 -> 474[label="",style="dashed", color="magenta", weight=3];
445 -> 475[label="",style="dashed", color="magenta", weight=3];
446 -> 364[label="",style="dashed", color="red", weight=0];
446[label="ww240 == ww250",fontsize=16,color="magenta"];446 -> 476[label="",style="dashed", color="magenta", weight=3];
446 -> 477[label="",style="dashed", color="magenta", weight=3];
447 -> 365[label="",style="dashed", color="red", weight=0];
447[label="ww240 == ww250",fontsize=16,color="magenta"];447 -> 478[label="",style="dashed", color="magenta", weight=3];
447 -> 479[label="",style="dashed", color="magenta", weight=3];
448[label="primEqNat (Succ ww15000) ww18000",fontsize=16,color="burlywood",shape="box"];560[label="ww18000/Succ ww180000",fontsize=10,color="white",style="solid",shape="box"];448 -> 560[label="",style="solid", color="burlywood", weight=9];
560 -> 480[label="",style="solid", color="burlywood", weight=3];
561[label="ww18000/Zero",fontsize=10,color="white",style="solid",shape="box"];448 -> 561[label="",style="solid", color="burlywood", weight=9];
561 -> 481[label="",style="solid", color="burlywood", weight=3];
449[label="primEqNat Zero ww18000",fontsize=16,color="burlywood",shape="box"];562[label="ww18000/Succ ww180000",fontsize=10,color="white",style="solid",shape="box"];449 -> 562[label="",style="solid", color="burlywood", weight=9];
562 -> 482[label="",style="solid", color="burlywood", weight=3];
563[label="ww18000/Zero",fontsize=10,color="white",style="solid",shape="box"];449 -> 563[label="",style="solid", color="burlywood", weight=9];
563 -> 483[label="",style="solid", color="burlywood", weight=3];
450[label="ww18000",fontsize=16,color="green",shape="box"];451[label="ww1500",fontsize=16,color="green",shape="box"];452[label="ww240",fontsize=16,color="green",shape="box"];453[label="ww250",fontsize=16,color="green",shape="box"];454[label="ww240",fontsize=16,color="green",shape="box"];455[label="ww250",fontsize=16,color="green",shape="box"];456[label="ww240",fontsize=16,color="green",shape="box"];457[label="ww250",fontsize=16,color="green",shape="box"];458[label="ww240",fontsize=16,color="green",shape="box"];459[label="ww250",fontsize=16,color="green",shape="box"];460[label="ww240",fontsize=16,color="green",shape="box"];461[label="ww250",fontsize=16,color="green",shape="box"];462[label="ww240",fontsize=16,color="green",shape="box"];463[label="ww250",fontsize=16,color="green",shape="box"];464[label="ww240",fontsize=16,color="green",shape="box"];465[label="ww250",fontsize=16,color="green",shape="box"];466[label="ww240",fontsize=16,color="green",shape="box"];467[label="ww250",fontsize=16,color="green",shape="box"];468[label="ww240",fontsize=16,color="green",shape="box"];469[label="ww250",fontsize=16,color="green",shape="box"];470[label="ww240",fontsize=16,color="green",shape="box"];471[label="ww250",fontsize=16,color="green",shape="box"];472[label="ww240",fontsize=16,color="green",shape="box"];473[label="ww250",fontsize=16,color="green",shape="box"];474[label="ww240",fontsize=16,color="green",shape="box"];475[label="ww250",fontsize=16,color="green",shape="box"];476[label="ww240",fontsize=16,color="green",shape="box"];477[label="ww250",fontsize=16,color="green",shape="box"];478[label="ww240",fontsize=16,color="green",shape="box"];479[label="ww250",fontsize=16,color="green",shape="box"];480[label="primEqNat (Succ ww15000) (Succ ww180000)",fontsize=16,color="black",shape="box"];480 -> 484[label="",style="solid", color="black", weight=3];
481[label="primEqNat (Succ ww15000) Zero",fontsize=16,color="black",shape="box"];481 -> 485[label="",style="solid", color="black", weight=3];
482[label="primEqNat Zero (Succ ww180000)",fontsize=16,color="black",shape="box"];482 -> 486[label="",style="solid", color="black", weight=3];
483[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];483 -> 487[label="",style="solid", color="black", weight=3];
484 -> 422[label="",style="dashed", color="red", weight=0];
484[label="primEqNat ww15000 ww180000",fontsize=16,color="magenta"];484 -> 488[label="",style="dashed", color="magenta", weight=3];
484 -> 489[label="",style="dashed", color="magenta", weight=3];
485[label="False",fontsize=16,color="green",shape="box"];486[label="False",fontsize=16,color="green",shape="box"];487[label="True",fontsize=16,color="green",shape="box"];488[label="ww180000",fontsize=16,color="green",shape="box"];489[label="ww15000",fontsize=16,color="green",shape="box"];}

----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_asAs(True, :(ww240, ww241), :(ww250, ww251), ba) -> new_asAs(new_esEs(ww240, ww250, ba), ww241, ww251, ba)

The TRS R consists of the following rules:

   new_esEs1(Pos(Zero), Neg(Zero)) -> True
   new_esEs1(Neg(Zero), Pos(Zero)) -> True
   new_esEs(ww240, ww250, ty_Int) -> new_esEs1(ww240, ww250)
   new_esEs1(Neg(Zero), Neg(Zero)) -> True
   new_esEs(ww240, ww250, ty_Float) -> new_esEs8(ww240, ww250)
   new_esEs1(Neg(Succ(ww1500)), Neg(Succ(ww18000))) -> new_primEqNat0(ww1500, ww18000)
   new_primEqNat0(Zero, Zero) -> True
   new_esEs1(Pos(Succ(ww1500)), Pos(Succ(ww18000))) -> new_primEqNat0(ww1500, ww18000)
   new_esEs13(ww15, ww180) -> error([])
   new_esEs(ww240, ww250, app(ty_Maybe, cd)) -> new_esEs0(ww240, ww250, cd)
   new_esEs(ww240, ww250, ty_Char) -> new_esEs6(ww240, ww250)
   new_esEs7(ww15, ww180) -> error([])
   new_esEs(ww240, ww250, ty_Ordering) -> new_esEs4(ww240, ww250)
   new_esEs1(Pos(Succ(ww1500)), Pos(Zero)) -> False
   new_esEs1(Pos(Zero), Pos(Succ(ww18000))) -> False
   new_esEs1(Pos(Zero), Pos(Zero)) -> True
   new_esEs(ww240, ww250, ty_Bool) -> new_esEs12(ww240, ww250)
   new_esEs12(ww15, ww180) -> error([])
   new_esEs0(ww15, ww180, bb) -> error([])
   new_esEs(ww240, ww250, app(app(app(ty_@3, ce), cf), cg)) -> new_esEs11(ww240, ww250, ce, cf, cg)
   new_esEs4(ww15, ww180) -> error([])
   new_esEs10(ww15, ww180, bh) -> error([])
   new_esEs(ww240, ww250, app(app(ty_Either, de), df)) -> new_esEs9(ww240, ww250, de, df)
   new_esEs(ww240, ww250, ty_@0) -> new_esEs7(ww240, ww250)
   new_esEs(ww240, ww250, ty_Double) -> new_esEs13(ww240, ww250)
   new_esEs11(ww15, ww180, ca, cb, cc) -> error([])
   new_primEqNat0(Succ(ww15000), Zero) -> False
   new_primEqNat0(Zero, Succ(ww180000)) -> False
   new_esEs1(Pos(Zero), Neg(Succ(ww18000))) -> False
   new_esEs1(Neg(Zero), Pos(Succ(ww18000))) -> False
   new_esEs(ww240, ww250, ty_Integer) -> new_esEs2(ww240, ww250)
   new_esEs1(Neg(Succ(ww1500)), Neg(Zero)) -> False
   new_esEs1(Neg(Zero), Neg(Succ(ww18000))) -> False
   new_esEs2(ww15, ww180) -> error([])
   new_primEqNat0(Succ(ww15000), Succ(ww180000)) -> new_primEqNat0(ww15000, ww180000)
   new_esEs3(ww15, ww180, bc, bd) -> error([])
   new_esEs5(ww15, ww180, be) -> error([])
   new_esEs(ww240, ww250, app(ty_[], dc)) -> new_esEs10(ww240, ww250, dc)
   new_esEs(ww240, ww250, app(ty_Ratio, dd)) -> new_esEs5(ww240, ww250, dd)
   new_esEs6(ww15, ww180) -> error([])
   new_esEs1(Pos(Succ(ww1500)), Neg(ww1800)) -> False
   new_esEs1(Neg(Succ(ww1500)), Pos(ww1800)) -> False
   new_esEs8(ww15, ww180) -> error([])
   new_esEs9(ww15, ww180, bf, bg) -> error([])
   new_esEs(ww240, ww250, app(app(ty_@2, da), db)) -> new_esEs3(ww240, ww250, da, db)

The set Q consists of the following terms:

   new_primEqNat0(Zero, Zero)
   new_esEs1(Pos(Zero), Pos(Succ(x0)))
   new_esEs1(Pos(Zero), Neg(Zero))
   new_esEs1(Neg(Zero), Pos(Zero))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
   new_esEs8(x0, x1)
   new_esEs1(Pos(Succ(x0)), Pos(Succ(x1)))
   new_esEs(x0, x1, ty_Integer)
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs1(Pos(Succ(x0)), Pos(Zero))
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs1(Pos(Zero), Neg(Succ(x0)))
   new_esEs1(Neg(Zero), Pos(Succ(x0)))
   new_esEs1(Neg(Zero), Neg(Zero))
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs1(Neg(Succ(x0)), Neg(Zero))
   new_esEs7(x0, x1)
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_esEs(x0, x1, ty_Int)
   new_esEs1(Pos(Zero), Pos(Zero))
   new_esEs2(x0, x1)
   new_esEs6(x0, x1)
   new_esEs1(Neg(Succ(x0)), Neg(Succ(x1)))
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_esEs3(x0, x1, x2, x3)
   new_esEs9(x0, x1, x2, x3)
   new_primEqNat0(Zero, Succ(x0))
   new_esEs5(x0, x1, x2)
   new_esEs1(Neg(Zero), Neg(Succ(x0)))
   new_esEs12(x0, x1)
   new_esEs(x0, x1, ty_Double)
   new_esEs(x0, x1, ty_Float)
   new_esEs10(x0, x1, x2)
   new_esEs0(x0, x1, x2)
   new_esEs4(x0, x1)
   new_esEs1(Pos(Succ(x0)), Neg(x1))
   new_esEs1(Neg(Succ(x0)), Pos(x1))
   new_esEs(x0, x1, ty_Char)
   new_esEs13(x0, x1)
   new_esEs(x0, x1, ty_Bool)
   new_esEs11(x0, x1, x2, x3, x4)
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(ty_[], x2))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_asAs(True, :(ww240, ww241), :(ww250, ww251), ba) -> new_asAs(new_esEs(ww240, ww250, ba), ww241, ww251, ba)
The graph contains the following edges 2 > 2, 3 > 3, 4 >= 4


----------------------------------------

(9)
YES

----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, new_flip(ww18, ww1710, ba), ww1711, ba)

The TRS R consists of the following rules:

   new_flip(ww14, ww15, ba) -> :(ww15, ww14)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, new_flip(ww18, ww1710, ba), ww1711, ba) at position [2] we obtained the following new rules [LPAR04]:

   (new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba),new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba))


----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba)

The TRS R consists of the following rules:

   new_flip(ww14, ww15, ba) -> :(ww15, ww14)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba)

R is empty.
The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_flip(x0, x1, x2)


----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_isPrefixOf(ww15, ww14, ww18, :(ww1710, ww1711), ba) -> new_isPrefixOf(ww15, ww14, :(ww1710, ww18), ww1711, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 > 4, 5 >= 5


----------------------------------------

(18)
YES

----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primEqNat(Succ(ww15000), Succ(ww180000)) -> new_primEqNat(ww15000, ww180000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(20) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_primEqNat(Succ(ww15000), Succ(ww180000)) -> new_primEqNat(ww15000, ww180000)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(21)
YES

----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(new_flip(ww14, ww15, ba), ww160, ww161, ww17, ba)

The TRS R consists of the following rules:

   new_flip(ww14, ww15, ba) -> :(ww15, ww14)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(23) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(new_flip(ww14, ww15, ba), ww160, ww161, ww17, ba) at position [0] we obtained the following new rules [LPAR04]:

   (new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba),new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba))


----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba)

The TRS R consists of the following rules:

   new_flip(ww14, ww15, ba) -> :(ww15, ww14)

The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(25) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba)

R is empty.
The set Q consists of the following terms:

   new_flip(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(27) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_flip(x0, x1, x2)


----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_isPrefixOf0(ww14, ww15, :(ww160, ww161), ww17, ba) -> new_isPrefixOf0(:(ww15, ww14), ww160, ww161, ww17, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


----------------------------------------

(30)
YES