/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 19 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 [] vy = True;
isPrefixOf vz [] = 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 [] vy = True;
isPrefixOf vz [] = 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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291 -> 285[label="",style="dashed", color="red", weight=0];
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291 -> 294[label="",style="dashed", color="magenta", weight=3];
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410 -> 295[label="",style="solid", color="burlywood", weight=3];
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411 -> 296[label="",style="solid", color="burlywood", weight=3];
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296[label="List.isPrefixOf ((:) wu15 wu14) []",fontsize=16,color="black",shape="box"];296 -> 300[label="",style="solid", color="black", weight=3];
297[label="wu18",fontsize=16,color="green",shape="box"];298[label="wu1710",fontsize=16,color="green",shape="box"];299 -> 301[label="",style="dashed", color="red", weight=0];
299[label="wu15 == wu180 && List.isPrefixOf wu14 wu181",fontsize=16,color="magenta"];299 -> 302[label="",style="dashed", color="magenta", weight=3];
299 -> 303[label="",style="dashed", color="magenta", weight=3];
299 -> 304[label="",style="dashed", color="magenta", weight=3];
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412 -> 305[label="",style="solid", color="blue", weight=3];
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414 -> 307[label="",style="solid", color="blue", weight=3];
415[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 415[label="",style="solid", color="blue", weight=9];
415 -> 308[label="",style="solid", color="blue", weight=3];
416[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 416[label="",style="solid", color="blue", weight=9];
416 -> 309[label="",style="solid", color="blue", weight=3];
417[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 417[label="",style="solid", color="blue", weight=9];
417 -> 310[label="",style="solid", color="blue", weight=3];
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418 -> 311[label="",style="solid", color="blue", weight=3];
419[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 419[label="",style="solid", color="blue", weight=9];
419 -> 312[label="",style="solid", color="blue", weight=3];
420[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 420[label="",style="solid", color="blue", weight=9];
420 -> 313[label="",style="solid", color="blue", weight=3];
421[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 421[label="",style="solid", color="blue", weight=9];
421 -> 314[label="",style="solid", color="blue", weight=3];
422[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 422[label="",style="solid", color="blue", weight=9];
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423[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 423[label="",style="solid", color="blue", weight=9];
423 -> 316[label="",style="solid", color="blue", weight=3];
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424 -> 317[label="",style="solid", color="blue", weight=3];
425[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];303 -> 425[label="",style="solid", color="blue", weight=9];
425 -> 318[label="",style="solid", color="blue", weight=3];
304[label="wu181",fontsize=16,color="green",shape="box"];301[label="wu23 && List.isPrefixOf wu24 wu25",fontsize=16,color="burlywood",shape="triangle"];426[label="wu23/False",fontsize=10,color="white",style="solid",shape="box"];301 -> 426[label="",style="solid", color="burlywood", weight=9];
426 -> 319[label="",style="solid", color="burlywood", weight=3];
427[label="wu23/True",fontsize=10,color="white",style="solid",shape="box"];301 -> 427[label="",style="solid", color="burlywood", weight=9];
427 -> 320[label="",style="solid", color="burlywood", weight=3];
305[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];305 -> 321[label="",style="solid", color="black", weight=3];
306[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];306 -> 322[label="",style="solid", color="black", weight=3];
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308[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];308 -> 324[label="",style="solid", color="black", weight=3];
309[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];309 -> 325[label="",style="solid", color="black", weight=3];
310[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];310 -> 326[label="",style="solid", color="black", weight=3];
311[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];311 -> 327[label="",style="solid", color="black", weight=3];
312[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];312 -> 328[label="",style="solid", color="black", weight=3];
313[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];313 -> 329[label="",style="solid", color="black", weight=3];
314[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];314 -> 330[label="",style="solid", color="black", weight=3];
315[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];315 -> 331[label="",style="solid", color="black", weight=3];
316[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];316 -> 332[label="",style="solid", color="black", weight=3];
317[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];317 -> 333[label="",style="solid", color="black", weight=3];
318[label="wu15 == wu180",fontsize=16,color="black",shape="triangle"];318 -> 334[label="",style="solid", color="black", weight=3];
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320[label="True && List.isPrefixOf wu24 wu25",fontsize=16,color="black",shape="box"];320 -> 336[label="",style="solid", color="black", weight=3];
321[label="error []",fontsize=16,color="red",shape="box"];322[label="error []",fontsize=16,color="red",shape="box"];323[label="error []",fontsize=16,color="red",shape="box"];324[label="error []",fontsize=16,color="red",shape="box"];325[label="error []",fontsize=16,color="red",shape="box"];326[label="error []",fontsize=16,color="red",shape="box"];327[label="error []",fontsize=16,color="red",shape="box"];328[label="error []",fontsize=16,color="red",shape="box"];329[label="error []",fontsize=16,color="red",shape="box"];330[label="error []",fontsize=16,color="red",shape="box"];331[label="error []",fontsize=16,color="red",shape="box"];332[label="error []",fontsize=16,color="red",shape="box"];333[label="error []",fontsize=16,color="red",shape="box"];334[label="primEqChar wu15 wu180",fontsize=16,color="burlywood",shape="box"];428[label="wu15/Char wu150",fontsize=10,color="white",style="solid",shape="box"];334 -> 428[label="",style="solid", color="burlywood", weight=9];
428 -> 337[label="",style="solid", color="burlywood", weight=3];
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429 -> 338[label="",style="solid", color="burlywood", weight=3];
430[label="wu24/[]",fontsize=10,color="white",style="solid",shape="box"];336 -> 430[label="",style="solid", color="burlywood", weight=9];
430 -> 339[label="",style="solid", color="burlywood", weight=3];
337[label="primEqChar (Char wu150) wu180",fontsize=16,color="burlywood",shape="box"];431[label="wu180/Char wu1800",fontsize=10,color="white",style="solid",shape="box"];337 -> 431[label="",style="solid", color="burlywood", weight=9];
431 -> 340[label="",style="solid", color="burlywood", weight=3];
338[label="List.isPrefixOf (wu240 : wu241) wu25",fontsize=16,color="burlywood",shape="box"];432[label="wu25/wu250 : wu251",fontsize=10,color="white",style="solid",shape="box"];338 -> 432[label="",style="solid", color="burlywood", weight=9];
432 -> 341[label="",style="solid", color="burlywood", weight=3];
433[label="wu25/[]",fontsize=10,color="white",style="solid",shape="box"];338 -> 433[label="",style="solid", color="burlywood", weight=9];
433 -> 342[label="",style="solid", color="burlywood", weight=3];
339[label="List.isPrefixOf [] wu25",fontsize=16,color="black",shape="box"];339 -> 343[label="",style="solid", color="black", weight=3];
340[label="primEqChar (Char wu150) (Char wu1800)",fontsize=16,color="black",shape="box"];340 -> 344[label="",style="solid", color="black", weight=3];
341[label="List.isPrefixOf (wu240 : wu241) (wu250 : wu251)",fontsize=16,color="black",shape="box"];341 -> 345[label="",style="solid", color="black", weight=3];
342[label="List.isPrefixOf (wu240 : wu241) []",fontsize=16,color="black",shape="box"];342 -> 346[label="",style="solid", color="black", weight=3];
343[label="True",fontsize=16,color="green",shape="box"];344[label="primEqNat wu150 wu1800",fontsize=16,color="burlywood",shape="triangle"];434[label="wu150/Succ wu1500",fontsize=10,color="white",style="solid",shape="box"];344 -> 434[label="",style="solid", color="burlywood", weight=9];
434 -> 347[label="",style="solid", color="burlywood", weight=3];
435[label="wu150/Zero",fontsize=10,color="white",style="solid",shape="box"];344 -> 435[label="",style="solid", color="burlywood", weight=9];
435 -> 348[label="",style="solid", color="burlywood", weight=3];
345 -> 301[label="",style="dashed", color="red", weight=0];
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345 -> 350[label="",style="dashed", color="magenta", weight=3];
345 -> 351[label="",style="dashed", color="magenta", weight=3];
346[label="False",fontsize=16,color="green",shape="box"];347[label="primEqNat (Succ wu1500) wu1800",fontsize=16,color="burlywood",shape="box"];436[label="wu1800/Succ wu18000",fontsize=10,color="white",style="solid",shape="box"];347 -> 436[label="",style="solid", color="burlywood", weight=9];
436 -> 352[label="",style="solid", color="burlywood", weight=3];
437[label="wu1800/Zero",fontsize=10,color="white",style="solid",shape="box"];347 -> 437[label="",style="solid", color="burlywood", weight=9];
437 -> 353[label="",style="solid", color="burlywood", weight=3];
348[label="primEqNat Zero wu1800",fontsize=16,color="burlywood",shape="box"];438[label="wu1800/Succ wu18000",fontsize=10,color="white",style="solid",shape="box"];348 -> 438[label="",style="solid", color="burlywood", weight=9];
438 -> 354[label="",style="solid", color="burlywood", weight=3];
439[label="wu1800/Zero",fontsize=10,color="white",style="solid",shape="box"];348 -> 439[label="",style="solid", color="burlywood", weight=9];
439 -> 355[label="",style="solid", color="burlywood", weight=3];
349[label="wu241",fontsize=16,color="green",shape="box"];350[label="wu240 == wu250",fontsize=16,color="blue",shape="box"];440[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 440[label="",style="solid", color="blue", weight=9];
440 -> 356[label="",style="solid", color="blue", weight=3];
441[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 441[label="",style="solid", color="blue", weight=9];
441 -> 357[label="",style="solid", color="blue", weight=3];
442[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 442[label="",style="solid", color="blue", weight=9];
442 -> 358[label="",style="solid", color="blue", weight=3];
443[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 443[label="",style="solid", color="blue", weight=9];
443 -> 359[label="",style="solid", color="blue", weight=3];
444[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 444[label="",style="solid", color="blue", weight=9];
444 -> 360[label="",style="solid", color="blue", weight=3];
445[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 445[label="",style="solid", color="blue", weight=9];
445 -> 361[label="",style="solid", color="blue", weight=3];
446[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 446[label="",style="solid", color="blue", weight=9];
446 -> 362[label="",style="solid", color="blue", weight=3];
447[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 447[label="",style="solid", color="blue", weight=9];
447 -> 363[label="",style="solid", color="blue", weight=3];
448[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 448[label="",style="solid", color="blue", weight=9];
448 -> 364[label="",style="solid", color="blue", weight=3];
449[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 449[label="",style="solid", color="blue", weight=9];
449 -> 365[label="",style="solid", color="blue", weight=3];
450[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 450[label="",style="solid", color="blue", weight=9];
450 -> 366[label="",style="solid", color="blue", weight=3];
451[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 451[label="",style="solid", color="blue", weight=9];
451 -> 367[label="",style="solid", color="blue", weight=3];
452[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 452[label="",style="solid", color="blue", weight=9];
452 -> 368[label="",style="solid", color="blue", weight=3];
453[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];350 -> 453[label="",style="solid", color="blue", weight=9];
453 -> 369[label="",style="solid", color="blue", weight=3];
351[label="wu251",fontsize=16,color="green",shape="box"];352[label="primEqNat (Succ wu1500) (Succ wu18000)",fontsize=16,color="black",shape="box"];352 -> 370[label="",style="solid", color="black", weight=3];
353[label="primEqNat (Succ wu1500) Zero",fontsize=16,color="black",shape="box"];353 -> 371[label="",style="solid", color="black", weight=3];
354[label="primEqNat Zero (Succ wu18000)",fontsize=16,color="black",shape="box"];354 -> 372[label="",style="solid", color="black", weight=3];
355[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];355 -> 373[label="",style="solid", color="black", weight=3];
356 -> 305[label="",style="dashed", color="red", weight=0];
356[label="wu240 == wu250",fontsize=16,color="magenta"];356 -> 374[label="",style="dashed", color="magenta", weight=3];
356 -> 375[label="",style="dashed", color="magenta", weight=3];
357 -> 306[label="",style="dashed", color="red", weight=0];
357[label="wu240 == wu250",fontsize=16,color="magenta"];357 -> 376[label="",style="dashed", color="magenta", weight=3];
357 -> 377[label="",style="dashed", color="magenta", weight=3];
358 -> 307[label="",style="dashed", color="red", weight=0];
358[label="wu240 == wu250",fontsize=16,color="magenta"];358 -> 378[label="",style="dashed", color="magenta", weight=3];
358 -> 379[label="",style="dashed", color="magenta", weight=3];
359 -> 308[label="",style="dashed", color="red", weight=0];
359[label="wu240 == wu250",fontsize=16,color="magenta"];359 -> 380[label="",style="dashed", color="magenta", weight=3];
359 -> 381[label="",style="dashed", color="magenta", weight=3];
360 -> 309[label="",style="dashed", color="red", weight=0];
360[label="wu240 == wu250",fontsize=16,color="magenta"];360 -> 382[label="",style="dashed", color="magenta", weight=3];
360 -> 383[label="",style="dashed", color="magenta", weight=3];
361 -> 310[label="",style="dashed", color="red", weight=0];
361[label="wu240 == wu250",fontsize=16,color="magenta"];361 -> 384[label="",style="dashed", color="magenta", weight=3];
361 -> 385[label="",style="dashed", color="magenta", weight=3];
362 -> 311[label="",style="dashed", color="red", weight=0];
362[label="wu240 == wu250",fontsize=16,color="magenta"];362 -> 386[label="",style="dashed", color="magenta", weight=3];
362 -> 387[label="",style="dashed", color="magenta", weight=3];
363 -> 312[label="",style="dashed", color="red", weight=0];
363[label="wu240 == wu250",fontsize=16,color="magenta"];363 -> 388[label="",style="dashed", color="magenta", weight=3];
363 -> 389[label="",style="dashed", color="magenta", weight=3];
364 -> 313[label="",style="dashed", color="red", weight=0];
364[label="wu240 == wu250",fontsize=16,color="magenta"];364 -> 390[label="",style="dashed", color="magenta", weight=3];
364 -> 391[label="",style="dashed", color="magenta", weight=3];
365 -> 314[label="",style="dashed", color="red", weight=0];
365[label="wu240 == wu250",fontsize=16,color="magenta"];365 -> 392[label="",style="dashed", color="magenta", weight=3];
365 -> 393[label="",style="dashed", color="magenta", weight=3];
366 -> 315[label="",style="dashed", color="red", weight=0];
366[label="wu240 == wu250",fontsize=16,color="magenta"];366 -> 394[label="",style="dashed", color="magenta", weight=3];
366 -> 395[label="",style="dashed", color="magenta", weight=3];
367 -> 316[label="",style="dashed", color="red", weight=0];
367[label="wu240 == wu250",fontsize=16,color="magenta"];367 -> 396[label="",style="dashed", color="magenta", weight=3];
367 -> 397[label="",style="dashed", color="magenta", weight=3];
368 -> 317[label="",style="dashed", color="red", weight=0];
368[label="wu240 == wu250",fontsize=16,color="magenta"];368 -> 398[label="",style="dashed", color="magenta", weight=3];
368 -> 399[label="",style="dashed", color="magenta", weight=3];
369 -> 318[label="",style="dashed", color="red", weight=0];
369[label="wu240 == wu250",fontsize=16,color="magenta"];369 -> 400[label="",style="dashed", color="magenta", weight=3];
369 -> 401[label="",style="dashed", color="magenta", weight=3];
370 -> 344[label="",style="dashed", color="red", weight=0];
370[label="primEqNat wu1500 wu18000",fontsize=16,color="magenta"];370 -> 402[label="",style="dashed", color="magenta", weight=3];
370 -> 403[label="",style="dashed", color="magenta", weight=3];
371[label="False",fontsize=16,color="green",shape="box"];372[label="False",fontsize=16,color="green",shape="box"];373[label="True",fontsize=16,color="green",shape="box"];374[label="wu250",fontsize=16,color="green",shape="box"];375[label="wu240",fontsize=16,color="green",shape="box"];376[label="wu250",fontsize=16,color="green",shape="box"];377[label="wu240",fontsize=16,color="green",shape="box"];378[label="wu250",fontsize=16,color="green",shape="box"];379[label="wu240",fontsize=16,color="green",shape="box"];380[label="wu250",fontsize=16,color="green",shape="box"];381[label="wu240",fontsize=16,color="green",shape="box"];382[label="wu250",fontsize=16,color="green",shape="box"];383[label="wu240",fontsize=16,color="green",shape="box"];384[label="wu250",fontsize=16,color="green",shape="box"];385[label="wu240",fontsize=16,color="green",shape="box"];386[label="wu250",fontsize=16,color="green",shape="box"];387[label="wu240",fontsize=16,color="green",shape="box"];388[label="wu250",fontsize=16,color="green",shape="box"];389[label="wu240",fontsize=16,color="green",shape="box"];390[label="wu250",fontsize=16,color="green",shape="box"];391[label="wu240",fontsize=16,color="green",shape="box"];392[label="wu250",fontsize=16,color="green",shape="box"];393[label="wu240",fontsize=16,color="green",shape="box"];394[label="wu250",fontsize=16,color="green",shape="box"];395[label="wu240",fontsize=16,color="green",shape="box"];396[label="wu250",fontsize=16,color="green",shape="box"];397[label="wu240",fontsize=16,color="green",shape="box"];398[label="wu250",fontsize=16,color="green",shape="box"];399[label="wu240",fontsize=16,color="green",shape="box"];400[label="wu250",fontsize=16,color="green",shape="box"];401[label="wu240",fontsize=16,color="green",shape="box"];402[label="wu18000",fontsize=16,color="green",shape="box"];403[label="wu1500",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, :(wu240, wu241), :(wu250, wu251), ba) -> new_asAs(new_esEs(wu240, wu250, ba), wu241, wu251, ba)

The TRS R consists of the following rules:

   new_esEs(wu240, wu250, ty_Int) -> new_esEs1(wu240, wu250)
   new_esEs(wu240, wu250, app(ty_Ratio, ca)) -> new_esEs3(wu240, wu250, ca)
   new_esEs(wu240, wu250, ty_Double) -> new_esEs4(wu240, wu250)
   new_primEqNat0(Zero, Zero) -> True
   new_esEs5(wu15, wu180) -> error([])
   new_esEs1(wu15, wu180) -> error([])
   new_esEs(wu240, wu250, app(app(ty_Either, cf), cg)) -> new_esEs6(wu240, wu250, cf, cg)
   new_esEs(wu240, wu250, app(app(ty_@2, bg), bh)) -> new_esEs11(wu240, wu250, bg, bh)
   new_esEs13(wu15, wu180, dd, de, df) -> error([])
   new_esEs(wu240, wu250, ty_Bool) -> new_esEs5(wu240, wu250)
   new_esEs(wu240, wu250, app(ty_[], ce)) -> new_esEs2(wu240, wu250, ce)
   new_esEs7(wu15, wu180) -> error([])
   new_esEs(wu240, wu250, app(ty_Maybe, bf)) -> new_esEs10(wu240, wu250, bf)
   new_esEs(wu240, wu250, ty_Integer) -> new_esEs12(wu240, wu250)
   new_esEs12(wu15, wu180) -> error([])
   new_esEs4(wu15, wu180) -> error([])
   new_esEs6(wu15, wu180, bd, be) -> error([])
   new_esEs(wu240, wu250, ty_@0) -> new_esEs8(wu240, wu250)
   new_esEs10(wu15, wu180, da) -> error([])
   new_esEs2(wu15, wu180, bb) -> error([])
   new_primEqNat0(Succ(wu1500), Zero) -> False
   new_primEqNat0(Zero, Succ(wu18000)) -> False
   new_primEqNat0(Succ(wu1500), Succ(wu18000)) -> new_primEqNat0(wu1500, wu18000)
   new_esEs0(wu15, wu180) -> error([])
   new_esEs(wu240, wu250, ty_Ordering) -> new_esEs7(wu240, wu250)
   new_esEs(wu240, wu250, ty_Char) -> new_esEs9(wu240, wu250)
   new_esEs11(wu15, wu180, db, dc) -> error([])
   new_esEs3(wu15, wu180, bc) -> error([])
   new_esEs(wu240, wu250, ty_Float) -> new_esEs0(wu240, wu250)
   new_esEs(wu240, wu250, app(app(app(ty_@3, cb), cc), cd)) -> new_esEs13(wu240, wu250, cb, cc, cd)
   new_esEs8(wu15, wu180) -> error([])
   new_esEs9(Char(wu150), Char(wu1800)) -> new_primEqNat0(wu150, wu1800)

The set Q consists of the following terms:

   new_esEs5(x0, x1)
   new_primEqNat0(Zero, Zero)
   new_esEs(x0, x1, ty_Double)
   new_esEs(x0, x1, ty_Float)
   new_esEs9(Char(x0), Char(x1))
   new_esEs(x0, x1, ty_Char)
   new_esEs11(x0, x1, x2, x3)
   new_esEs(x0, x1, ty_Int)
   new_esEs2(x0, x1, x2)
   new_esEs3(x0, x1, x2)
   new_esEs1(x0, x1)
   new_esEs4(x0, x1)
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_esEs(x0, x1, ty_Bool)
   new_esEs13(x0, x1, x2, x3, x4)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs0(x0, x1)
   new_esEs6(x0, x1, x2, x3)
   new_esEs(x0, x1, ty_Integer)
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs8(x0, x1)
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, app(ty_[], x2))
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs10(x0, x1, x2)
   new_esEs12(x0, x1)
   new_esEs7(x0, x1)
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_Ratio, 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, :(wu240, wu241), :(wu250, wu251), ba) -> new_asAs(new_esEs(wu240, wu250, ba), wu241, wu251, 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(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, new_flip(wu18, wu1710, ba), wu1711, ba)

The TRS R consists of the following rules:

   new_flip(wu14, wu15, ba) -> :(wu15, wu14)

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(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, new_flip(wu18, wu1710, ba), wu1711, ba) at position [2] we obtained the following new rules [LPAR04]:

   (new_isPrefixOf(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, ba),new_isPrefixOf(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, ba))


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

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

   new_isPrefixOf(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, ba)

The TRS R consists of the following rules:

   new_flip(wu14, wu15, ba) -> :(wu15, wu14)

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(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, 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(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, 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(wu15, wu14, wu18, :(wu1710, wu1711), ba) -> new_isPrefixOf(wu15, wu14, :(wu1710, wu18), wu1711, 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(wu1500), Succ(wu18000)) -> new_primEqNat(wu1500, wu18000)

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(wu1500), Succ(wu18000)) -> new_primEqNat(wu1500, wu18000)
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(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(new_flip(wu14, wu15, ba), wu160, wu161, wu17, ba)

The TRS R consists of the following rules:

   new_flip(wu14, wu15, ba) -> :(wu15, wu14)

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(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(new_flip(wu14, wu15, ba), wu160, wu161, wu17, ba) at position [0] we obtained the following new rules [LPAR04]:

   (new_isPrefixOf0(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, ba),new_isPrefixOf0(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, ba))


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

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

   new_isPrefixOf0(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, ba)

The TRS R consists of the following rules:

   new_flip(wu14, wu15, ba) -> :(wu15, wu14)

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(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, 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(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, 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(wu14, wu15, :(wu160, wu161), wu17, ba) -> new_isPrefixOf0(:(wu15, wu14), wu160, wu161, wu17, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


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

(30)
YES