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


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


YES
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 be proven:

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


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;

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

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

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if eq x y then ys else y : deleteBy eq x ys"
is transformed to
"deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy wu wv [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

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

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy wu wv [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

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

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
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58[label="List.deleteBy0 ww31 (Pos Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];58 -> 74[label="",style="solid", color="black", weight=3];
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60[label="List.deleteBy0 ww31 (Neg Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];60 -> 76[label="",style="solid", color="black", weight=3];
484[label="ww4000",fontsize=16,color="green",shape="box"];485[label="ww3000",fontsize=16,color="green",shape="box"];486[label="ww3000",fontsize=16,color="green",shape="box"];487[label="ww4000",fontsize=16,color="green",shape="box"];488[label="ww31",fontsize=16,color="green",shape="box"];483[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww48 ww49)",fontsize=16,color="burlywood",shape="triangle"];649[label="ww48/Succ ww480",fontsize=10,color="white",style="solid",shape="box"];483 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 534[label="",style="solid", color="burlywood", weight=3];
650[label="ww48/Zero",fontsize=10,color="white",style="solid",shape="box"];483 -> 650[label="",style="solid", color="burlywood", weight=9];
650 -> 535[label="",style="solid", color="burlywood", weight=3];
63[label="Pos Zero : List.deleteBy primEqInt (Pos (Succ ww4000)) ww31",fontsize=16,color="green",shape="box"];63 -> 81[label="",style="dashed", color="green", weight=3];
64[label="List.deleteBy primEqInt (Pos (Succ ww4000)) ww31",fontsize=16,color="burlywood",shape="triangle"];651[label="ww31/ww310 : ww311",fontsize=10,color="white",style="solid",shape="box"];64 -> 651[label="",style="solid", color="burlywood", weight=9];
651 -> 82[label="",style="solid", color="burlywood", weight=3];
652[label="ww31/[]",fontsize=10,color="white",style="solid",shape="box"];64 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 83[label="",style="solid", color="burlywood", weight=3];
65[label="Pos (Succ ww3000) : List.deleteBy primEqInt (Pos Zero) ww31",fontsize=16,color="green",shape="box"];65 -> 84[label="",style="dashed", color="green", weight=3];
66[label="ww31",fontsize=16,color="green",shape="box"];67[label="Neg (Succ ww3000) : List.deleteBy primEqInt (Pos Zero) ww31",fontsize=16,color="green",shape="box"];67 -> 85[label="",style="dashed", color="green", weight=3];
68[label="ww31",fontsize=16,color="green",shape="box"];69[label="List.deleteBy primEqInt (Neg (Succ ww4000)) ww31",fontsize=16,color="burlywood",shape="triangle"];653[label="ww31/ww310 : ww311",fontsize=10,color="white",style="solid",shape="box"];69 -> 653[label="",style="solid", color="burlywood", weight=9];
653 -> 86[label="",style="solid", color="burlywood", weight=3];
654[label="ww31/[]",fontsize=10,color="white",style="solid",shape="box"];69 -> 654[label="",style="solid", color="burlywood", weight=9];
654 -> 87[label="",style="solid", color="burlywood", weight=3];
537[label="ww3000",fontsize=16,color="green",shape="box"];538[label="ww3000",fontsize=16,color="green",shape="box"];539[label="ww4000",fontsize=16,color="green",shape="box"];540[label="ww31",fontsize=16,color="green",shape="box"];541[label="ww4000",fontsize=16,color="green",shape="box"];536[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww54 ww55)",fontsize=16,color="burlywood",shape="triangle"];655[label="ww54/Succ ww540",fontsize=10,color="white",style="solid",shape="box"];536 -> 655[label="",style="solid", color="burlywood", weight=9];
655 -> 587[label="",style="solid", color="burlywood", weight=3];
656[label="ww54/Zero",fontsize=10,color="white",style="solid",shape="box"];536 -> 656[label="",style="solid", color="burlywood", weight=9];
656 -> 588[label="",style="solid", color="burlywood", weight=3];
72[label="Neg Zero : List.deleteBy primEqInt (Neg (Succ ww4000)) ww31",fontsize=16,color="green",shape="box"];72 -> 92[label="",style="dashed", color="green", weight=3];
73[label="Pos (Succ ww3000) : List.deleteBy primEqInt (Neg Zero) ww31",fontsize=16,color="green",shape="box"];73 -> 93[label="",style="dashed", color="green", weight=3];
74[label="ww31",fontsize=16,color="green",shape="box"];75[label="Neg (Succ ww3000) : List.deleteBy primEqInt (Neg Zero) ww31",fontsize=16,color="green",shape="box"];75 -> 94[label="",style="dashed", color="green", weight=3];
76[label="ww31",fontsize=16,color="green",shape="box"];534[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) ww49)",fontsize=16,color="burlywood",shape="box"];657[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];534 -> 657[label="",style="solid", color="burlywood", weight=9];
657 -> 589[label="",style="solid", color="burlywood", weight=3];
658[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];534 -> 658[label="",style="solid", color="burlywood", weight=9];
658 -> 590[label="",style="solid", color="burlywood", weight=3];
535[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero ww49)",fontsize=16,color="burlywood",shape="box"];659[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];535 -> 659[label="",style="solid", color="burlywood", weight=9];
659 -> 591[label="",style="solid", color="burlywood", weight=3];
660[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];535 -> 660[label="",style="solid", color="burlywood", weight=9];
660 -> 592[label="",style="solid", color="burlywood", weight=3];
81 -> 64[label="",style="dashed", color="red", weight=0];
81[label="List.deleteBy primEqInt (Pos (Succ ww4000)) ww31",fontsize=16,color="magenta"];82[label="List.deleteBy primEqInt (Pos (Succ ww4000)) (ww310 : ww311)",fontsize=16,color="black",shape="box"];82 -> 99[label="",style="solid", color="black", weight=3];
83[label="List.deleteBy primEqInt (Pos (Succ ww4000)) []",fontsize=16,color="black",shape="box"];83 -> 100[label="",style="solid", color="black", weight=3];
84[label="List.deleteBy primEqInt (Pos Zero) ww31",fontsize=16,color="burlywood",shape="triangle"];661[label="ww31/ww310 : ww311",fontsize=10,color="white",style="solid",shape="box"];84 -> 661[label="",style="solid", color="burlywood", weight=9];
661 -> 101[label="",style="solid", color="burlywood", weight=3];
662[label="ww31/[]",fontsize=10,color="white",style="solid",shape="box"];84 -> 662[label="",style="solid", color="burlywood", weight=9];
662 -> 102[label="",style="solid", color="burlywood", weight=3];
85 -> 84[label="",style="dashed", color="red", weight=0];
85[label="List.deleteBy primEqInt (Pos Zero) ww31",fontsize=16,color="magenta"];86[label="List.deleteBy primEqInt (Neg (Succ ww4000)) (ww310 : ww311)",fontsize=16,color="black",shape="box"];86 -> 103[label="",style="solid", color="black", weight=3];
87[label="List.deleteBy primEqInt (Neg (Succ ww4000)) []",fontsize=16,color="black",shape="box"];87 -> 104[label="",style="solid", color="black", weight=3];
587[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) ww55)",fontsize=16,color="burlywood",shape="box"];663[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];587 -> 663[label="",style="solid", color="burlywood", weight=9];
663 -> 593[label="",style="solid", color="burlywood", weight=3];
664[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];587 -> 664[label="",style="solid", color="burlywood", weight=9];
664 -> 594[label="",style="solid", color="burlywood", weight=3];
588[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero ww55)",fontsize=16,color="burlywood",shape="box"];665[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];588 -> 665[label="",style="solid", color="burlywood", weight=9];
665 -> 595[label="",style="solid", color="burlywood", weight=3];
666[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];588 -> 666[label="",style="solid", color="burlywood", weight=9];
666 -> 596[label="",style="solid", color="burlywood", weight=3];
92 -> 69[label="",style="dashed", color="red", weight=0];
92[label="List.deleteBy primEqInt (Neg (Succ ww4000)) ww31",fontsize=16,color="magenta"];93[label="List.deleteBy primEqInt (Neg Zero) ww31",fontsize=16,color="burlywood",shape="triangle"];667[label="ww31/ww310 : ww311",fontsize=10,color="white",style="solid",shape="box"];93 -> 667[label="",style="solid", color="burlywood", weight=9];
667 -> 109[label="",style="solid", color="burlywood", weight=3];
668[label="ww31/[]",fontsize=10,color="white",style="solid",shape="box"];93 -> 668[label="",style="solid", color="burlywood", weight=9];
668 -> 110[label="",style="solid", color="burlywood", weight=3];
94 -> 93[label="",style="dashed", color="red", weight=0];
94[label="List.deleteBy primEqInt (Neg Zero) ww31",fontsize=16,color="magenta"];589[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) (Succ ww490))",fontsize=16,color="black",shape="box"];589 -> 597[label="",style="solid", color="black", weight=3];
590[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) Zero)",fontsize=16,color="black",shape="box"];590 -> 598[label="",style="solid", color="black", weight=3];
591[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero (Succ ww490))",fontsize=16,color="black",shape="box"];591 -> 599[label="",style="solid", color="black", weight=3];
592[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];592 -> 600[label="",style="solid", color="black", weight=3];
99 -> 18[label="",style="dashed", color="red", weight=0];
99[label="List.deleteBy0 ww311 ww310 primEqInt (Pos (Succ ww4000)) (primEqInt (Pos (Succ ww4000)) ww310)",fontsize=16,color="magenta"];99 -> 116[label="",style="dashed", color="magenta", weight=3];
99 -> 117[label="",style="dashed", color="magenta", weight=3];
99 -> 118[label="",style="dashed", color="magenta", weight=3];
100[label="[]",fontsize=16,color="green",shape="box"];101[label="List.deleteBy primEqInt (Pos Zero) (ww310 : ww311)",fontsize=16,color="black",shape="box"];101 -> 119[label="",style="solid", color="black", weight=3];
102[label="List.deleteBy primEqInt (Pos Zero) []",fontsize=16,color="black",shape="box"];102 -> 120[label="",style="solid", color="black", weight=3];
103 -> 18[label="",style="dashed", color="red", weight=0];
103[label="List.deleteBy0 ww311 ww310 primEqInt (Neg (Succ ww4000)) (primEqInt (Neg (Succ ww4000)) ww310)",fontsize=16,color="magenta"];103 -> 121[label="",style="dashed", color="magenta", weight=3];
103 -> 122[label="",style="dashed", color="magenta", weight=3];
103 -> 123[label="",style="dashed", color="magenta", weight=3];
104[label="[]",fontsize=16,color="green",shape="box"];593[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) (Succ ww550))",fontsize=16,color="black",shape="box"];593 -> 601[label="",style="solid", color="black", weight=3];
594[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) Zero)",fontsize=16,color="black",shape="box"];594 -> 602[label="",style="solid", color="black", weight=3];
595[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero (Succ ww550))",fontsize=16,color="black",shape="box"];595 -> 603[label="",style="solid", color="black", weight=3];
596[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];596 -> 604[label="",style="solid", color="black", weight=3];
109[label="List.deleteBy primEqInt (Neg Zero) (ww310 : ww311)",fontsize=16,color="black",shape="box"];109 -> 129[label="",style="solid", color="black", weight=3];
110[label="List.deleteBy primEqInt (Neg Zero) []",fontsize=16,color="black",shape="box"];110 -> 130[label="",style="solid", color="black", weight=3];
597 -> 483[label="",style="dashed", color="red", weight=0];
597[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww480 ww490)",fontsize=16,color="magenta"];597 -> 605[label="",style="dashed", color="magenta", weight=3];
597 -> 606[label="",style="dashed", color="magenta", weight=3];
598[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="black",shape="triangle"];598 -> 607[label="",style="solid", color="black", weight=3];
599 -> 598[label="",style="dashed", color="red", weight=0];
599[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="magenta"];600[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) True",fontsize=16,color="black",shape="box"];600 -> 608[label="",style="solid", color="black", weight=3];
116[label="ww310",fontsize=16,color="green",shape="box"];117[label="Pos (Succ ww4000)",fontsize=16,color="green",shape="box"];118[label="ww311",fontsize=16,color="green",shape="box"];119 -> 18[label="",style="dashed", color="red", weight=0];
119[label="List.deleteBy0 ww311 ww310 primEqInt (Pos Zero) (primEqInt (Pos Zero) ww310)",fontsize=16,color="magenta"];119 -> 137[label="",style="dashed", color="magenta", weight=3];
119 -> 138[label="",style="dashed", color="magenta", weight=3];
119 -> 139[label="",style="dashed", color="magenta", weight=3];
120[label="[]",fontsize=16,color="green",shape="box"];121[label="ww310",fontsize=16,color="green",shape="box"];122[label="Neg (Succ ww4000)",fontsize=16,color="green",shape="box"];123[label="ww311",fontsize=16,color="green",shape="box"];601 -> 536[label="",style="dashed", color="red", weight=0];
601[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww540 ww550)",fontsize=16,color="magenta"];601 -> 609[label="",style="dashed", color="magenta", weight=3];
601 -> 610[label="",style="dashed", color="magenta", weight=3];
602[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="black",shape="triangle"];602 -> 611[label="",style="solid", color="black", weight=3];
603 -> 602[label="",style="dashed", color="red", weight=0];
603[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="magenta"];604[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) True",fontsize=16,color="black",shape="box"];604 -> 612[label="",style="solid", color="black", weight=3];
129 -> 18[label="",style="dashed", color="red", weight=0];
129[label="List.deleteBy0 ww311 ww310 primEqInt (Neg Zero) (primEqInt (Neg Zero) ww310)",fontsize=16,color="magenta"];129 -> 146[label="",style="dashed", color="magenta", weight=3];
129 -> 147[label="",style="dashed", color="magenta", weight=3];
129 -> 148[label="",style="dashed", color="magenta", weight=3];
130[label="[]",fontsize=16,color="green",shape="box"];605[label="ww480",fontsize=16,color="green",shape="box"];606[label="ww490",fontsize=16,color="green",shape="box"];607[label="Pos (Succ ww46) : List.deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="green",shape="box"];607 -> 613[label="",style="dashed", color="green", weight=3];
608[label="ww45",fontsize=16,color="green",shape="box"];137[label="ww310",fontsize=16,color="green",shape="box"];138[label="Pos Zero",fontsize=16,color="green",shape="box"];139[label="ww311",fontsize=16,color="green",shape="box"];609[label="ww550",fontsize=16,color="green",shape="box"];610[label="ww540",fontsize=16,color="green",shape="box"];611[label="Neg (Succ ww52) : List.deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="green",shape="box"];611 -> 614[label="",style="dashed", color="green", weight=3];
612[label="ww51",fontsize=16,color="green",shape="box"];146[label="ww310",fontsize=16,color="green",shape="box"];147[label="Neg Zero",fontsize=16,color="green",shape="box"];148[label="ww311",fontsize=16,color="green",shape="box"];613 -> 64[label="",style="dashed", color="red", weight=0];
613[label="List.deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="magenta"];613 -> 615[label="",style="dashed", color="magenta", weight=3];
613 -> 616[label="",style="dashed", color="magenta", weight=3];
614 -> 69[label="",style="dashed", color="red", weight=0];
614[label="List.deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="magenta"];614 -> 617[label="",style="dashed", color="magenta", weight=3];
614 -> 618[label="",style="dashed", color="magenta", weight=3];
615[label="ww47",fontsize=16,color="green",shape="box"];616[label="ww45",fontsize=16,color="green",shape="box"];617[label="ww51",fontsize=16,color="green",shape="box"];618[label="ww53",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) -> new_deleteBy(ww47, ww45)
   new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) -> new_deleteBy2(ww53, ww51)
   new_deleteBy(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
   new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) -> new_deleteBy03(ww51, ww52, ww53)
   new_deleteBy03(ww51, ww52, ww53) -> new_deleteBy2(ww53, ww51)
   new_deleteBy00(ww45, ww46, ww47) -> new_deleteBy(ww47, ww45)
   new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Pos(Zero)) -> new_deleteBy01(ww311, ww310, Pos(Zero))
   new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Neg(Zero)) -> new_deleteBy01(ww311, ww310, Neg(Zero))
   new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Zero)) -> new_deleteBy3(ww31)
   new_deleteBy01(ww31, Pos(Succ(ww3000)), Pos(Succ(ww4000))) -> new_deleteBy0(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy01(:(ww310, ww311), Neg(ww300), Pos(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
   new_deleteBy01(ww31, Neg(Succ(ww3000)), Pos(Zero)) -> new_deleteBy1(ww31)
   new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) -> new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
   new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) -> new_deleteBy00(ww45, ww46, ww47)
   new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Succ(ww4000))) -> new_deleteBy02(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy2(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))
   new_deleteBy3(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Zero))
   new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) -> new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
   new_deleteBy1(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Zero))
   new_deleteBy01(ww31, Neg(Zero), Neg(Succ(ww4000))) -> new_deleteBy2(ww4000, ww31)
   new_deleteBy01(ww31, Pos(Zero), Pos(Succ(ww4000))) -> new_deleteBy(ww4000, ww31)
   new_deleteBy01(:(ww310, ww311), Pos(ww300), Neg(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))

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

(10) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs.
----------------------------------------

(11)
Complex Obligation (AND)

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

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

   new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Zero)) -> new_deleteBy3(ww31)
   new_deleteBy3(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Zero))
   new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Neg(Zero)) -> new_deleteBy01(ww311, ww310, Neg(Zero))

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

(13) 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_deleteBy3(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2


*new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Neg(Zero)) -> new_deleteBy01(ww311, ww310, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Zero)) -> new_deleteBy3(ww31)
The graph contains the following edges 1 >= 1


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

(14)
YES

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

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

   new_deleteBy01(ww31, Neg(Succ(ww3000)), Pos(Zero)) -> new_deleteBy1(ww31)
   new_deleteBy1(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Zero))
   new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Pos(Zero)) -> new_deleteBy01(ww311, ww310, Pos(Zero))

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

(16) 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_deleteBy1(:(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2


*new_deleteBy01(:(ww310, ww311), Pos(Succ(ww3000)), Pos(Zero)) -> new_deleteBy01(ww311, ww310, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww31, Neg(Succ(ww3000)), Pos(Zero)) -> new_deleteBy1(ww31)
The graph contains the following edges 1 >= 1


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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy2(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))
   new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Succ(ww4000))) -> new_deleteBy02(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) -> new_deleteBy2(ww53, ww51)
   new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) -> new_deleteBy03(ww51, ww52, ww53)
   new_deleteBy03(ww51, ww52, ww53) -> new_deleteBy2(ww53, ww51)
   new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) -> new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
   new_deleteBy01(ww31, Neg(Zero), Neg(Succ(ww4000))) -> new_deleteBy2(ww4000, ww31)
   new_deleteBy01(:(ww310, ww311), Pos(ww300), Neg(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))

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

(19) 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_deleteBy01(ww31, Neg(Zero), Neg(Succ(ww4000))) -> new_deleteBy2(ww4000, ww31)
The graph contains the following edges 3 > 1, 1 >= 2


*new_deleteBy01(:(ww310, ww311), Pos(ww300), Neg(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww31, Neg(Succ(ww3000)), Neg(Succ(ww4000))) -> new_deleteBy02(ww31, ww3000, ww4000, ww4000, ww3000)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5


*new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Zero) -> new_deleteBy2(ww53, ww51)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy03(ww51, ww52, ww53) -> new_deleteBy2(ww53, ww51)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy2(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Neg(Succ(ww4000)))
The graph contains the following edges 2 > 1, 2 > 2


*new_deleteBy02(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) -> new_deleteBy02(ww51, ww52, ww53, ww540, ww550)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


*new_deleteBy02(ww51, ww52, ww53, Zero, Succ(ww550)) -> new_deleteBy03(ww51, ww52, ww53)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3


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

(20)
YES

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
   new_deleteBy01(ww31, Pos(Succ(ww3000)), Pos(Succ(ww4000))) -> new_deleteBy0(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) -> new_deleteBy(ww47, ww45)
   new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) -> new_deleteBy00(ww45, ww46, ww47)
   new_deleteBy00(ww45, ww46, ww47) -> new_deleteBy(ww47, ww45)
   new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) -> new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
   new_deleteBy01(:(ww310, ww311), Neg(ww300), Pos(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
   new_deleteBy01(ww31, Pos(Zero), Pos(Succ(ww4000))) -> new_deleteBy(ww4000, ww31)

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

(22) 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_deleteBy01(ww31, Pos(Zero), Pos(Succ(ww4000))) -> new_deleteBy(ww4000, ww31)
The graph contains the following edges 3 > 1, 1 >= 2


*new_deleteBy01(:(ww310, ww311), Neg(ww300), Pos(Succ(ww4000))) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww31, Pos(Succ(ww3000)), Pos(Succ(ww4000))) -> new_deleteBy0(ww31, ww3000, ww4000, ww4000, ww3000)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5


*new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Zero) -> new_deleteBy(ww47, ww45)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy00(ww45, ww46, ww47) -> new_deleteBy(ww47, ww45)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy(ww4000, :(ww310, ww311)) -> new_deleteBy01(ww311, ww310, Pos(Succ(ww4000)))
The graph contains the following edges 2 > 1, 2 > 2


*new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) -> new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


*new_deleteBy0(ww45, ww46, ww47, Zero, Succ(ww490)) -> new_deleteBy00(ww45, ww46, ww47)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3


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

(23)
YES

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

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

   new_foldl(ww3, :(ww40, ww41)) -> new_foldl(new_deleteBy4(ww40, ww3), ww41)

The TRS R consists of the following rules:

   new_deleteBy04(ww45, ww46, ww47, Zero, Zero) -> ww45
   new_deleteBy07(ww51, ww52, ww53, Succ(ww540), Zero) -> new_deleteBy08(ww51, ww52, ww53)
   new_deleteBy07(ww51, ww52, ww53, Zero, Succ(ww550)) -> new_deleteBy08(ww51, ww52, ww53)
   new_deleteBy4(ww40, []) -> []
   new_deleteBy06(ww31, Neg(Zero), Neg(Succ(ww4000))) -> :(Neg(Zero), new_deleteBy8(ww4000, ww31))
   new_deleteBy06(ww31, Pos(Succ(ww3000)), Pos(Succ(ww4000))) -> new_deleteBy04(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy7(:(ww310, ww311)) -> new_deleteBy06(ww311, ww310, Pos(Zero))
   new_deleteBy06(ww31, Neg(Succ(ww3000)), Pos(Zero)) -> :(Neg(Succ(ww3000)), new_deleteBy7(ww31))
   new_deleteBy6(ww4000, []) -> []
   new_deleteBy06(ww31, Pos(Zero), Pos(Succ(ww4000))) -> :(Pos(Zero), new_deleteBy6(ww4000, ww31))
   new_deleteBy4(ww40, :(ww30, ww31)) -> new_deleteBy06(ww31, ww30, ww40)
   new_deleteBy8(ww4000, []) -> []
   new_deleteBy04(ww45, ww46, ww47, Succ(ww480), Zero) -> new_deleteBy05(ww45, ww46, ww47)
   new_deleteBy04(ww45, ww46, ww47, Zero, Succ(ww490)) -> new_deleteBy05(ww45, ww46, ww47)
   new_deleteBy06(ww31, Pos(Zero), Pos(Zero)) -> ww31
   new_deleteBy06(ww31, Neg(Zero), Pos(Zero)) -> ww31
   new_deleteBy06(ww31, Pos(Zero), Neg(Zero)) -> ww31
   new_deleteBy06(ww31, Neg(Succ(ww3000)), Neg(Zero)) -> :(Neg(Succ(ww3000)), new_deleteBy5(ww31))
   new_deleteBy5(:(ww310, ww311)) -> new_deleteBy06(ww311, ww310, Neg(Zero))
   new_deleteBy06(ww31, Neg(ww300), Pos(Succ(ww4000))) -> :(Neg(ww300), new_deleteBy6(ww4000, ww31))
   new_deleteBy06(ww31, Pos(Succ(ww3000)), Pos(Zero)) -> :(Pos(Succ(ww3000)), new_deleteBy7(ww31))
   new_deleteBy04(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) -> new_deleteBy04(ww45, ww46, ww47, ww480, ww490)
   new_deleteBy06(ww31, Pos(Succ(ww3000)), Neg(Zero)) -> :(Pos(Succ(ww3000)), new_deleteBy5(ww31))
   new_deleteBy06(ww31, Neg(Zero), Neg(Zero)) -> ww31
   new_deleteBy06(ww31, Neg(Succ(ww3000)), Neg(Succ(ww4000))) -> new_deleteBy07(ww31, ww3000, ww4000, ww4000, ww3000)
   new_deleteBy7([]) -> []
   new_deleteBy5([]) -> []
   new_deleteBy07(ww51, ww52, ww53, Succ(ww540), Succ(ww550)) -> new_deleteBy07(ww51, ww52, ww53, ww540, ww550)
   new_deleteBy07(ww51, ww52, ww53, Zero, Zero) -> ww51
   new_deleteBy06(ww31, Pos(ww300), Neg(Succ(ww4000))) -> :(Pos(ww300), new_deleteBy8(ww4000, ww31))
   new_deleteBy05(ww45, ww46, ww47) -> :(Pos(Succ(ww46)), new_deleteBy6(ww47, ww45))
   new_deleteBy8(ww4000, :(ww310, ww311)) -> new_deleteBy06(ww311, ww310, Neg(Succ(ww4000)))
   new_deleteBy6(ww4000, :(ww310, ww311)) -> new_deleteBy06(ww311, ww310, Pos(Succ(ww4000)))
   new_deleteBy08(ww51, ww52, ww53) -> :(Neg(Succ(ww52)), new_deleteBy8(ww53, ww51))

The set Q consists of the following terms:

   new_deleteBy06(x0, Neg(Succ(x1)), Neg(Zero))
   new_deleteBy04(x0, x1, x2, Succ(x3), Zero)
   new_deleteBy07(x0, x1, x2, Zero, Succ(x3))
   new_deleteBy6(x0, :(x1, x2))
   new_deleteBy08(x0, x1, x2)
   new_deleteBy05(x0, x1, x2)
   new_deleteBy07(x0, x1, x2, Succ(x3), Zero)
   new_deleteBy04(x0, x1, x2, Succ(x3), Succ(x4))
   new_deleteBy5(:(x0, x1))
   new_deleteBy7([])
   new_deleteBy6(x0, [])
   new_deleteBy04(x0, x1, x2, Zero, Zero)
   new_deleteBy8(x0, [])
   new_deleteBy06(x0, Pos(Zero), Pos(Succ(x1)))
   new_deleteBy06(x0, Neg(Zero), Neg(Zero))
   new_deleteBy4(x0, [])
   new_deleteBy06(x0, Neg(Zero), Pos(Zero))
   new_deleteBy06(x0, Pos(Zero), Neg(Zero))
   new_deleteBy06(x0, Pos(Succ(x1)), Pos(Zero))
   new_deleteBy8(x0, :(x1, x2))
   new_deleteBy04(x0, x1, x2, Zero, Succ(x3))
   new_deleteBy06(x0, Pos(Zero), Pos(Zero))
   new_deleteBy06(x0, Neg(Succ(x1)), Neg(Succ(x2)))
   new_deleteBy06(x0, Neg(Succ(x1)), Pos(Zero))
   new_deleteBy06(x0, Pos(Succ(x1)), Neg(Zero))
   new_deleteBy5([])
   new_deleteBy7(:(x0, x1))
   new_deleteBy06(x0, Pos(Succ(x1)), Pos(Succ(x2)))
   new_deleteBy07(x0, x1, x2, Zero, Zero)
   new_deleteBy06(x0, Neg(x1), Pos(Succ(x2)))
   new_deleteBy06(x0, Pos(x1), Neg(Succ(x2)))
   new_deleteBy4(x0, :(x1, x2))
   new_deleteBy07(x0, x1, x2, Succ(x3), Succ(x4))
   new_deleteBy06(x0, Neg(Zero), Neg(Succ(x1)))

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

(25) 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_foldl(ww3, :(ww40, ww41)) -> new_foldl(new_deleteBy4(ww40, ww3), ww41)
The graph contains the following edges 2 > 2


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

(26)
YES