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


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


MAYBE
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) IFR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) NumRed [SOUND, 0 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 45 ms]
        (17) QDP
        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QDP
        (23) DependencyGraphProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) TransformationProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) MNOCProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) InductionCalculusProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) TransformationProof [EQUIVALENT, 0 ms]
        (36) QDP
            (37) DependencyGraphProof [EQUIVALENT, 0 ms]
            (38) QDP
            (39) TransformationProof [EQUIVALENT, 0 ms]
            (40) QDP
            (41) DependencyGraphProof [EQUIVALENT, 0 ms]
            (42) QDP
            (43) TransformationProof [EQUIVALENT, 0 ms]
            (44) QDP
            (45) DependencyGraphProof [EQUIVALENT, 0 ms]
            (46) QDP
            (47) TransformationProof [EQUIVALENT, 0 ms]
            (48) QDP
            (49) DependencyGraphProof [EQUIVALENT, 0 ms]
            (50) QDP
            (51) MNOCProof [EQUIVALENT, 0 ms]
            (52) QDP
            (53) InductionCalculusProof [EQUIVALENT, 0 ms]
            (54) QDP
    (55) QDP
        (56) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (57) YES
    (58) QDP
        (59) DependencyGraphProof [EQUIVALENT, 0 ms]
        (60) QDP
        (61) QDPOrderProof [EQUIVALENT, 0 ms]
        (62) QDP
        (63) DependencyGraphProof [EQUIVALENT, 0 ms]
        (64) QDP
        (65) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (66) YES
    (67) QDP
        (68) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (69) YES
(70) Narrow [COMPLETE, 0 ms]
(71) TRUE


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\_->q"
is transformed to
"gtGt0 q _ = q;
"

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(3) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero"
is transformed to
"primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y));
primDivNatS0 x y False = Zero;
"
The following If expression
"if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x"
is transformed to
"primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y);
primModNatS0 x y False = Succ x;
"
The following If expression
"if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x"
is transformed to
"primModNatP0 x y True = primModNatP (primMinusNatS x y) (Succ y);
primModNatP0 x y False = primMinusNatS y x;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"randomSelect (x : []) = x;
randomSelect (x : xs)|terminatorrandomSelect xs|otherwisex;
"
is transformed to
"randomSelect (x : []) = randomSelect3 (x : []);
randomSelect (x : xs) = randomSelect2 (x : xs);
"
"randomSelect0 x xs True = x;
"
"randomSelect1 x xs True = randomSelect xs;
randomSelect1 x xs False = randomSelect0 x xs otherwise;
"
"randomSelect2 (x : xs) = randomSelect1 x xs terminator;
"
"randomSelect3 (x : []) = x;
randomSelect3 wz = randomSelect2 wz;
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(9) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(10)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="print",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="print xu3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="putStrLn . show",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="putStrLn (show xu3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6 -> 7[label="",style="dashed", color="red", weight=0];
6[label="putStr (show xu3) >> putChar (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];6 -> 8[label="",style="dashed", color="magenta", weight=3];
6 -> 9[label="",style="dashed", color="magenta", weight=3];
8[label="xu3",fontsize=16,color="green",shape="box"];9[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];7[label="putStr (show xu5) >> putChar (Char (Succ xu6))",fontsize=16,color="black",shape="triangle"];7 -> 10[label="",style="solid", color="black", weight=3];
10[label="putStr (show xu5) >>= gtGt0 (putChar (Char (Succ xu6)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11 -> 321[label="",style="dashed", color="red", weight=0];
11[label="primbindIO (putStr (show xu5)) (gtGt0 (putChar (Char (Succ xu6))))",fontsize=16,color="magenta"];11 -> 322[label="",style="dashed", color="magenta", weight=3];
11 -> 323[label="",style="dashed", color="magenta", weight=3];
322[label="putChar (Char (Succ xu6))",fontsize=16,color="black",shape="box"];322 -> 426[label="",style="solid", color="black", weight=3];
323 -> 427[label="",style="dashed", color="red", weight=0];
323[label="putStr (show xu5)",fontsize=16,color="magenta"];323 -> 428[label="",style="dashed", color="magenta", weight=3];
321[label="primbindIO xu25 (gtGt0 xu24)",fontsize=16,color="burlywood",shape="triangle"];1364[label="xu25/IO xu250",fontsize=10,color="white",style="solid",shape="box"];321 -> 1364[label="",style="solid", color="burlywood", weight=9];
1364 -> 429[label="",style="solid", color="burlywood", weight=3];
1365[label="xu25/AProVE_IO xu250",fontsize=10,color="white",style="solid",shape="box"];321 -> 1365[label="",style="solid", color="burlywood", weight=9];
1365 -> 430[label="",style="solid", color="burlywood", weight=3];
1366[label="xu25/AProVE_Exception xu250",fontsize=10,color="white",style="solid",shape="box"];321 -> 1366[label="",style="solid", color="burlywood", weight=9];
1366 -> 431[label="",style="solid", color="burlywood", weight=3];
1367[label="xu25/AProVE_Error xu250",fontsize=10,color="white",style="solid",shape="box"];321 -> 1367[label="",style="solid", color="burlywood", weight=9];
1367 -> 432[label="",style="solid", color="burlywood", weight=3];
426 -> 528[label="",style="dashed", color="red", weight=0];
426[label="(seq Char (Succ xu6) output)",fontsize=16,color="magenta"];426 -> 529[label="",style="dashed", color="magenta", weight=3];
426 -> 530[label="",style="dashed", color="magenta", weight=3];
428[label="show xu5",fontsize=16,color="blue",shape="box"];1368[label="show :: IOErrorKind -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1368[label="",style="solid", color="blue", weight=9];
1368 -> 434[label="",style="solid", color="blue", weight=3];
1369[label="show :: Double -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1369[label="",style="solid", color="blue", weight=9];
1369 -> 435[label="",style="solid", color="blue", weight=3];
1370[label="show :: (Maybe a) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1370[label="",style="solid", color="blue", weight=9];
1370 -> 436[label="",style="solid", color="blue", weight=3];
1371[label="show :: ((@2) a b) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1371[label="",style="solid", color="blue", weight=9];
1371 -> 437[label="",style="solid", color="blue", weight=3];
1372[label="show :: (IO a) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1372[label="",style="solid", color="blue", weight=9];
1372 -> 438[label="",style="solid", color="blue", weight=3];
1373[label="show :: IOError -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1373[label="",style="solid", color="blue", weight=9];
1373 -> 439[label="",style="solid", color="blue", weight=3];
1374[label="show :: Char -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1374[label="",style="solid", color="blue", weight=9];
1374 -> 440[label="",style="solid", color="blue", weight=3];
1375[label="show :: Integer -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1375[label="",style="solid", color="blue", weight=9];
1375 -> 441[label="",style="solid", color="blue", weight=3];
1376[label="show :: Bool -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1376[label="",style="solid", color="blue", weight=9];
1376 -> 442[label="",style="solid", color="blue", weight=3];
1377[label="show :: Int -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1377[label="",style="solid", color="blue", weight=9];
1377 -> 443[label="",style="solid", color="blue", weight=3];
1378[label="show :: () -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1378[label="",style="solid", color="blue", weight=9];
1378 -> 444[label="",style="solid", color="blue", weight=3];
1379[label="show :: ([] a) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1379[label="",style="solid", color="blue", weight=9];
1379 -> 445[label="",style="solid", color="blue", weight=3];
1380[label="show :: Float -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1380[label="",style="solid", color="blue", weight=9];
1380 -> 446[label="",style="solid", color="blue", weight=3];
1381[label="show :: HugsException -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1381[label="",style="solid", color="blue", weight=9];
1381 -> 447[label="",style="solid", color="blue", weight=3];
1382[label="show :: ((@3) a b c) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1382[label="",style="solid", color="blue", weight=9];
1382 -> 448[label="",style="solid", color="blue", weight=3];
1383[label="show :: Ordering -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1383[label="",style="solid", color="blue", weight=9];
1383 -> 449[label="",style="solid", color="blue", weight=3];
1384[label="show :: (Ratio a) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1384[label="",style="solid", color="blue", weight=9];
1384 -> 450[label="",style="solid", color="blue", weight=3];
1385[label="show :: (Either a b) -> [] Char",fontsize=10,color="white",style="solid",shape="box"];428 -> 1385[label="",style="solid", color="blue", weight=9];
1385 -> 451[label="",style="solid", color="blue", weight=3];
427[label="putStr xu28",fontsize=16,color="burlywood",shape="triangle"];1386[label="xu28/xu280 : xu281",fontsize=10,color="white",style="solid",shape="box"];427 -> 1386[label="",style="solid", color="burlywood", weight=9];
1386 -> 452[label="",style="solid", color="burlywood", weight=3];
1387[label="xu28/[]",fontsize=10,color="white",style="solid",shape="box"];427 -> 1387[label="",style="solid", color="burlywood", weight=9];
1387 -> 453[label="",style="solid", color="burlywood", weight=3];
429[label="primbindIO (IO xu250) (gtGt0 xu24)",fontsize=16,color="black",shape="box"];429 -> 454[label="",style="solid", color="black", weight=3];
430[label="primbindIO (AProVE_IO xu250) (gtGt0 xu24)",fontsize=16,color="black",shape="box"];430 -> 455[label="",style="solid", color="black", weight=3];
431[label="primbindIO (AProVE_Exception xu250) (gtGt0 xu24)",fontsize=16,color="black",shape="box"];431 -> 456[label="",style="solid", color="black", weight=3];
432[label="primbindIO (AProVE_Error xu250) (gtGt0 xu24)",fontsize=16,color="black",shape="box"];432 -> 457[label="",style="solid", color="black", weight=3];
529[label="Char (Succ xu6)",fontsize=16,color="green",shape="box"];530 -> 458[label="",style="dashed", color="red", weight=0];
530[label="output",fontsize=16,color="magenta"];528[label="(seq xu280 xu41)",fontsize=16,color="black",shape="triangle"];528 -> 532[label="",style="solid", color="black", weight=3];
434[label="show xu5",fontsize=16,color="black",shape="box"];434 -> 459[label="",style="solid", color="black", weight=3];
435[label="show xu5",fontsize=16,color="black",shape="box"];435 -> 460[label="",style="solid", color="black", weight=3];
436[label="show xu5",fontsize=16,color="black",shape="box"];436 -> 461[label="",style="solid", color="black", weight=3];
437[label="show xu5",fontsize=16,color="black",shape="box"];437 -> 462[label="",style="solid", color="black", weight=3];
438[label="show xu5",fontsize=16,color="black",shape="box"];438 -> 463[label="",style="solid", color="black", weight=3];
439[label="show xu5",fontsize=16,color="black",shape="box"];439 -> 464[label="",style="solid", color="black", weight=3];
440[label="show xu5",fontsize=16,color="black",shape="box"];440 -> 465[label="",style="solid", color="black", weight=3];
441[label="show xu5",fontsize=16,color="black",shape="box"];441 -> 466[label="",style="solid", color="black", weight=3];
442[label="show xu5",fontsize=16,color="black",shape="box"];442 -> 467[label="",style="solid", color="black", weight=3];
443[label="show xu5",fontsize=16,color="black",shape="box"];443 -> 468[label="",style="solid", color="black", weight=3];
444[label="show xu5",fontsize=16,color="black",shape="box"];444 -> 469[label="",style="solid", color="black", weight=3];
445[label="show xu5",fontsize=16,color="black",shape="box"];445 -> 470[label="",style="solid", color="black", weight=3];
446[label="show xu5",fontsize=16,color="black",shape="box"];446 -> 471[label="",style="solid", color="black", weight=3];
447[label="show xu5",fontsize=16,color="black",shape="box"];447 -> 472[label="",style="solid", color="black", weight=3];
448[label="show xu5",fontsize=16,color="black",shape="box"];448 -> 473[label="",style="solid", color="black", weight=3];
449[label="show xu5",fontsize=16,color="black",shape="box"];449 -> 474[label="",style="solid", color="black", weight=3];
450[label="show xu5",fontsize=16,color="black",shape="box"];450 -> 475[label="",style="solid", color="black", weight=3];
451[label="show xu5",fontsize=16,color="black",shape="box"];451 -> 476[label="",style="solid", color="black", weight=3];
452[label="putStr (xu280 : xu281)",fontsize=16,color="black",shape="box"];452 -> 477[label="",style="solid", color="black", weight=3];
453[label="putStr []",fontsize=16,color="black",shape="box"];453 -> 478[label="",style="solid", color="black", weight=3];
454[label="error []",fontsize=16,color="red",shape="box"];455[label="gtGt0 xu24 xu250",fontsize=16,color="black",shape="box"];455 -> 479[label="",style="solid", color="black", weight=3];
456[label="AProVE_Exception xu250",fontsize=16,color="green",shape="box"];457[label="AProVE_Error xu250",fontsize=16,color="green",shape="box"];458[label="output",fontsize=16,color="black",shape="triangle"];458 -> 480[label="",style="solid", color="black", weight=3];
532[label="enforceWHNF (WHNF xu280) xu41",fontsize=16,color="black",shape="box"];532 -> 538[label="",style="solid", color="black", weight=3];
459[label="error []",fontsize=16,color="red",shape="box"];460[label="error []",fontsize=16,color="red",shape="box"];461[label="error []",fontsize=16,color="red",shape="box"];462[label="error []",fontsize=16,color="red",shape="box"];463[label="error []",fontsize=16,color="red",shape="box"];464[label="error []",fontsize=16,color="red",shape="box"];465[label="error []",fontsize=16,color="red",shape="box"];466[label="error []",fontsize=16,color="red",shape="box"];467[label="error []",fontsize=16,color="red",shape="box"];468[label="primShowInt xu5",fontsize=16,color="burlywood",shape="triangle"];1388[label="xu5/Pos xu50",fontsize=10,color="white",style="solid",shape="box"];468 -> 1388[label="",style="solid", color="burlywood", weight=9];
1388 -> 481[label="",style="solid", color="burlywood", weight=3];
1389[label="xu5/Neg xu50",fontsize=10,color="white",style="solid",shape="box"];468 -> 1389[label="",style="solid", color="burlywood", weight=9];
1389 -> 482[label="",style="solid", color="burlywood", weight=3];
469[label="error []",fontsize=16,color="red",shape="box"];470[label="error []",fontsize=16,color="red",shape="box"];471[label="error []",fontsize=16,color="red",shape="box"];472[label="error []",fontsize=16,color="red",shape="box"];473[label="error []",fontsize=16,color="red",shape="box"];474[label="error []",fontsize=16,color="red",shape="box"];475[label="error []",fontsize=16,color="red",shape="box"];476[label="error []",fontsize=16,color="red",shape="box"];477 -> 483[label="",style="dashed", color="red", weight=0];
477[label="putChar xu280 >> putStr xu281",fontsize=16,color="magenta"];477 -> 484[label="",style="dashed", color="magenta", weight=3];
478 -> 458[label="",style="dashed", color="red", weight=0];
478[label="output",fontsize=16,color="magenta"];479[label="xu24",fontsize=16,color="green",shape="box"];480[label="randomSelect (aIOE IOError_FullError : aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];480 -> 485[label="",style="solid", color="black", weight=3];
538[label="xu41",fontsize=16,color="green",shape="box"];481[label="primShowInt (Pos xu50)",fontsize=16,color="burlywood",shape="box"];1390[label="xu50/Succ xu500",fontsize=10,color="white",style="solid",shape="box"];481 -> 1390[label="",style="solid", color="burlywood", weight=9];
1390 -> 486[label="",style="solid", color="burlywood", weight=3];
1391[label="xu50/Zero",fontsize=10,color="white",style="solid",shape="box"];481 -> 1391[label="",style="solid", color="burlywood", weight=9];
1391 -> 487[label="",style="solid", color="burlywood", weight=3];
482[label="primShowInt (Neg xu50)",fontsize=16,color="black",shape="box"];482 -> 488[label="",style="solid", color="black", weight=3];
484 -> 427[label="",style="dashed", color="red", weight=0];
484[label="putStr xu281",fontsize=16,color="magenta"];484 -> 489[label="",style="dashed", color="magenta", weight=3];
483[label="putChar xu280 >> xu29",fontsize=16,color="black",shape="triangle"];483 -> 490[label="",style="solid", color="black", weight=3];
485[label="randomSelect2 (aIOE IOError_FullError : aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];485 -> 491[label="",style="solid", color="black", weight=3];
486[label="primShowInt (Pos (Succ xu500))",fontsize=16,color="black",shape="box"];486 -> 492[label="",style="solid", color="black", weight=3];
487[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];487 -> 493[label="",style="solid", color="black", weight=3];
488[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))) : primShowInt (Pos xu50)",fontsize=16,color="green",shape="box"];488 -> 494[label="",style="dashed", color="green", weight=3];
489[label="xu281",fontsize=16,color="green",shape="box"];490[label="putChar xu280 >>= gtGt0 xu29",fontsize=16,color="black",shape="box"];490 -> 495[label="",style="solid", color="black", weight=3];
491[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) terminator",fontsize=16,color="black",shape="box"];491 -> 496[label="",style="solid", color="black", weight=3];
492 -> 512[label="",style="dashed", color="red", weight=0];
492[label="primShowInt (div Pos (Succ xu500) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) ++ toEnum (mod Pos (Succ xu500) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) : []",fontsize=16,color="magenta"];492 -> 513[label="",style="dashed", color="magenta", weight=3];
492 -> 514[label="",style="dashed", color="magenta", weight=3];
492 -> 515[label="",style="dashed", color="magenta", weight=3];
493[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))) : []",fontsize=16,color="green",shape="box"];494 -> 468[label="",style="dashed", color="red", weight=0];
494[label="primShowInt (Pos xu50)",fontsize=16,color="magenta"];494 -> 500[label="",style="dashed", color="magenta", weight=3];
495 -> 321[label="",style="dashed", color="red", weight=0];
495[label="primbindIO (putChar xu280) (gtGt0 xu29)",fontsize=16,color="magenta"];495 -> 501[label="",style="dashed", color="magenta", weight=3];
495 -> 502[label="",style="dashed", color="magenta", weight=3];
496[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) ter5m",fontsize=16,color="burlywood",shape="box"];1392[label="ter5m/False",fontsize=10,color="white",style="solid",shape="box"];496 -> 1392[label="",style="solid", color="burlywood", weight=9];
1392 -> 503[label="",style="solid", color="burlywood", weight=3];
1393[label="ter5m/True",fontsize=10,color="white",style="solid",shape="box"];496 -> 1393[label="",style="solid", color="burlywood", weight=9];
1393 -> 504[label="",style="solid", color="burlywood", weight=3];
513[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];514[label="xu500",fontsize=16,color="green",shape="box"];515 -> 468[label="",style="dashed", color="red", weight=0];
515[label="primShowInt (div Pos (Succ xu500) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];515 -> 517[label="",style="dashed", color="magenta", weight=3];
512[label="xu37 ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="burlywood",shape="triangle"];1394[label="xu37/xu370 : xu371",fontsize=10,color="white",style="solid",shape="box"];512 -> 1394[label="",style="solid", color="burlywood", weight=9];
1394 -> 518[label="",style="solid", color="burlywood", weight=3];
1395[label="xu37/[]",fontsize=10,color="white",style="solid",shape="box"];512 -> 1395[label="",style="solid", color="burlywood", weight=9];
1395 -> 519[label="",style="solid", color="burlywood", weight=3];
500[label="Pos xu50",fontsize=16,color="green",shape="box"];501[label="xu29",fontsize=16,color="green",shape="box"];502[label="putChar xu280",fontsize=16,color="black",shape="box"];502 -> 520[label="",style="solid", color="black", weight=3];
503[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];503 -> 521[label="",style="solid", color="black", weight=3];
504[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];504 -> 522[label="",style="solid", color="black", weight=3];
517 -> 523[label="",style="dashed", color="red", weight=0];
517[label="div Pos (Succ xu500) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))",fontsize=16,color="magenta"];517 -> 524[label="",style="dashed", color="magenta", weight=3];
517 -> 525[label="",style="dashed", color="magenta", weight=3];
518[label="(xu370 : xu371) ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="black",shape="box"];518 -> 526[label="",style="solid", color="black", weight=3];
519[label="[] ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="black",shape="box"];519 -> 527[label="",style="solid", color="black", weight=3];
520 -> 528[label="",style="dashed", color="red", weight=0];
520[label="(seq xu280 output)",fontsize=16,color="magenta"];520 -> 531[label="",style="dashed", color="magenta", weight=3];
521[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];521 -> 536[label="",style="solid", color="black", weight=3];
522[label="randomSelect (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];522 -> 537[label="",style="solid", color="black", weight=3];
524[label="xu500",fontsize=16,color="green",shape="box"];525[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];523[label="div Pos (Succ xu39) Pos (Succ xu40)",fontsize=16,color="black",shape="triangle"];523 -> 533[label="",style="solid", color="black", weight=3];
526[label="xu370 : xu371 ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="green",shape="box"];526 -> 534[label="",style="dashed", color="green", weight=3];
527[label="toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="green",shape="box"];527 -> 535[label="",style="dashed", color="green", weight=3];
531 -> 458[label="",style="dashed", color="red", weight=0];
531[label="output",fontsize=16,color="magenta"];536[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];536 -> 542[label="",style="solid", color="black", weight=3];
537[label="randomSelect2 (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];537 -> 543[label="",style="solid", color="black", weight=3];
533[label="primDivInt (Pos (Succ xu39)) (Pos (Succ xu40))",fontsize=16,color="black",shape="box"];533 -> 539[label="",style="solid", color="black", weight=3];
534 -> 512[label="",style="dashed", color="red", weight=0];
534[label="xu371 ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="magenta"];534 -> 540[label="",style="dashed", color="magenta", weight=3];
535[label="toEnum (mod Pos (Succ xu34) Pos (Succ xu36))",fontsize=16,color="black",shape="box"];535 -> 541[label="",style="solid", color="black", weight=3];
542[label="aIOE IOError_FullError",fontsize=16,color="black",shape="box"];542 -> 546[label="",style="solid", color="black", weight=3];
543[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) terminator",fontsize=16,color="black",shape="box"];543 -> 547[label="",style="solid", color="black", weight=3];
539[label="Pos (primDivNatS (Succ xu39) (Succ xu40))",fontsize=16,color="green",shape="box"];539 -> 544[label="",style="dashed", color="green", weight=3];
540[label="xu371",fontsize=16,color="green",shape="box"];541[label="primIntToChar (mod Pos (Succ xu34) Pos (Succ xu36))",fontsize=16,color="black",shape="box"];541 -> 545[label="",style="solid", color="black", weight=3];
546[label="AProVE_Exception (AET_IOError (IOError IOError_FullError [] [] Nothing))",fontsize=16,color="green",shape="box"];547[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) ter6m",fontsize=16,color="burlywood",shape="box"];1396[label="ter6m/False",fontsize=10,color="white",style="solid",shape="box"];547 -> 1396[label="",style="solid", color="burlywood", weight=9];
1396 -> 550[label="",style="solid", color="burlywood", weight=3];
1397[label="ter6m/True",fontsize=10,color="white",style="solid",shape="box"];547 -> 1397[label="",style="solid", color="burlywood", weight=9];
1397 -> 551[label="",style="solid", color="burlywood", weight=3];
544[label="primDivNatS (Succ xu39) (Succ xu40)",fontsize=16,color="black",shape="triangle"];544 -> 548[label="",style="solid", color="black", weight=3];
545[label="primIntToChar (primModInt (Pos (Succ xu34)) (Pos (Succ xu36)))",fontsize=16,color="black",shape="box"];545 -> 549[label="",style="solid", color="black", weight=3];
550[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];550 -> 555[label="",style="solid", color="black", weight=3];
551[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];551 -> 556[label="",style="solid", color="black", weight=3];
548[label="primDivNatS0 xu39 xu40 (primGEqNatS xu39 xu40)",fontsize=16,color="burlywood",shape="box"];1398[label="xu39/Succ xu390",fontsize=10,color="white",style="solid",shape="box"];548 -> 1398[label="",style="solid", color="burlywood", weight=9];
1398 -> 552[label="",style="solid", color="burlywood", weight=3];
1399[label="xu39/Zero",fontsize=10,color="white",style="solid",shape="box"];548 -> 1399[label="",style="solid", color="burlywood", weight=9];
1399 -> 553[label="",style="solid", color="burlywood", weight=3];
549[label="primIntToChar (Pos (primModNatS (Succ xu34) (Succ xu36)))",fontsize=16,color="black",shape="box"];549 -> 554[label="",style="solid", color="black", weight=3];
555[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];555 -> 562[label="",style="solid", color="black", weight=3];
556[label="randomSelect (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];556 -> 563[label="",style="solid", color="black", weight=3];
552[label="primDivNatS0 (Succ xu390) xu40 (primGEqNatS (Succ xu390) xu40)",fontsize=16,color="burlywood",shape="box"];1400[label="xu40/Succ xu400",fontsize=10,color="white",style="solid",shape="box"];552 -> 1400[label="",style="solid", color="burlywood", weight=9];
1400 -> 557[label="",style="solid", color="burlywood", weight=3];
1401[label="xu40/Zero",fontsize=10,color="white",style="solid",shape="box"];552 -> 1401[label="",style="solid", color="burlywood", weight=9];
1401 -> 558[label="",style="solid", color="burlywood", weight=3];
553[label="primDivNatS0 Zero xu40 (primGEqNatS Zero xu40)",fontsize=16,color="burlywood",shape="box"];1402[label="xu40/Succ xu400",fontsize=10,color="white",style="solid",shape="box"];553 -> 1402[label="",style="solid", color="burlywood", weight=9];
1402 -> 559[label="",style="solid", color="burlywood", weight=3];
1403[label="xu40/Zero",fontsize=10,color="white",style="solid",shape="box"];553 -> 1403[label="",style="solid", color="burlywood", weight=9];
1403 -> 560[label="",style="solid", color="burlywood", weight=3];
554[label="Char (primModNatS (Succ xu34) (Succ xu36))",fontsize=16,color="green",shape="box"];554 -> 561[label="",style="dashed", color="green", weight=3];
562[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];562 -> 569[label="",style="solid", color="black", weight=3];
563[label="randomSelect3 (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];563 -> 570[label="",style="solid", color="black", weight=3];
557[label="primDivNatS0 (Succ xu390) (Succ xu400) (primGEqNatS (Succ xu390) (Succ xu400))",fontsize=16,color="black",shape="box"];557 -> 564[label="",style="solid", color="black", weight=3];
558[label="primDivNatS0 (Succ xu390) Zero (primGEqNatS (Succ xu390) Zero)",fontsize=16,color="black",shape="box"];558 -> 565[label="",style="solid", color="black", weight=3];
559[label="primDivNatS0 Zero (Succ xu400) (primGEqNatS Zero (Succ xu400))",fontsize=16,color="black",shape="box"];559 -> 566[label="",style="solid", color="black", weight=3];
560[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];560 -> 567[label="",style="solid", color="black", weight=3];
561[label="primModNatS (Succ xu34) (Succ xu36)",fontsize=16,color="black",shape="triangle"];561 -> 568[label="",style="solid", color="black", weight=3];
569[label="aIOE IOError_PermDenied",fontsize=16,color="black",shape="box"];569 -> 578[label="",style="solid", color="black", weight=3];
570[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];564 -> 1038[label="",style="dashed", color="red", weight=0];
564[label="primDivNatS0 (Succ xu390) (Succ xu400) (primGEqNatS xu390 xu400)",fontsize=16,color="magenta"];564 -> 1039[label="",style="dashed", color="magenta", weight=3];
564 -> 1040[label="",style="dashed", color="magenta", weight=3];
564 -> 1041[label="",style="dashed", color="magenta", weight=3];
564 -> 1042[label="",style="dashed", color="magenta", weight=3];
565[label="primDivNatS0 (Succ xu390) Zero True",fontsize=16,color="black",shape="box"];565 -> 573[label="",style="solid", color="black", weight=3];
566[label="primDivNatS0 Zero (Succ xu400) False",fontsize=16,color="black",shape="box"];566 -> 574[label="",style="solid", color="black", weight=3];
567[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];567 -> 575[label="",style="solid", color="black", weight=3];
568[label="primModNatS0 xu34 xu36 (primGEqNatS xu34 xu36)",fontsize=16,color="burlywood",shape="box"];1404[label="xu34/Succ xu340",fontsize=10,color="white",style="solid",shape="box"];568 -> 1404[label="",style="solid", color="burlywood", weight=9];
1404 -> 576[label="",style="solid", color="burlywood", weight=3];
1405[label="xu34/Zero",fontsize=10,color="white",style="solid",shape="box"];568 -> 1405[label="",style="solid", color="burlywood", weight=9];
1405 -> 577[label="",style="solid", color="burlywood", weight=3];
578[label="AProVE_Exception (AET_IOError (IOError IOError_PermDenied [] [] Nothing))",fontsize=16,color="green",shape="box"];1039[label="xu400",fontsize=16,color="green",shape="box"];1040[label="xu390",fontsize=16,color="green",shape="box"];1041[label="xu400",fontsize=16,color="green",shape="box"];1042[label="xu390",fontsize=16,color="green",shape="box"];1038[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS xu83 xu84)",fontsize=16,color="burlywood",shape="triangle"];1406[label="xu83/Succ xu830",fontsize=10,color="white",style="solid",shape="box"];1038 -> 1406[label="",style="solid", color="burlywood", weight=9];
1406 -> 1079[label="",style="solid", color="burlywood", weight=3];
1407[label="xu83/Zero",fontsize=10,color="white",style="solid",shape="box"];1038 -> 1407[label="",style="solid", color="burlywood", weight=9];
1407 -> 1080[label="",style="solid", color="burlywood", weight=3];
573[label="Succ (primDivNatS (primMinusNatS (Succ xu390) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];573 -> 583[label="",style="dashed", color="green", weight=3];
574[label="Zero",fontsize=16,color="green",shape="box"];575[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];575 -> 584[label="",style="dashed", color="green", weight=3];
576[label="primModNatS0 (Succ xu340) xu36 (primGEqNatS (Succ xu340) xu36)",fontsize=16,color="burlywood",shape="box"];1408[label="xu36/Succ xu360",fontsize=10,color="white",style="solid",shape="box"];576 -> 1408[label="",style="solid", color="burlywood", weight=9];
1408 -> 585[label="",style="solid", color="burlywood", weight=3];
1409[label="xu36/Zero",fontsize=10,color="white",style="solid",shape="box"];576 -> 1409[label="",style="solid", color="burlywood", weight=9];
1409 -> 586[label="",style="solid", color="burlywood", weight=3];
577[label="primModNatS0 Zero xu36 (primGEqNatS Zero xu36)",fontsize=16,color="burlywood",shape="box"];1410[label="xu36/Succ xu360",fontsize=10,color="white",style="solid",shape="box"];577 -> 1410[label="",style="solid", color="burlywood", weight=9];
1410 -> 587[label="",style="solid", color="burlywood", weight=3];
1411[label="xu36/Zero",fontsize=10,color="white",style="solid",shape="box"];577 -> 1411[label="",style="solid", color="burlywood", weight=9];
1411 -> 588[label="",style="solid", color="burlywood", weight=3];
1079[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) xu84)",fontsize=16,color="burlywood",shape="box"];1412[label="xu84/Succ xu840",fontsize=10,color="white",style="solid",shape="box"];1079 -> 1412[label="",style="solid", color="burlywood", weight=9];
1412 -> 1109[label="",style="solid", color="burlywood", weight=3];
1413[label="xu84/Zero",fontsize=10,color="white",style="solid",shape="box"];1079 -> 1413[label="",style="solid", color="burlywood", weight=9];
1413 -> 1110[label="",style="solid", color="burlywood", weight=3];
1080[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero xu84)",fontsize=16,color="burlywood",shape="box"];1414[label="xu84/Succ xu840",fontsize=10,color="white",style="solid",shape="box"];1080 -> 1414[label="",style="solid", color="burlywood", weight=9];
1414 -> 1111[label="",style="solid", color="burlywood", weight=3];
1415[label="xu84/Zero",fontsize=10,color="white",style="solid",shape="box"];1080 -> 1415[label="",style="solid", color="burlywood", weight=9];
1415 -> 1112[label="",style="solid", color="burlywood", weight=3];
583 -> 1324[label="",style="dashed", color="red", weight=0];
583[label="primDivNatS (primMinusNatS (Succ xu390) Zero) (Succ Zero)",fontsize=16,color="magenta"];583 -> 1325[label="",style="dashed", color="magenta", weight=3];
583 -> 1326[label="",style="dashed", color="magenta", weight=3];
583 -> 1327[label="",style="dashed", color="magenta", weight=3];
584 -> 1324[label="",style="dashed", color="red", weight=0];
584[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];584 -> 1328[label="",style="dashed", color="magenta", weight=3];
584 -> 1329[label="",style="dashed", color="magenta", weight=3];
584 -> 1330[label="",style="dashed", color="magenta", weight=3];
585[label="primModNatS0 (Succ xu340) (Succ xu360) (primGEqNatS (Succ xu340) (Succ xu360))",fontsize=16,color="black",shape="box"];585 -> 595[label="",style="solid", color="black", weight=3];
586[label="primModNatS0 (Succ xu340) Zero (primGEqNatS (Succ xu340) Zero)",fontsize=16,color="black",shape="box"];586 -> 596[label="",style="solid", color="black", weight=3];
587[label="primModNatS0 Zero (Succ xu360) (primGEqNatS Zero (Succ xu360))",fontsize=16,color="black",shape="box"];587 -> 597[label="",style="solid", color="black", weight=3];
588[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];588 -> 598[label="",style="solid", color="black", weight=3];
1109[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) (Succ xu840))",fontsize=16,color="black",shape="box"];1109 -> 1125[label="",style="solid", color="black", weight=3];
1110[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) Zero)",fontsize=16,color="black",shape="box"];1110 -> 1126[label="",style="solid", color="black", weight=3];
1111[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero (Succ xu840))",fontsize=16,color="black",shape="box"];1111 -> 1127[label="",style="solid", color="black", weight=3];
1112[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1112 -> 1128[label="",style="solid", color="black", weight=3];
1325[label="Zero",fontsize=16,color="green",shape="box"];1326[label="Zero",fontsize=16,color="green",shape="box"];1327[label="Succ xu390",fontsize=16,color="green",shape="box"];1324[label="primDivNatS (primMinusNatS xu98 xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="triangle"];1416[label="xu98/Succ xu980",fontsize=10,color="white",style="solid",shape="box"];1324 -> 1416[label="",style="solid", color="burlywood", weight=9];
1416 -> 1349[label="",style="solid", color="burlywood", weight=3];
1417[label="xu98/Zero",fontsize=10,color="white",style="solid",shape="box"];1324 -> 1417[label="",style="solid", color="burlywood", weight=9];
1417 -> 1350[label="",style="solid", color="burlywood", weight=3];
1328[label="Zero",fontsize=16,color="green",shape="box"];1329[label="Zero",fontsize=16,color="green",shape="box"];1330[label="Zero",fontsize=16,color="green",shape="box"];595 -> 1149[label="",style="dashed", color="red", weight=0];
595[label="primModNatS0 (Succ xu340) (Succ xu360) (primGEqNatS xu340 xu360)",fontsize=16,color="magenta"];595 -> 1150[label="",style="dashed", color="magenta", weight=3];
595 -> 1151[label="",style="dashed", color="magenta", weight=3];
595 -> 1152[label="",style="dashed", color="magenta", weight=3];
595 -> 1153[label="",style="dashed", color="magenta", weight=3];
596[label="primModNatS0 (Succ xu340) Zero True",fontsize=16,color="black",shape="box"];596 -> 609[label="",style="solid", color="black", weight=3];
597[label="primModNatS0 Zero (Succ xu360) False",fontsize=16,color="black",shape="box"];597 -> 610[label="",style="solid", color="black", weight=3];
598[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];598 -> 611[label="",style="solid", color="black", weight=3];
1125 -> 1038[label="",style="dashed", color="red", weight=0];
1125[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS xu830 xu840)",fontsize=16,color="magenta"];1125 -> 1141[label="",style="dashed", color="magenta", weight=3];
1125 -> 1142[label="",style="dashed", color="magenta", weight=3];
1126[label="primDivNatS0 (Succ xu81) (Succ xu82) True",fontsize=16,color="black",shape="triangle"];1126 -> 1143[label="",style="solid", color="black", weight=3];
1127[label="primDivNatS0 (Succ xu81) (Succ xu82) False",fontsize=16,color="black",shape="box"];1127 -> 1144[label="",style="solid", color="black", weight=3];
1128 -> 1126[label="",style="dashed", color="red", weight=0];
1128[label="primDivNatS0 (Succ xu81) (Succ xu82) True",fontsize=16,color="magenta"];1349[label="primDivNatS (primMinusNatS (Succ xu980) xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="box"];1418[label="xu99/Succ xu990",fontsize=10,color="white",style="solid",shape="box"];1349 -> 1418[label="",style="solid", color="burlywood", weight=9];
1418 -> 1351[label="",style="solid", color="burlywood", weight=3];
1419[label="xu99/Zero",fontsize=10,color="white",style="solid",shape="box"];1349 -> 1419[label="",style="solid", color="burlywood", weight=9];
1419 -> 1352[label="",style="solid", color="burlywood", weight=3];
1350[label="primDivNatS (primMinusNatS Zero xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="box"];1420[label="xu99/Succ xu990",fontsize=10,color="white",style="solid",shape="box"];1350 -> 1420[label="",style="solid", color="burlywood", weight=9];
1420 -> 1353[label="",style="solid", color="burlywood", weight=3];
1421[label="xu99/Zero",fontsize=10,color="white",style="solid",shape="box"];1350 -> 1421[label="",style="solid", color="burlywood", weight=9];
1421 -> 1354[label="",style="solid", color="burlywood", weight=3];
1150[label="xu340",fontsize=16,color="green",shape="box"];1151[label="xu340",fontsize=16,color="green",shape="box"];1152[label="xu360",fontsize=16,color="green",shape="box"];1153[label="xu360",fontsize=16,color="green",shape="box"];1149[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS xu91 xu92)",fontsize=16,color="burlywood",shape="triangle"];1422[label="xu91/Succ xu910",fontsize=10,color="white",style="solid",shape="box"];1149 -> 1422[label="",style="solid", color="burlywood", weight=9];
1422 -> 1190[label="",style="solid", color="burlywood", weight=3];
1423[label="xu91/Zero",fontsize=10,color="white",style="solid",shape="box"];1149 -> 1423[label="",style="solid", color="burlywood", weight=9];
1423 -> 1191[label="",style="solid", color="burlywood", weight=3];
609 -> 1236[label="",style="dashed", color="red", weight=0];
609[label="primModNatS (primMinusNatS (Succ xu340) Zero) (Succ Zero)",fontsize=16,color="magenta"];609 -> 1237[label="",style="dashed", color="magenta", weight=3];
609 -> 1238[label="",style="dashed", color="magenta", weight=3];
609 -> 1239[label="",style="dashed", color="magenta", weight=3];
610[label="Succ Zero",fontsize=16,color="green",shape="box"];611 -> 1236[label="",style="dashed", color="red", weight=0];
611[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];611 -> 1240[label="",style="dashed", color="magenta", weight=3];
611 -> 1241[label="",style="dashed", color="magenta", weight=3];
611 -> 1242[label="",style="dashed", color="magenta", weight=3];
1141[label="xu840",fontsize=16,color="green",shape="box"];1142[label="xu830",fontsize=16,color="green",shape="box"];1143[label="Succ (primDivNatS (primMinusNatS (Succ xu81) (Succ xu82)) (Succ (Succ xu82)))",fontsize=16,color="green",shape="box"];1143 -> 1192[label="",style="dashed", color="green", weight=3];
1144[label="Zero",fontsize=16,color="green",shape="box"];1351[label="primDivNatS (primMinusNatS (Succ xu980) (Succ xu990)) (Succ xu100)",fontsize=16,color="black",shape="box"];1351 -> 1355[label="",style="solid", color="black", weight=3];
1352[label="primDivNatS (primMinusNatS (Succ xu980) Zero) (Succ xu100)",fontsize=16,color="black",shape="box"];1352 -> 1356[label="",style="solid", color="black", weight=3];
1353[label="primDivNatS (primMinusNatS Zero (Succ xu990)) (Succ xu100)",fontsize=16,color="black",shape="box"];1353 -> 1357[label="",style="solid", color="black", weight=3];
1354[label="primDivNatS (primMinusNatS Zero Zero) (Succ xu100)",fontsize=16,color="black",shape="box"];1354 -> 1358[label="",style="solid", color="black", weight=3];
1190[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) xu92)",fontsize=16,color="burlywood",shape="box"];1424[label="xu92/Succ xu920",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1424[label="",style="solid", color="burlywood", weight=9];
1424 -> 1197[label="",style="solid", color="burlywood", weight=3];
1425[label="xu92/Zero",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1425[label="",style="solid", color="burlywood", weight=9];
1425 -> 1198[label="",style="solid", color="burlywood", weight=3];
1191[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero xu92)",fontsize=16,color="burlywood",shape="box"];1426[label="xu92/Succ xu920",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1426[label="",style="solid", color="burlywood", weight=9];
1426 -> 1199[label="",style="solid", color="burlywood", weight=3];
1427[label="xu92/Zero",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1427[label="",style="solid", color="burlywood", weight=9];
1427 -> 1200[label="",style="solid", color="burlywood", weight=3];
1237[label="Zero",fontsize=16,color="green",shape="box"];1238[label="Succ xu340",fontsize=16,color="green",shape="box"];1239[label="Zero",fontsize=16,color="green",shape="box"];1236[label="primModNatS (primMinusNatS xu94 xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="triangle"];1428[label="xu94/Succ xu940",fontsize=10,color="white",style="solid",shape="box"];1236 -> 1428[label="",style="solid", color="burlywood", weight=9];
1428 -> 1267[label="",style="solid", color="burlywood", weight=3];
1429[label="xu94/Zero",fontsize=10,color="white",style="solid",shape="box"];1236 -> 1429[label="",style="solid", color="burlywood", weight=9];
1429 -> 1268[label="",style="solid", color="burlywood", weight=3];
1240[label="Zero",fontsize=16,color="green",shape="box"];1241[label="Zero",fontsize=16,color="green",shape="box"];1242[label="Zero",fontsize=16,color="green",shape="box"];1192 -> 1324[label="",style="dashed", color="red", weight=0];
1192[label="primDivNatS (primMinusNatS (Succ xu81) (Succ xu82)) (Succ (Succ xu82))",fontsize=16,color="magenta"];1192 -> 1331[label="",style="dashed", color="magenta", weight=3];
1192 -> 1332[label="",style="dashed", color="magenta", weight=3];
1192 -> 1333[label="",style="dashed", color="magenta", weight=3];
1355 -> 1324[label="",style="dashed", color="red", weight=0];
1355[label="primDivNatS (primMinusNatS xu980 xu990) (Succ xu100)",fontsize=16,color="magenta"];1355 -> 1359[label="",style="dashed", color="magenta", weight=3];
1355 -> 1360[label="",style="dashed", color="magenta", weight=3];
1356 -> 544[label="",style="dashed", color="red", weight=0];
1356[label="primDivNatS (Succ xu980) (Succ xu100)",fontsize=16,color="magenta"];1356 -> 1361[label="",style="dashed", color="magenta", weight=3];
1356 -> 1362[label="",style="dashed", color="magenta", weight=3];
1357[label="primDivNatS Zero (Succ xu100)",fontsize=16,color="black",shape="triangle"];1357 -> 1363[label="",style="solid", color="black", weight=3];
1358 -> 1357[label="",style="dashed", color="red", weight=0];
1358[label="primDivNatS Zero (Succ xu100)",fontsize=16,color="magenta"];1197[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) (Succ xu920))",fontsize=16,color="black",shape="box"];1197 -> 1208[label="",style="solid", color="black", weight=3];
1198[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) Zero)",fontsize=16,color="black",shape="box"];1198 -> 1209[label="",style="solid", color="black", weight=3];
1199[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero (Succ xu920))",fontsize=16,color="black",shape="box"];1199 -> 1210[label="",style="solid", color="black", weight=3];
1200[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1200 -> 1211[label="",style="solid", color="black", weight=3];
1267[label="primModNatS (primMinusNatS (Succ xu940) xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="box"];1430[label="xu95/Succ xu950",fontsize=10,color="white",style="solid",shape="box"];1267 -> 1430[label="",style="solid", color="burlywood", weight=9];
1430 -> 1275[label="",style="solid", color="burlywood", weight=3];
1431[label="xu95/Zero",fontsize=10,color="white",style="solid",shape="box"];1267 -> 1431[label="",style="solid", color="burlywood", weight=9];
1431 -> 1276[label="",style="solid", color="burlywood", weight=3];
1268[label="primModNatS (primMinusNatS Zero xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="box"];1432[label="xu95/Succ xu950",fontsize=10,color="white",style="solid",shape="box"];1268 -> 1432[label="",style="solid", color="burlywood", weight=9];
1432 -> 1277[label="",style="solid", color="burlywood", weight=3];
1433[label="xu95/Zero",fontsize=10,color="white",style="solid",shape="box"];1268 -> 1433[label="",style="solid", color="burlywood", weight=9];
1433 -> 1278[label="",style="solid", color="burlywood", weight=3];
1331[label="Succ xu82",fontsize=16,color="green",shape="box"];1332[label="Succ xu82",fontsize=16,color="green",shape="box"];1333[label="Succ xu81",fontsize=16,color="green",shape="box"];1359[label="xu990",fontsize=16,color="green",shape="box"];1360[label="xu980",fontsize=16,color="green",shape="box"];1361[label="xu980",fontsize=16,color="green",shape="box"];1362[label="xu100",fontsize=16,color="green",shape="box"];1363[label="Zero",fontsize=16,color="green",shape="box"];1208 -> 1149[label="",style="dashed", color="red", weight=0];
1208[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS xu910 xu920)",fontsize=16,color="magenta"];1208 -> 1218[label="",style="dashed", color="magenta", weight=3];
1208 -> 1219[label="",style="dashed", color="magenta", weight=3];
1209[label="primModNatS0 (Succ xu89) (Succ xu90) True",fontsize=16,color="black",shape="triangle"];1209 -> 1220[label="",style="solid", color="black", weight=3];
1210[label="primModNatS0 (Succ xu89) (Succ xu90) False",fontsize=16,color="black",shape="box"];1210 -> 1221[label="",style="solid", color="black", weight=3];
1211 -> 1209[label="",style="dashed", color="red", weight=0];
1211[label="primModNatS0 (Succ xu89) (Succ xu90) True",fontsize=16,color="magenta"];1275[label="primModNatS (primMinusNatS (Succ xu940) (Succ xu950)) (Succ xu96)",fontsize=16,color="black",shape="box"];1275 -> 1283[label="",style="solid", color="black", weight=3];
1276[label="primModNatS (primMinusNatS (Succ xu940) Zero) (Succ xu96)",fontsize=16,color="black",shape="box"];1276 -> 1284[label="",style="solid", color="black", weight=3];
1277[label="primModNatS (primMinusNatS Zero (Succ xu950)) (Succ xu96)",fontsize=16,color="black",shape="box"];1277 -> 1285[label="",style="solid", color="black", weight=3];
1278[label="primModNatS (primMinusNatS Zero Zero) (Succ xu96)",fontsize=16,color="black",shape="box"];1278 -> 1286[label="",style="solid", color="black", weight=3];
1218[label="xu910",fontsize=16,color="green",shape="box"];1219[label="xu920",fontsize=16,color="green",shape="box"];1220 -> 1236[label="",style="dashed", color="red", weight=0];
1220[label="primModNatS (primMinusNatS (Succ xu89) (Succ xu90)) (Succ (Succ xu90))",fontsize=16,color="magenta"];1220 -> 1249[label="",style="dashed", color="magenta", weight=3];
1220 -> 1250[label="",style="dashed", color="magenta", weight=3];
1220 -> 1251[label="",style="dashed", color="magenta", weight=3];
1221[label="Succ (Succ xu89)",fontsize=16,color="green",shape="box"];1283 -> 1236[label="",style="dashed", color="red", weight=0];
1283[label="primModNatS (primMinusNatS xu940 xu950) (Succ xu96)",fontsize=16,color="magenta"];1283 -> 1291[label="",style="dashed", color="magenta", weight=3];
1283 -> 1292[label="",style="dashed", color="magenta", weight=3];
1284 -> 561[label="",style="dashed", color="red", weight=0];
1284[label="primModNatS (Succ xu940) (Succ xu96)",fontsize=16,color="magenta"];1284 -> 1293[label="",style="dashed", color="magenta", weight=3];
1284 -> 1294[label="",style="dashed", color="magenta", weight=3];
1285[label="primModNatS Zero (Succ xu96)",fontsize=16,color="black",shape="triangle"];1285 -> 1295[label="",style="solid", color="black", weight=3];
1286 -> 1285[label="",style="dashed", color="red", weight=0];
1286[label="primModNatS Zero (Succ xu96)",fontsize=16,color="magenta"];1249[label="Succ xu90",fontsize=16,color="green",shape="box"];1250[label="Succ xu89",fontsize=16,color="green",shape="box"];1251[label="Succ xu90",fontsize=16,color="green",shape="box"];1291[label="xu940",fontsize=16,color="green",shape="box"];1292[label="xu950",fontsize=16,color="green",shape="box"];1293[label="xu96",fontsize=16,color="green",shape="box"];1294[label="xu940",fontsize=16,color="green",shape="box"];1295[label="Zero",fontsize=16,color="green",shape="box"];}

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

(12)
Complex Obligation (AND)

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS0(xu81, xu82, Zero, Zero) -> new_primDivNatS00(xu81, xu82)
   new_primDivNatS00(xu81, xu82) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))
   new_primDivNatS(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS(xu980, xu990, xu100)
   new_primDivNatS1(Succ(xu390), Zero) -> new_primDivNatS(Succ(xu390), Zero, Zero)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS0(xu81, xu82, xu830, xu840)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))
   new_primDivNatS1(Succ(xu390), Succ(xu400)) -> new_primDivNatS0(xu390, xu400, xu390, xu400)
   new_primDivNatS1(Zero, Zero) -> new_primDivNatS(Zero, Zero, Zero)
   new_primDivNatS(Succ(xu980), Zero, xu100) -> new_primDivNatS1(xu980, xu100)

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

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS00(xu81, xu82) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))
   new_primDivNatS(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS(xu980, xu990, xu100)
   new_primDivNatS(Succ(xu980), Zero, xu100) -> new_primDivNatS1(xu980, xu100)
   new_primDivNatS1(Succ(xu390), Zero) -> new_primDivNatS(Succ(xu390), Zero, Zero)
   new_primDivNatS1(Succ(xu390), Succ(xu400)) -> new_primDivNatS0(xu390, xu400, xu390, xu400)
   new_primDivNatS0(xu81, xu82, Zero, Zero) -> new_primDivNatS00(xu81, xu82)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS0(xu81, xu82, xu830, xu840)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))

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

(16) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_primDivNatS(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS(xu980, xu990, xu100)
   new_primDivNatS1(Succ(xu390), Zero) -> new_primDivNatS(Succ(xu390), Zero, Zero)
   new_primDivNatS1(Succ(xu390), Succ(xu400)) -> new_primDivNatS0(xu390, xu400, xu390, xu400)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 0
   POL(new_primDivNatS(x_1, x_2, x_3)) = x_1
   POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
   POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1
   POL(new_primDivNatS1(x_1, x_2)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primDivNatS00(xu81, xu82) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))
   new_primDivNatS(Succ(xu980), Zero, xu100) -> new_primDivNatS1(xu980, xu100)
   new_primDivNatS0(xu81, xu82, Zero, Zero) -> new_primDivNatS00(xu81, xu82)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS0(xu81, xu82, xu830, xu840)
   new_primDivNatS0(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS(Succ(xu81), Succ(xu82), Succ(xu82))

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

(18) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

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

   new_primDivNatS0(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS0(xu81, xu82, xu830, xu840)

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_primDivNatS0(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS0(xu81, xu82, xu830, xu840)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(21)
YES

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

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

   new_primShowInt(Neg(xu50)) -> new_primShowInt(Pos(xu50))
   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(new_div(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))

The TRS R consists of the following rules:

   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_div(xu39, xu40) -> Pos(new_primDivNatS4(xu39, xu40))
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_div(x0, x1)
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(23) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

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

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(new_div(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))

The TRS R consists of the following rules:

   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_div(xu39, xu40) -> Pos(new_primDivNatS4(xu39, xu40))
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_div(x0, x1)
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(25) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(new_div(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0] we obtained the following new rules [LPAR04]:

   (new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))),new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))


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

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

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

The TRS R consists of the following rules:

   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_div(xu39, xu40) -> Pos(new_primDivNatS4(xu39, xu40))
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_div(x0, x1)
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(27) 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.
----------------------------------------

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

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_div(x0, x1)
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(29) 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_div(x0, x1)


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

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(31) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

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

(33) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) the following chains were created:
*We consider the chain new_primShowInt(Pos(Succ(x0))) -> new_primShowInt(Pos(new_primDivNatS4(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))), new_primShowInt(Pos(Succ(x1))) -> new_primShowInt(Pos(new_primDivNatS4(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) which results in the following constraint:

(1)    (new_primShowInt(Pos(new_primDivNatS4(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))=new_primShowInt(Pos(Succ(x1)))  ==>  new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS4(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))



We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:

(2)    (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=x2 & new_primDivNatS4(x0, x2)=Succ(x1)  ==>  new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS4(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS4(x0, x2)=Succ(x1) which results in the following new constraints:

(3)    (new_primDivNatS01(x4, x3, x4, x3)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Succ(x3)  ==>  new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))

(4)    (Succ(new_primDivNatS3(Zero, Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero  ==>  new_primShowInt(Pos(Succ(Zero)))_>=_new_primShowInt(Pos(new_primDivNatS4(Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))

(5)    (Succ(new_primDivNatS3(Succ(x6), Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(x6))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x6), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))



We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint:

(6)    (x4=x7 & x3=x8 & new_primDivNatS01(x4, x3, x7, x8)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3  ==>  new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))



We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x4, x3, x7, x8)=Succ(x1) which results in the following new constraints:

(7)    (new_primDivNatS02(x10, x9)=Succ(x1) & x10=Zero & x9=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x9  ==>  new_primShowInt(Pos(Succ(Succ(x10))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x10), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))

(8)    (new_primDivNatS02(x16, x15)=Succ(x1) & x16=Succ(x14) & x15=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15  ==>  new_primShowInt(Pos(Succ(Succ(x16))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x16), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))

(9)    (new_primDivNatS01(x20, x19, x18, x17)=Succ(x1) & x20=Succ(x18) & x19=Succ(x17) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x19 & (\/x21:new_primDivNatS01(x20, x19, x18, x17)=Succ(x21) & x20=x18 & x19=x17 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x19  ==>  new_primShowInt(Pos(Succ(Succ(x20))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x20), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))  ==>  new_primShowInt(Pos(Succ(Succ(x20))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(x20), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))



We solved constraint (7) using rules (I), (II), (III).We solved constraint (8) using rules (I), (II), (III).We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:

(10)    (new_primShowInt(Pos(Succ(Succ(Succ(x18)))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(Succ(x18)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

*(new_primShowInt(Pos(Succ(Succ(Succ(x18)))))_>=_new_primShowInt(Pos(new_primDivNatS4(Succ(Succ(x18)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(35) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(xu500))) -> new_primShowInt(Pos(new_primDivNatS4(xu500, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))),new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))
   (new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)))


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

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))
   new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(37) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(39) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)))
   (new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))),new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))


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

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero))
   new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(43) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)))
   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))


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

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero))
   new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(45) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(47) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) at position [0,0] we obtained the following new rules [LPAR04]:

   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)))
   (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))))


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

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero))
   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(49) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(51) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

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

(53) InductionCalculusProof (EQUIVALENT)
Note that final constraints are written in bold face.



For Pair new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) the following chains were created:
*We consider the chain new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x1))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Zero)))))))) which results in the following constraint:

(1)    (new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero))))))))=new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1)))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))))



We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint:

(2)    (Succ(Succ(Succ(x0)))=x2 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x4 & new_primDivNatS01(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1)))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))))



We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints:

(3)    (new_primDivNatS02(x6, x5)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Zero)))=x6 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x5 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Succ(Succ(Succ(Succ(Succ(Zero)))))))))

(4)    (new_primDivNatS02(x12, x11)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x10))))=x12 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x11 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x10))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x10)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x10), Succ(Succ(Succ(Succ(Succ(Zero)))))))))

(5)    (new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x14))))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Succ(x13) & (\/x17:new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x17))))) & Succ(Succ(Succ(x14)))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x13  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x14)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x14))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x14, Succ(Succ(Succ(Succ(Succ(Zero)))))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x14))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x14)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x14), Succ(Succ(Succ(Succ(Succ(Zero)))))))))



We solved constraint (3) using rules (I), (II).We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:

(6)    (new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x14))))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Zero))))=x13  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x14))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x14)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x14), Succ(Succ(Succ(Succ(Succ(Zero)))))))))



We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints:

(7)    (new_primDivNatS02(x19, x18)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Zero))))=x19 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x18 & Succ(Succ(Succ(Succ(Zero))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Zero)))))))))

(8)    (new_primDivNatS02(x25, x24)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x23)))))=x25 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x24 & Succ(Succ(Succ(Succ(Zero))))=Zero  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x23)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x23))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x23)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))

(9)    (new_primDivNatS01(x29, x28, x27, x26)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x27)))))=x29 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x28 & Succ(Succ(Succ(Succ(Zero))))=Succ(x26) & (\/x30:new_primDivNatS01(x29, x28, x27, x26)=Succ(Succ(Succ(Succ(Succ(x30))))) & Succ(Succ(Succ(Succ(x27))))=x29 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x28 & Succ(Succ(Succ(Succ(Zero))))=x26  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x27))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x27)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x27), Succ(Succ(Succ(Succ(Succ(Zero)))))))))  ==>  new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))



We solved constraint (7) using rules (I), (II).We solved constraint (8) using rules (I), (II).We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint:

(10)    (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))








To summarize, we get the following constraints P__>=_ for the following pairs.

*new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))

*(new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero)))))))))








The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))

The TRS R consists of the following rules:

   new_primDivNatS4(Succ(xu390), Succ(xu400)) -> new_primDivNatS01(xu390, xu400, xu390, xu400)
   new_primDivNatS4(Zero, Succ(xu400)) -> Zero
   new_primDivNatS01(xu81, xu82, Zero, Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Zero, Succ(xu840)) -> Zero
   new_primDivNatS01(xu81, xu82, Succ(xu830), Zero) -> new_primDivNatS02(xu81, xu82)
   new_primDivNatS01(xu81, xu82, Succ(xu830), Succ(xu840)) -> new_primDivNatS01(xu81, xu82, xu830, xu840)
   new_primDivNatS02(xu81, xu82) -> Succ(new_primDivNatS3(Succ(xu81), Succ(xu82), Succ(xu82)))
   new_primDivNatS3(Succ(xu980), Succ(xu990), xu100) -> new_primDivNatS3(xu980, xu990, xu100)
   new_primDivNatS3(Succ(xu980), Zero, xu100) -> new_primDivNatS4(xu980, xu100)
   new_primDivNatS3(Zero, Succ(xu990), xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS3(Zero, Zero, xu100) -> new_primDivNatS2(xu100)
   new_primDivNatS2(xu100) -> Zero
   new_primDivNatS4(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero))
   new_primDivNatS4(Succ(xu390), Zero) -> Succ(new_primDivNatS3(Succ(xu390), Zero, Zero))

The set Q consists of the following terms:

   new_primDivNatS01(x0, x1, Succ(x2), Zero)
   new_primDivNatS4(Zero, Succ(x0))
   new_primDivNatS4(Succ(x0), Zero)
   new_primDivNatS02(x0, x1)
   new_primDivNatS2(x0)
   new_primDivNatS3(Succ(x0), Succ(x1), x2)
   new_primDivNatS01(x0, x1, Zero, Succ(x2))
   new_primDivNatS4(Succ(x0), Succ(x1))
   new_primDivNatS3(Zero, Succ(x0), x1)
   new_primDivNatS01(x0, x1, Zero, Zero)
   new_primDivNatS3(Zero, Zero, x0)
   new_primDivNatS4(Zero, Zero)
   new_primDivNatS3(Succ(x0), Zero, x1)
   new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))

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

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_putStr(:(xu280, xu281)) -> new_putStr(xu281)

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

(56) 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_putStr(:(xu280, xu281)) -> new_putStr(xu281)
The graph contains the following edges 1 > 1


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

(57)
YES

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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS(Succ(xu940), Zero, xu96) -> new_primModNatS1(xu940, xu96)
   new_primModNatS1(Zero, Zero) -> new_primModNatS(Zero, Zero, Zero)
   new_primModNatS00(xu89, xu90) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))
   new_primModNatS0(xu89, xu90, Succ(xu910), Zero) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))
   new_primModNatS(Succ(xu940), Succ(xu950), xu96) -> new_primModNatS(xu940, xu950, xu96)
   new_primModNatS0(xu89, xu90, Succ(xu910), Succ(xu920)) -> new_primModNatS0(xu89, xu90, xu910, xu920)
   new_primModNatS1(Succ(xu340), Succ(xu360)) -> new_primModNatS0(xu340, xu360, xu340, xu360)
   new_primModNatS0(xu89, xu90, Zero, Zero) -> new_primModNatS00(xu89, xu90)
   new_primModNatS1(Succ(xu340), Zero) -> new_primModNatS(Succ(xu340), Zero, Zero)

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

(59) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS1(Succ(xu340), Succ(xu360)) -> new_primModNatS0(xu340, xu360, xu340, xu360)
   new_primModNatS0(xu89, xu90, Succ(xu910), Zero) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))
   new_primModNatS(Succ(xu940), Succ(xu950), xu96) -> new_primModNatS(xu940, xu950, xu96)
   new_primModNatS(Succ(xu940), Zero, xu96) -> new_primModNatS1(xu940, xu96)
   new_primModNatS1(Succ(xu340), Zero) -> new_primModNatS(Succ(xu340), Zero, Zero)
   new_primModNatS0(xu89, xu90, Succ(xu910), Succ(xu920)) -> new_primModNatS0(xu89, xu90, xu910, xu920)
   new_primModNatS0(xu89, xu90, Zero, Zero) -> new_primModNatS00(xu89, xu90)
   new_primModNatS00(xu89, xu90) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))

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

(61) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_primModNatS1(Succ(xu340), Succ(xu360)) -> new_primModNatS0(xu340, xu360, xu340, xu360)
   new_primModNatS(Succ(xu940), Succ(xu950), xu96) -> new_primModNatS(xu940, xu950, xu96)
   new_primModNatS1(Succ(xu340), Zero) -> new_primModNatS(Succ(xu340), Zero, Zero)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(Succ(x_1)) = 1 + x_1
   POL(Zero) = 0
   POL(new_primModNatS(x_1, x_2, x_3)) = x_1
   POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1
   POL(new_primModNatS00(x_1, x_2)) = 1 + x_1
   POL(new_primModNatS1(x_1, x_2)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(xu89, xu90, Succ(xu910), Zero) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))
   new_primModNatS(Succ(xu940), Zero, xu96) -> new_primModNatS1(xu940, xu96)
   new_primModNatS0(xu89, xu90, Succ(xu910), Succ(xu920)) -> new_primModNatS0(xu89, xu90, xu910, xu920)
   new_primModNatS0(xu89, xu90, Zero, Zero) -> new_primModNatS00(xu89, xu90)
   new_primModNatS00(xu89, xu90) -> new_primModNatS(Succ(xu89), Succ(xu90), Succ(xu90))

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

(63) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primModNatS0(xu89, xu90, Succ(xu910), Succ(xu920)) -> new_primModNatS0(xu89, xu90, xu910, xu920)

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

(65) 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_primModNatS0(xu89, xu90, Succ(xu910), Succ(xu920)) -> new_primModNatS0(xu89, xu90, xu910, xu920)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4


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

(66)
YES

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

(67)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(xu370, xu371), xu34, xu36) -> new_psPs(xu371, xu34, xu36)

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

(68) 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_psPs(:(xu370, xu371), xu34, xu36) -> new_psPs(xu371, xu34, xu36)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(69)
YES

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

(70) Narrow (COMPLETE)
Haskell To QDPs

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1395 -> 519[label="",style="solid", color="burlywood", weight=3];
500[label="Pos xu50",fontsize=16,color="green",shape="box"];501[label="xu29",fontsize=16,color="green",shape="box"];502[label="putChar xu280",fontsize=16,color="black",shape="box"];502 -> 520[label="",style="solid", color="black", weight=3];
503[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];503 -> 521[label="",style="solid", color="black", weight=3];
504[label="randomSelect1 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];504 -> 522[label="",style="solid", color="black", weight=3];
517 -> 523[label="",style="dashed", color="red", weight=0];
517[label="div Pos (Succ xu500) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))",fontsize=16,color="magenta"];517 -> 524[label="",style="dashed", color="magenta", weight=3];
517 -> 525[label="",style="dashed", color="magenta", weight=3];
518[label="(xu370 : xu371) ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="black",shape="box"];518 -> 526[label="",style="solid", color="black", weight=3];
519[label="[] ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="black",shape="box"];519 -> 527[label="",style="solid", color="black", weight=3];
520 -> 528[label="",style="dashed", color="red", weight=0];
520[label="(seq xu280 output)",fontsize=16,color="magenta"];520 -> 531[label="",style="dashed", color="magenta", weight=3];
521[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];521 -> 536[label="",style="solid", color="black", weight=3];
522[label="randomSelect (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];522 -> 537[label="",style="solid", color="black", weight=3];
524[label="xu500",fontsize=16,color="green",shape="box"];525[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];523[label="div Pos (Succ xu39) Pos (Succ xu40)",fontsize=16,color="black",shape="triangle"];523 -> 533[label="",style="solid", color="black", weight=3];
526[label="xu370 : xu371 ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="green",shape="box"];526 -> 534[label="",style="dashed", color="green", weight=3];
527[label="toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="green",shape="box"];527 -> 535[label="",style="dashed", color="green", weight=3];
531 -> 458[label="",style="dashed", color="red", weight=0];
531[label="output",fontsize=16,color="magenta"];536[label="randomSelect0 (aIOE IOError_FullError) (aIOE IOError_PermDenied : AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];536 -> 542[label="",style="solid", color="black", weight=3];
537[label="randomSelect2 (aIOE IOError_PermDenied : AProVE_IO () : [])",fontsize=16,color="black",shape="box"];537 -> 543[label="",style="solid", color="black", weight=3];
533[label="primDivInt (Pos (Succ xu39)) (Pos (Succ xu40))",fontsize=16,color="black",shape="box"];533 -> 539[label="",style="solid", color="black", weight=3];
534 -> 512[label="",style="dashed", color="red", weight=0];
534[label="xu371 ++ toEnum (mod Pos (Succ xu34) Pos (Succ xu36)) : []",fontsize=16,color="magenta"];534 -> 540[label="",style="dashed", color="magenta", weight=3];
535[label="toEnum (mod Pos (Succ xu34) Pos (Succ xu36))",fontsize=16,color="black",shape="box"];535 -> 541[label="",style="solid", color="black", weight=3];
542[label="aIOE IOError_FullError",fontsize=16,color="black",shape="box"];542 -> 546[label="",style="solid", color="black", weight=3];
543[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) terminator",fontsize=16,color="black",shape="box"];543 -> 547[label="",style="solid", color="black", weight=3];
539[label="Pos (primDivNatS (Succ xu39) (Succ xu40))",fontsize=16,color="green",shape="box"];539 -> 544[label="",style="dashed", color="green", weight=3];
540[label="xu371",fontsize=16,color="green",shape="box"];541[label="primIntToChar (mod Pos (Succ xu34) Pos (Succ xu36))",fontsize=16,color="black",shape="box"];541 -> 545[label="",style="solid", color="black", weight=3];
546[label="AProVE_Exception (AET_IOError (IOError IOError_FullError [] [] Nothing))",fontsize=16,color="green",shape="box"];547[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) ter6m",fontsize=16,color="burlywood",shape="box"];1396[label="ter6m/False",fontsize=10,color="white",style="solid",shape="box"];547 -> 1396[label="",style="solid", color="burlywood", weight=9];
1396 -> 550[label="",style="solid", color="burlywood", weight=3];
1397[label="ter6m/True",fontsize=10,color="white",style="solid",shape="box"];547 -> 1397[label="",style="solid", color="burlywood", weight=9];
1397 -> 551[label="",style="solid", color="burlywood", weight=3];
544[label="primDivNatS (Succ xu39) (Succ xu40)",fontsize=16,color="black",shape="triangle"];544 -> 548[label="",style="solid", color="black", weight=3];
545[label="primIntToChar (primModInt (Pos (Succ xu34)) (Pos (Succ xu36)))",fontsize=16,color="black",shape="box"];545 -> 549[label="",style="solid", color="black", weight=3];
550[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) False",fontsize=16,color="black",shape="box"];550 -> 555[label="",style="solid", color="black", weight=3];
551[label="randomSelect1 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];551 -> 556[label="",style="solid", color="black", weight=3];
548[label="primDivNatS0 xu39 xu40 (primGEqNatS xu39 xu40)",fontsize=16,color="burlywood",shape="box"];1398[label="xu39/Succ xu390",fontsize=10,color="white",style="solid",shape="box"];548 -> 1398[label="",style="solid", color="burlywood", weight=9];
1398 -> 552[label="",style="solid", color="burlywood", weight=3];
1399[label="xu39/Zero",fontsize=10,color="white",style="solid",shape="box"];548 -> 1399[label="",style="solid", color="burlywood", weight=9];
1399 -> 553[label="",style="solid", color="burlywood", weight=3];
549[label="primIntToChar (Pos (primModNatS (Succ xu34) (Succ xu36)))",fontsize=16,color="black",shape="box"];549 -> 554[label="",style="solid", color="black", weight=3];
555[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) otherwise",fontsize=16,color="black",shape="box"];555 -> 562[label="",style="solid", color="black", weight=3];
556[label="randomSelect (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];556 -> 563[label="",style="solid", color="black", weight=3];
552[label="primDivNatS0 (Succ xu390) xu40 (primGEqNatS (Succ xu390) xu40)",fontsize=16,color="burlywood",shape="box"];1400[label="xu40/Succ xu400",fontsize=10,color="white",style="solid",shape="box"];552 -> 1400[label="",style="solid", color="burlywood", weight=9];
1400 -> 557[label="",style="solid", color="burlywood", weight=3];
1401[label="xu40/Zero",fontsize=10,color="white",style="solid",shape="box"];552 -> 1401[label="",style="solid", color="burlywood", weight=9];
1401 -> 558[label="",style="solid", color="burlywood", weight=3];
553[label="primDivNatS0 Zero xu40 (primGEqNatS Zero xu40)",fontsize=16,color="burlywood",shape="box"];1402[label="xu40/Succ xu400",fontsize=10,color="white",style="solid",shape="box"];553 -> 1402[label="",style="solid", color="burlywood", weight=9];
1402 -> 559[label="",style="solid", color="burlywood", weight=3];
1403[label="xu40/Zero",fontsize=10,color="white",style="solid",shape="box"];553 -> 1403[label="",style="solid", color="burlywood", weight=9];
1403 -> 560[label="",style="solid", color="burlywood", weight=3];
554[label="Char (primModNatS (Succ xu34) (Succ xu36))",fontsize=16,color="green",shape="box"];554 -> 561[label="",style="dashed", color="green", weight=3];
562[label="randomSelect0 (aIOE IOError_PermDenied) (AProVE_IO () : []) True",fontsize=16,color="black",shape="box"];562 -> 569[label="",style="solid", color="black", weight=3];
563[label="randomSelect3 (AProVE_IO () : [])",fontsize=16,color="black",shape="box"];563 -> 570[label="",style="solid", color="black", weight=3];
557[label="primDivNatS0 (Succ xu390) (Succ xu400) (primGEqNatS (Succ xu390) (Succ xu400))",fontsize=16,color="black",shape="box"];557 -> 564[label="",style="solid", color="black", weight=3];
558[label="primDivNatS0 (Succ xu390) Zero (primGEqNatS (Succ xu390) Zero)",fontsize=16,color="black",shape="box"];558 -> 565[label="",style="solid", color="black", weight=3];
559[label="primDivNatS0 Zero (Succ xu400) (primGEqNatS Zero (Succ xu400))",fontsize=16,color="black",shape="box"];559 -> 566[label="",style="solid", color="black", weight=3];
560[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];560 -> 567[label="",style="solid", color="black", weight=3];
561[label="primModNatS (Succ xu34) (Succ xu36)",fontsize=16,color="black",shape="triangle"];561 -> 568[label="",style="solid", color="black", weight=3];
569[label="aIOE IOError_PermDenied",fontsize=16,color="black",shape="box"];569 -> 578[label="",style="solid", color="black", weight=3];
570[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];564 -> 1038[label="",style="dashed", color="red", weight=0];
564[label="primDivNatS0 (Succ xu390) (Succ xu400) (primGEqNatS xu390 xu400)",fontsize=16,color="magenta"];564 -> 1039[label="",style="dashed", color="magenta", weight=3];
564 -> 1040[label="",style="dashed", color="magenta", weight=3];
564 -> 1041[label="",style="dashed", color="magenta", weight=3];
564 -> 1042[label="",style="dashed", color="magenta", weight=3];
565[label="primDivNatS0 (Succ xu390) Zero True",fontsize=16,color="black",shape="box"];565 -> 573[label="",style="solid", color="black", weight=3];
566[label="primDivNatS0 Zero (Succ xu400) False",fontsize=16,color="black",shape="box"];566 -> 574[label="",style="solid", color="black", weight=3];
567[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];567 -> 575[label="",style="solid", color="black", weight=3];
568[label="primModNatS0 xu34 xu36 (primGEqNatS xu34 xu36)",fontsize=16,color="burlywood",shape="box"];1404[label="xu34/Succ xu340",fontsize=10,color="white",style="solid",shape="box"];568 -> 1404[label="",style="solid", color="burlywood", weight=9];
1404 -> 576[label="",style="solid", color="burlywood", weight=3];
1405[label="xu34/Zero",fontsize=10,color="white",style="solid",shape="box"];568 -> 1405[label="",style="solid", color="burlywood", weight=9];
1405 -> 577[label="",style="solid", color="burlywood", weight=3];
578[label="AProVE_Exception (AET_IOError (IOError IOError_PermDenied [] [] Nothing))",fontsize=16,color="green",shape="box"];1039[label="xu400",fontsize=16,color="green",shape="box"];1040[label="xu390",fontsize=16,color="green",shape="box"];1041[label="xu400",fontsize=16,color="green",shape="box"];1042[label="xu390",fontsize=16,color="green",shape="box"];1038[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS xu83 xu84)",fontsize=16,color="burlywood",shape="triangle"];1406[label="xu83/Succ xu830",fontsize=10,color="white",style="solid",shape="box"];1038 -> 1406[label="",style="solid", color="burlywood", weight=9];
1406 -> 1079[label="",style="solid", color="burlywood", weight=3];
1407[label="xu83/Zero",fontsize=10,color="white",style="solid",shape="box"];1038 -> 1407[label="",style="solid", color="burlywood", weight=9];
1407 -> 1080[label="",style="solid", color="burlywood", weight=3];
573[label="Succ (primDivNatS (primMinusNatS (Succ xu390) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];573 -> 583[label="",style="dashed", color="green", weight=3];
574[label="Zero",fontsize=16,color="green",shape="box"];575[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];575 -> 584[label="",style="dashed", color="green", weight=3];
576[label="primModNatS0 (Succ xu340) xu36 (primGEqNatS (Succ xu340) xu36)",fontsize=16,color="burlywood",shape="box"];1408[label="xu36/Succ xu360",fontsize=10,color="white",style="solid",shape="box"];576 -> 1408[label="",style="solid", color="burlywood", weight=9];
1408 -> 585[label="",style="solid", color="burlywood", weight=3];
1409[label="xu36/Zero",fontsize=10,color="white",style="solid",shape="box"];576 -> 1409[label="",style="solid", color="burlywood", weight=9];
1409 -> 586[label="",style="solid", color="burlywood", weight=3];
577[label="primModNatS0 Zero xu36 (primGEqNatS Zero xu36)",fontsize=16,color="burlywood",shape="box"];1410[label="xu36/Succ xu360",fontsize=10,color="white",style="solid",shape="box"];577 -> 1410[label="",style="solid", color="burlywood", weight=9];
1410 -> 587[label="",style="solid", color="burlywood", weight=3];
1411[label="xu36/Zero",fontsize=10,color="white",style="solid",shape="box"];577 -> 1411[label="",style="solid", color="burlywood", weight=9];
1411 -> 588[label="",style="solid", color="burlywood", weight=3];
1079[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) xu84)",fontsize=16,color="burlywood",shape="box"];1412[label="xu84/Succ xu840",fontsize=10,color="white",style="solid",shape="box"];1079 -> 1412[label="",style="solid", color="burlywood", weight=9];
1412 -> 1109[label="",style="solid", color="burlywood", weight=3];
1413[label="xu84/Zero",fontsize=10,color="white",style="solid",shape="box"];1079 -> 1413[label="",style="solid", color="burlywood", weight=9];
1413 -> 1110[label="",style="solid", color="burlywood", weight=3];
1080[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero xu84)",fontsize=16,color="burlywood",shape="box"];1414[label="xu84/Succ xu840",fontsize=10,color="white",style="solid",shape="box"];1080 -> 1414[label="",style="solid", color="burlywood", weight=9];
1414 -> 1111[label="",style="solid", color="burlywood", weight=3];
1415[label="xu84/Zero",fontsize=10,color="white",style="solid",shape="box"];1080 -> 1415[label="",style="solid", color="burlywood", weight=9];
1415 -> 1112[label="",style="solid", color="burlywood", weight=3];
583 -> 1324[label="",style="dashed", color="red", weight=0];
583[label="primDivNatS (primMinusNatS (Succ xu390) Zero) (Succ Zero)",fontsize=16,color="magenta"];583 -> 1325[label="",style="dashed", color="magenta", weight=3];
583 -> 1326[label="",style="dashed", color="magenta", weight=3];
583 -> 1327[label="",style="dashed", color="magenta", weight=3];
584 -> 1324[label="",style="dashed", color="red", weight=0];
584[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];584 -> 1328[label="",style="dashed", color="magenta", weight=3];
584 -> 1329[label="",style="dashed", color="magenta", weight=3];
584 -> 1330[label="",style="dashed", color="magenta", weight=3];
585[label="primModNatS0 (Succ xu340) (Succ xu360) (primGEqNatS (Succ xu340) (Succ xu360))",fontsize=16,color="black",shape="box"];585 -> 595[label="",style="solid", color="black", weight=3];
586[label="primModNatS0 (Succ xu340) Zero (primGEqNatS (Succ xu340) Zero)",fontsize=16,color="black",shape="box"];586 -> 596[label="",style="solid", color="black", weight=3];
587[label="primModNatS0 Zero (Succ xu360) (primGEqNatS Zero (Succ xu360))",fontsize=16,color="black",shape="box"];587 -> 597[label="",style="solid", color="black", weight=3];
588[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];588 -> 598[label="",style="solid", color="black", weight=3];
1109[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) (Succ xu840))",fontsize=16,color="black",shape="box"];1109 -> 1125[label="",style="solid", color="black", weight=3];
1110[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS (Succ xu830) Zero)",fontsize=16,color="black",shape="box"];1110 -> 1126[label="",style="solid", color="black", weight=3];
1111[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero (Succ xu840))",fontsize=16,color="black",shape="box"];1111 -> 1127[label="",style="solid", color="black", weight=3];
1112[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1112 -> 1128[label="",style="solid", color="black", weight=3];
1325[label="Zero",fontsize=16,color="green",shape="box"];1326[label="Zero",fontsize=16,color="green",shape="box"];1327[label="Succ xu390",fontsize=16,color="green",shape="box"];1324[label="primDivNatS (primMinusNatS xu98 xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="triangle"];1416[label="xu98/Succ xu980",fontsize=10,color="white",style="solid",shape="box"];1324 -> 1416[label="",style="solid", color="burlywood", weight=9];
1416 -> 1349[label="",style="solid", color="burlywood", weight=3];
1417[label="xu98/Zero",fontsize=10,color="white",style="solid",shape="box"];1324 -> 1417[label="",style="solid", color="burlywood", weight=9];
1417 -> 1350[label="",style="solid", color="burlywood", weight=3];
1328[label="Zero",fontsize=16,color="green",shape="box"];1329[label="Zero",fontsize=16,color="green",shape="box"];1330[label="Zero",fontsize=16,color="green",shape="box"];595 -> 1149[label="",style="dashed", color="red", weight=0];
595[label="primModNatS0 (Succ xu340) (Succ xu360) (primGEqNatS xu340 xu360)",fontsize=16,color="magenta"];595 -> 1150[label="",style="dashed", color="magenta", weight=3];
595 -> 1151[label="",style="dashed", color="magenta", weight=3];
595 -> 1152[label="",style="dashed", color="magenta", weight=3];
595 -> 1153[label="",style="dashed", color="magenta", weight=3];
596[label="primModNatS0 (Succ xu340) Zero True",fontsize=16,color="black",shape="box"];596 -> 609[label="",style="solid", color="black", weight=3];
597[label="primModNatS0 Zero (Succ xu360) False",fontsize=16,color="black",shape="box"];597 -> 610[label="",style="solid", color="black", weight=3];
598[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];598 -> 611[label="",style="solid", color="black", weight=3];
1125 -> 1038[label="",style="dashed", color="red", weight=0];
1125[label="primDivNatS0 (Succ xu81) (Succ xu82) (primGEqNatS xu830 xu840)",fontsize=16,color="magenta"];1125 -> 1141[label="",style="dashed", color="magenta", weight=3];
1125 -> 1142[label="",style="dashed", color="magenta", weight=3];
1126[label="primDivNatS0 (Succ xu81) (Succ xu82) True",fontsize=16,color="black",shape="triangle"];1126 -> 1143[label="",style="solid", color="black", weight=3];
1127[label="primDivNatS0 (Succ xu81) (Succ xu82) False",fontsize=16,color="black",shape="box"];1127 -> 1144[label="",style="solid", color="black", weight=3];
1128 -> 1126[label="",style="dashed", color="red", weight=0];
1128[label="primDivNatS0 (Succ xu81) (Succ xu82) True",fontsize=16,color="magenta"];1349[label="primDivNatS (primMinusNatS (Succ xu980) xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="box"];1418[label="xu99/Succ xu990",fontsize=10,color="white",style="solid",shape="box"];1349 -> 1418[label="",style="solid", color="burlywood", weight=9];
1418 -> 1351[label="",style="solid", color="burlywood", weight=3];
1419[label="xu99/Zero",fontsize=10,color="white",style="solid",shape="box"];1349 -> 1419[label="",style="solid", color="burlywood", weight=9];
1419 -> 1352[label="",style="solid", color="burlywood", weight=3];
1350[label="primDivNatS (primMinusNatS Zero xu99) (Succ xu100)",fontsize=16,color="burlywood",shape="box"];1420[label="xu99/Succ xu990",fontsize=10,color="white",style="solid",shape="box"];1350 -> 1420[label="",style="solid", color="burlywood", weight=9];
1420 -> 1353[label="",style="solid", color="burlywood", weight=3];
1421[label="xu99/Zero",fontsize=10,color="white",style="solid",shape="box"];1350 -> 1421[label="",style="solid", color="burlywood", weight=9];
1421 -> 1354[label="",style="solid", color="burlywood", weight=3];
1150[label="xu340",fontsize=16,color="green",shape="box"];1151[label="xu340",fontsize=16,color="green",shape="box"];1152[label="xu360",fontsize=16,color="green",shape="box"];1153[label="xu360",fontsize=16,color="green",shape="box"];1149[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS xu91 xu92)",fontsize=16,color="burlywood",shape="triangle"];1422[label="xu91/Succ xu910",fontsize=10,color="white",style="solid",shape="box"];1149 -> 1422[label="",style="solid", color="burlywood", weight=9];
1422 -> 1190[label="",style="solid", color="burlywood", weight=3];
1423[label="xu91/Zero",fontsize=10,color="white",style="solid",shape="box"];1149 -> 1423[label="",style="solid", color="burlywood", weight=9];
1423 -> 1191[label="",style="solid", color="burlywood", weight=3];
609 -> 1236[label="",style="dashed", color="red", weight=0];
609[label="primModNatS (primMinusNatS (Succ xu340) Zero) (Succ Zero)",fontsize=16,color="magenta"];609 -> 1237[label="",style="dashed", color="magenta", weight=3];
609 -> 1238[label="",style="dashed", color="magenta", weight=3];
609 -> 1239[label="",style="dashed", color="magenta", weight=3];
610[label="Succ Zero",fontsize=16,color="green",shape="box"];611 -> 1236[label="",style="dashed", color="red", weight=0];
611[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];611 -> 1240[label="",style="dashed", color="magenta", weight=3];
611 -> 1241[label="",style="dashed", color="magenta", weight=3];
611 -> 1242[label="",style="dashed", color="magenta", weight=3];
1141[label="xu840",fontsize=16,color="green",shape="box"];1142[label="xu830",fontsize=16,color="green",shape="box"];1143[label="Succ (primDivNatS (primMinusNatS (Succ xu81) (Succ xu82)) (Succ (Succ xu82)))",fontsize=16,color="green",shape="box"];1143 -> 1192[label="",style="dashed", color="green", weight=3];
1144[label="Zero",fontsize=16,color="green",shape="box"];1351[label="primDivNatS (primMinusNatS (Succ xu980) (Succ xu990)) (Succ xu100)",fontsize=16,color="black",shape="box"];1351 -> 1355[label="",style="solid", color="black", weight=3];
1352[label="primDivNatS (primMinusNatS (Succ xu980) Zero) (Succ xu100)",fontsize=16,color="black",shape="box"];1352 -> 1356[label="",style="solid", color="black", weight=3];
1353[label="primDivNatS (primMinusNatS Zero (Succ xu990)) (Succ xu100)",fontsize=16,color="black",shape="box"];1353 -> 1357[label="",style="solid", color="black", weight=3];
1354[label="primDivNatS (primMinusNatS Zero Zero) (Succ xu100)",fontsize=16,color="black",shape="box"];1354 -> 1358[label="",style="solid", color="black", weight=3];
1190[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) xu92)",fontsize=16,color="burlywood",shape="box"];1424[label="xu92/Succ xu920",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1424[label="",style="solid", color="burlywood", weight=9];
1424 -> 1197[label="",style="solid", color="burlywood", weight=3];
1425[label="xu92/Zero",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1425[label="",style="solid", color="burlywood", weight=9];
1425 -> 1198[label="",style="solid", color="burlywood", weight=3];
1191[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero xu92)",fontsize=16,color="burlywood",shape="box"];1426[label="xu92/Succ xu920",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1426[label="",style="solid", color="burlywood", weight=9];
1426 -> 1199[label="",style="solid", color="burlywood", weight=3];
1427[label="xu92/Zero",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1427[label="",style="solid", color="burlywood", weight=9];
1427 -> 1200[label="",style="solid", color="burlywood", weight=3];
1237[label="Zero",fontsize=16,color="green",shape="box"];1238[label="Succ xu340",fontsize=16,color="green",shape="box"];1239[label="Zero",fontsize=16,color="green",shape="box"];1236[label="primModNatS (primMinusNatS xu94 xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="triangle"];1428[label="xu94/Succ xu940",fontsize=10,color="white",style="solid",shape="box"];1236 -> 1428[label="",style="solid", color="burlywood", weight=9];
1428 -> 1267[label="",style="solid", color="burlywood", weight=3];
1429[label="xu94/Zero",fontsize=10,color="white",style="solid",shape="box"];1236 -> 1429[label="",style="solid", color="burlywood", weight=9];
1429 -> 1268[label="",style="solid", color="burlywood", weight=3];
1240[label="Zero",fontsize=16,color="green",shape="box"];1241[label="Zero",fontsize=16,color="green",shape="box"];1242[label="Zero",fontsize=16,color="green",shape="box"];1192 -> 1324[label="",style="dashed", color="red", weight=0];
1192[label="primDivNatS (primMinusNatS (Succ xu81) (Succ xu82)) (Succ (Succ xu82))",fontsize=16,color="magenta"];1192 -> 1331[label="",style="dashed", color="magenta", weight=3];
1192 -> 1332[label="",style="dashed", color="magenta", weight=3];
1192 -> 1333[label="",style="dashed", color="magenta", weight=3];
1355 -> 1324[label="",style="dashed", color="red", weight=0];
1355[label="primDivNatS (primMinusNatS xu980 xu990) (Succ xu100)",fontsize=16,color="magenta"];1355 -> 1359[label="",style="dashed", color="magenta", weight=3];
1355 -> 1360[label="",style="dashed", color="magenta", weight=3];
1356 -> 544[label="",style="dashed", color="red", weight=0];
1356[label="primDivNatS (Succ xu980) (Succ xu100)",fontsize=16,color="magenta"];1356 -> 1361[label="",style="dashed", color="magenta", weight=3];
1356 -> 1362[label="",style="dashed", color="magenta", weight=3];
1357[label="primDivNatS Zero (Succ xu100)",fontsize=16,color="black",shape="triangle"];1357 -> 1363[label="",style="solid", color="black", weight=3];
1358 -> 1357[label="",style="dashed", color="red", weight=0];
1358[label="primDivNatS Zero (Succ xu100)",fontsize=16,color="magenta"];1197[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) (Succ xu920))",fontsize=16,color="black",shape="box"];1197 -> 1208[label="",style="solid", color="black", weight=3];
1198[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS (Succ xu910) Zero)",fontsize=16,color="black",shape="box"];1198 -> 1209[label="",style="solid", color="black", weight=3];
1199[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero (Succ xu920))",fontsize=16,color="black",shape="box"];1199 -> 1210[label="",style="solid", color="black", weight=3];
1200[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1200 -> 1211[label="",style="solid", color="black", weight=3];
1267[label="primModNatS (primMinusNatS (Succ xu940) xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="box"];1430[label="xu95/Succ xu950",fontsize=10,color="white",style="solid",shape="box"];1267 -> 1430[label="",style="solid", color="burlywood", weight=9];
1430 -> 1275[label="",style="solid", color="burlywood", weight=3];
1431[label="xu95/Zero",fontsize=10,color="white",style="solid",shape="box"];1267 -> 1431[label="",style="solid", color="burlywood", weight=9];
1431 -> 1276[label="",style="solid", color="burlywood", weight=3];
1268[label="primModNatS (primMinusNatS Zero xu95) (Succ xu96)",fontsize=16,color="burlywood",shape="box"];1432[label="xu95/Succ xu950",fontsize=10,color="white",style="solid",shape="box"];1268 -> 1432[label="",style="solid", color="burlywood", weight=9];
1432 -> 1277[label="",style="solid", color="burlywood", weight=3];
1433[label="xu95/Zero",fontsize=10,color="white",style="solid",shape="box"];1268 -> 1433[label="",style="solid", color="burlywood", weight=9];
1433 -> 1278[label="",style="solid", color="burlywood", weight=3];
1331[label="Succ xu82",fontsize=16,color="green",shape="box"];1332[label="Succ xu82",fontsize=16,color="green",shape="box"];1333[label="Succ xu81",fontsize=16,color="green",shape="box"];1359[label="xu990",fontsize=16,color="green",shape="box"];1360[label="xu980",fontsize=16,color="green",shape="box"];1361[label="xu980",fontsize=16,color="green",shape="box"];1362[label="xu100",fontsize=16,color="green",shape="box"];1363[label="Zero",fontsize=16,color="green",shape="box"];1208 -> 1149[label="",style="dashed", color="red", weight=0];
1208[label="primModNatS0 (Succ xu89) (Succ xu90) (primGEqNatS xu910 xu920)",fontsize=16,color="magenta"];1208 -> 1218[label="",style="dashed", color="magenta", weight=3];
1208 -> 1219[label="",style="dashed", color="magenta", weight=3];
1209[label="primModNatS0 (Succ xu89) (Succ xu90) True",fontsize=16,color="black",shape="triangle"];1209 -> 1220[label="",style="solid", color="black", weight=3];
1210[label="primModNatS0 (Succ xu89) (Succ xu90) False",fontsize=16,color="black",shape="box"];1210 -> 1221[label="",style="solid", color="black", weight=3];
1211 -> 1209[label="",style="dashed", color="red", weight=0];
1211[label="primModNatS0 (Succ xu89) (Succ xu90) True",fontsize=16,color="magenta"];1275[label="primModNatS (primMinusNatS (Succ xu940) (Succ xu950)) (Succ xu96)",fontsize=16,color="black",shape="box"];1275 -> 1283[label="",style="solid", color="black", weight=3];
1276[label="primModNatS (primMinusNatS (Succ xu940) Zero) (Succ xu96)",fontsize=16,color="black",shape="box"];1276 -> 1284[label="",style="solid", color="black", weight=3];
1277[label="primModNatS (primMinusNatS Zero (Succ xu950)) (Succ xu96)",fontsize=16,color="black",shape="box"];1277 -> 1285[label="",style="solid", color="black", weight=3];
1278[label="primModNatS (primMinusNatS Zero Zero) (Succ xu96)",fontsize=16,color="black",shape="box"];1278 -> 1286[label="",style="solid", color="black", weight=3];
1218[label="xu910",fontsize=16,color="green",shape="box"];1219[label="xu920",fontsize=16,color="green",shape="box"];1220 -> 1236[label="",style="dashed", color="red", weight=0];
1220[label="primModNatS (primMinusNatS (Succ xu89) (Succ xu90)) (Succ (Succ xu90))",fontsize=16,color="magenta"];1220 -> 1249[label="",style="dashed", color="magenta", weight=3];
1220 -> 1250[label="",style="dashed", color="magenta", weight=3];
1220 -> 1251[label="",style="dashed", color="magenta", weight=3];
1221[label="Succ (Succ xu89)",fontsize=16,color="green",shape="box"];1283 -> 1236[label="",style="dashed", color="red", weight=0];
1283[label="primModNatS (primMinusNatS xu940 xu950) (Succ xu96)",fontsize=16,color="magenta"];1283 -> 1291[label="",style="dashed", color="magenta", weight=3];
1283 -> 1292[label="",style="dashed", color="magenta", weight=3];
1284 -> 561[label="",style="dashed", color="red", weight=0];
1284[label="primModNatS (Succ xu940) (Succ xu96)",fontsize=16,color="magenta"];1284 -> 1293[label="",style="dashed", color="magenta", weight=3];
1284 -> 1294[label="",style="dashed", color="magenta", weight=3];
1285[label="primModNatS Zero (Succ xu96)",fontsize=16,color="black",shape="triangle"];1285 -> 1295[label="",style="solid", color="black", weight=3];
1286 -> 1285[label="",style="dashed", color="red", weight=0];
1286[label="primModNatS Zero (Succ xu96)",fontsize=16,color="magenta"];1249[label="Succ xu90",fontsize=16,color="green",shape="box"];1250[label="Succ xu89",fontsize=16,color="green",shape="box"];1251[label="Succ xu90",fontsize=16,color="green",shape="box"];1291[label="xu940",fontsize=16,color="green",shape="box"];1292[label="xu950",fontsize=16,color="green",shape="box"];1293[label="xu96",fontsize=16,color="green",shape="box"];1294[label="xu940",fontsize=16,color="green",shape="box"];1295[label="Zero",fontsize=16,color="green",shape="box"];}

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

(71)
TRUE