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


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) HASKELL
(1) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 20 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) 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;
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;
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;
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;
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 {
node [outthreshold=100, inthreshold=100];1[label="List.delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.delete ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.delete ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.deleteBy (==) ww3 ww4",fontsize=16,color="burlywood",shape="box"];611[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 611[label="",style="solid", color="burlywood", weight=9];
611 -> 6[label="",style="solid", color="burlywood", weight=3];
612[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 612[label="",style="solid", color="burlywood", weight=9];
612 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.deleteBy (==) ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.deleteBy (==) ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="List.deleteBy0 ww41 ww40 (==) ww3 ((==) ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="List.deleteBy0 ww41 ww40 primEqInt ww3 (primEqInt ww3 ww40)",fontsize=16,color="burlywood",shape="triangle"];613[label="ww3/Pos ww30",fontsize=10,color="white",style="solid",shape="box"];10 -> 613[label="",style="solid", color="burlywood", weight=9];
613 -> 11[label="",style="solid", color="burlywood", weight=3];
614[label="ww3/Neg ww30",fontsize=10,color="white",style="solid",shape="box"];10 -> 614[label="",style="solid", color="burlywood", weight=9];
614 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="List.deleteBy0 ww41 ww40 primEqInt (Pos ww30) (primEqInt (Pos ww30) ww40)",fontsize=16,color="burlywood",shape="box"];615[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];11 -> 615[label="",style="solid", color="burlywood", weight=9];
615 -> 13[label="",style="solid", color="burlywood", weight=3];
616[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 616[label="",style="solid", color="burlywood", weight=9];
616 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="List.deleteBy0 ww41 ww40 primEqInt (Neg ww30) (primEqInt (Neg ww30) ww40)",fontsize=16,color="burlywood",shape="box"];617[label="ww30/Succ ww300",fontsize=10,color="white",style="solid",shape="box"];12 -> 617[label="",style="solid", color="burlywood", weight=9];
617 -> 15[label="",style="solid", color="burlywood", weight=3];
618[label="ww30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 618[label="",style="solid", color="burlywood", weight=9];
618 -> 16[label="",style="solid", color="burlywood", weight=3];
13[label="List.deleteBy0 ww41 ww40 primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) ww40)",fontsize=16,color="burlywood",shape="box"];619[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];13 -> 619[label="",style="solid", color="burlywood", weight=9];
619 -> 17[label="",style="solid", color="burlywood", weight=3];
620[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];13 -> 620[label="",style="solid", color="burlywood", weight=9];
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621 -> 19[label="",style="solid", color="burlywood", weight=3];
622[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];14 -> 622[label="",style="solid", color="burlywood", weight=9];
622 -> 20[label="",style="solid", color="burlywood", weight=3];
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623 -> 21[label="",style="solid", color="burlywood", weight=3];
624[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];15 -> 624[label="",style="solid", color="burlywood", weight=9];
624 -> 22[label="",style="solid", color="burlywood", weight=3];
16[label="List.deleteBy0 ww41 ww40 primEqInt (Neg Zero) (primEqInt (Neg Zero) ww40)",fontsize=16,color="burlywood",shape="box"];625[label="ww40/Pos ww400",fontsize=10,color="white",style="solid",shape="box"];16 -> 625[label="",style="solid", color="burlywood", weight=9];
625 -> 23[label="",style="solid", color="burlywood", weight=3];
626[label="ww40/Neg ww400",fontsize=10,color="white",style="solid",shape="box"];16 -> 626[label="",style="solid", color="burlywood", weight=9];
626 -> 24[label="",style="solid", color="burlywood", weight=3];
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627 -> 25[label="",style="solid", color="burlywood", weight=3];
628[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 628[label="",style="solid", color="burlywood", weight=9];
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18[label="List.deleteBy0 ww41 (Neg ww400) primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) (Neg ww400))",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
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630[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 630[label="",style="solid", color="burlywood", weight=9];
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631 -> 30[label="",style="solid", color="burlywood", weight=3];
632[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 632[label="",style="solid", color="burlywood", weight=9];
632 -> 31[label="",style="solid", color="burlywood", weight=3];
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633 -> 33[label="",style="solid", color="burlywood", weight=3];
634[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 634[label="",style="solid", color="burlywood", weight=9];
634 -> 34[label="",style="solid", color="burlywood", weight=3];
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635 -> 35[label="",style="solid", color="burlywood", weight=3];
636[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 636[label="",style="solid", color="burlywood", weight=9];
636 -> 36[label="",style="solid", color="burlywood", weight=3];
24[label="List.deleteBy0 ww41 (Neg ww400) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg ww400))",fontsize=16,color="burlywood",shape="box"];637[label="ww400/Succ ww4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 637[label="",style="solid", color="burlywood", weight=9];
637 -> 37[label="",style="solid", color="burlywood", weight=3];
638[label="ww400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 638[label="",style="solid", color="burlywood", weight=9];
638 -> 38[label="",style="solid", color="burlywood", weight=3];
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27[label="List.deleteBy0 ww41 (Neg ww400) primEqInt (Pos (Succ ww300)) False",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3];
28[label="List.deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Pos (Succ ww4000)))",fontsize=16,color="black",shape="box"];28 -> 42[label="",style="solid", color="black", weight=3];
29[label="List.deleteBy0 ww41 (Pos Zero) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
30[label="List.deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Neg (Succ ww4000)))",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
31[label="List.deleteBy0 ww41 (Neg Zero) primEqInt (Pos Zero) (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
32[label="List.deleteBy0 ww41 (Pos ww400) primEqInt (Neg (Succ ww300)) False",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
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35[label="List.deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Pos (Succ ww4000)))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="List.deleteBy0 ww41 (Pos Zero) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="List.deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg (Succ ww4000)))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="List.deleteBy0 ww41 (Neg Zero) primEqInt (Neg Zero) (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39 -> 475[label="",style="dashed", color="red", weight=0];
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39 -> 477[label="",style="dashed", color="magenta", weight=3];
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42[label="List.deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Pos Zero) False",fontsize=16,color="black",shape="box"];42 -> 57[label="",style="solid", color="black", weight=3];
43[label="List.deleteBy0 ww41 (Pos Zero) primEqInt (Pos Zero) True",fontsize=16,color="black",shape="box"];43 -> 58[label="",style="solid", color="black", weight=3];
44[label="List.deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Pos Zero) False",fontsize=16,color="black",shape="box"];44 -> 59[label="",style="solid", color="black", weight=3];
45[label="List.deleteBy0 ww41 (Neg Zero) primEqInt (Pos Zero) True",fontsize=16,color="black",shape="box"];45 -> 60[label="",style="solid", color="black", weight=3];
46[label="Pos ww400 : List.deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];46 -> 61[label="",style="dashed", color="green", weight=3];
47 -> 528[label="",style="dashed", color="red", weight=0];
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47 -> 530[label="",style="dashed", color="magenta", weight=3];
47 -> 531[label="",style="dashed", color="magenta", weight=3];
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49[label="List.deleteBy0 ww41 (Pos (Succ ww4000)) primEqInt (Neg Zero) False",fontsize=16,color="black",shape="box"];49 -> 65[label="",style="solid", color="black", weight=3];
50[label="List.deleteBy0 ww41 (Pos Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];50 -> 66[label="",style="solid", color="black", weight=3];
51[label="List.deleteBy0 ww41 (Neg (Succ ww4000)) primEqInt (Neg Zero) False",fontsize=16,color="black",shape="box"];51 -> 67[label="",style="solid", color="black", weight=3];
52[label="List.deleteBy0 ww41 (Neg Zero) primEqInt (Neg Zero) True",fontsize=16,color="black",shape="box"];52 -> 68[label="",style="solid", color="black", weight=3];
476[label="ww41",fontsize=16,color="green",shape="box"];477[label="ww4000",fontsize=16,color="green",shape="box"];478[label="ww300",fontsize=16,color="green",shape="box"];479[label="ww4000",fontsize=16,color="green",shape="box"];480[label="ww300",fontsize=16,color="green",shape="box"];475[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww48 ww49)",fontsize=16,color="burlywood",shape="triangle"];639[label="ww48/Succ ww480",fontsize=10,color="white",style="solid",shape="box"];475 -> 639[label="",style="solid", color="burlywood", weight=9];
639 -> 526[label="",style="solid", color="burlywood", weight=3];
640[label="ww48/Zero",fontsize=10,color="white",style="solid",shape="box"];475 -> 640[label="",style="solid", color="burlywood", weight=9];
640 -> 527[label="",style="solid", color="burlywood", weight=3];
55[label="Pos Zero : List.deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];55 -> 73[label="",style="dashed", color="green", weight=3];
56[label="List.deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="burlywood",shape="triangle"];641[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];56 -> 641[label="",style="solid", color="burlywood", weight=9];
641 -> 74[label="",style="solid", color="burlywood", weight=3];
642[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];56 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 75[label="",style="solid", color="burlywood", weight=3];
57[label="Pos (Succ ww4000) : List.deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="green",shape="box"];57 -> 76[label="",style="dashed", color="green", weight=3];
58[label="ww41",fontsize=16,color="green",shape="box"];59[label="Neg (Succ ww4000) : List.deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="green",shape="box"];59 -> 77[label="",style="dashed", color="green", weight=3];
60[label="ww41",fontsize=16,color="green",shape="box"];61[label="List.deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="burlywood",shape="triangle"];643[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];61 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 78[label="",style="solid", color="burlywood", weight=3];
644[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 644[label="",style="solid", color="burlywood", weight=9];
644 -> 79[label="",style="solid", color="burlywood", weight=3];
529[label="ww4000",fontsize=16,color="green",shape="box"];530[label="ww300",fontsize=16,color="green",shape="box"];531[label="ww300",fontsize=16,color="green",shape="box"];532[label="ww4000",fontsize=16,color="green",shape="box"];533[label="ww41",fontsize=16,color="green",shape="box"];528[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww54 ww55)",fontsize=16,color="burlywood",shape="triangle"];645[label="ww54/Succ ww540",fontsize=10,color="white",style="solid",shape="box"];528 -> 645[label="",style="solid", color="burlywood", weight=9];
645 -> 579[label="",style="solid", color="burlywood", weight=3];
646[label="ww54/Zero",fontsize=10,color="white",style="solid",shape="box"];528 -> 646[label="",style="solid", color="burlywood", weight=9];
646 -> 580[label="",style="solid", color="burlywood", weight=3];
64[label="Neg Zero : List.deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];64 -> 84[label="",style="dashed", color="green", weight=3];
65[label="Pos (Succ ww4000) : List.deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];65 -> 85[label="",style="dashed", color="green", weight=3];
66[label="ww41",fontsize=16,color="green",shape="box"];67[label="Neg (Succ ww4000) : List.deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];67 -> 86[label="",style="dashed", color="green", weight=3];
68[label="ww41",fontsize=16,color="green",shape="box"];526[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) ww49)",fontsize=16,color="burlywood",shape="box"];647[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];526 -> 647[label="",style="solid", color="burlywood", weight=9];
647 -> 581[label="",style="solid", color="burlywood", weight=3];
648[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 648[label="",style="solid", color="burlywood", weight=9];
648 -> 582[label="",style="solid", color="burlywood", weight=3];
527[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero ww49)",fontsize=16,color="burlywood",shape="box"];649[label="ww49/Succ ww490",fontsize=10,color="white",style="solid",shape="box"];527 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 583[label="",style="solid", color="burlywood", weight=3];
650[label="ww49/Zero",fontsize=10,color="white",style="solid",shape="box"];527 -> 650[label="",style="solid", color="burlywood", weight=9];
650 -> 584[label="",style="solid", color="burlywood", weight=3];
73 -> 56[label="",style="dashed", color="red", weight=0];
73[label="List.deleteBy primEqInt (Pos (Succ ww300)) ww41",fontsize=16,color="magenta"];74[label="List.deleteBy primEqInt (Pos (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];74 -> 91[label="",style="solid", color="black", weight=3];
75[label="List.deleteBy primEqInt (Pos (Succ ww300)) []",fontsize=16,color="black",shape="box"];75 -> 92[label="",style="solid", color="black", weight=3];
76[label="List.deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="burlywood",shape="triangle"];651[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];76 -> 651[label="",style="solid", color="burlywood", weight=9];
651 -> 93[label="",style="solid", color="burlywood", weight=3];
652[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];76 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 94[label="",style="solid", color="burlywood", weight=3];
77 -> 76[label="",style="dashed", color="red", weight=0];
77[label="List.deleteBy primEqInt (Pos Zero) ww41",fontsize=16,color="magenta"];78[label="List.deleteBy primEqInt (Neg (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
79[label="List.deleteBy primEqInt (Neg (Succ ww300)) []",fontsize=16,color="black",shape="box"];79 -> 96[label="",style="solid", color="black", weight=3];
579[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) ww55)",fontsize=16,color="burlywood",shape="box"];653[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];579 -> 653[label="",style="solid", color="burlywood", weight=9];
653 -> 585[label="",style="solid", color="burlywood", weight=3];
654[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];579 -> 654[label="",style="solid", color="burlywood", weight=9];
654 -> 586[label="",style="solid", color="burlywood", weight=3];
580[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero ww55)",fontsize=16,color="burlywood",shape="box"];655[label="ww55/Succ ww550",fontsize=10,color="white",style="solid",shape="box"];580 -> 655[label="",style="solid", color="burlywood", weight=9];
655 -> 587[label="",style="solid", color="burlywood", weight=3];
656[label="ww55/Zero",fontsize=10,color="white",style="solid",shape="box"];580 -> 656[label="",style="solid", color="burlywood", weight=9];
656 -> 588[label="",style="solid", color="burlywood", weight=3];
84 -> 61[label="",style="dashed", color="red", weight=0];
84[label="List.deleteBy primEqInt (Neg (Succ ww300)) ww41",fontsize=16,color="magenta"];85[label="List.deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="burlywood",shape="triangle"];657[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];85 -> 657[label="",style="solid", color="burlywood", weight=9];
657 -> 101[label="",style="solid", color="burlywood", weight=3];
658[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];85 -> 658[label="",style="solid", color="burlywood", weight=9];
658 -> 102[label="",style="solid", color="burlywood", weight=3];
86 -> 85[label="",style="dashed", color="red", weight=0];
86[label="List.deleteBy primEqInt (Neg Zero) ww41",fontsize=16,color="magenta"];581[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) (Succ ww490))",fontsize=16,color="black",shape="box"];581 -> 589[label="",style="solid", color="black", weight=3];
582[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat (Succ ww480) Zero)",fontsize=16,color="black",shape="box"];582 -> 590[label="",style="solid", color="black", weight=3];
583[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero (Succ ww490))",fontsize=16,color="black",shape="box"];583 -> 591[label="",style="solid", color="black", weight=3];
584[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];584 -> 592[label="",style="solid", color="black", weight=3];
91 -> 10[label="",style="dashed", color="red", weight=0];
91[label="List.deleteBy0 ww411 ww410 primEqInt (Pos (Succ ww300)) (primEqInt (Pos (Succ ww300)) ww410)",fontsize=16,color="magenta"];91 -> 108[label="",style="dashed", color="magenta", weight=3];
91 -> 109[label="",style="dashed", color="magenta", weight=3];
91 -> 110[label="",style="dashed", color="magenta", weight=3];
92[label="[]",fontsize=16,color="green",shape="box"];93[label="List.deleteBy primEqInt (Pos Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];93 -> 111[label="",style="solid", color="black", weight=3];
94[label="List.deleteBy primEqInt (Pos Zero) []",fontsize=16,color="black",shape="box"];94 -> 112[label="",style="solid", color="black", weight=3];
95 -> 10[label="",style="dashed", color="red", weight=0];
95[label="List.deleteBy0 ww411 ww410 primEqInt (Neg (Succ ww300)) (primEqInt (Neg (Succ ww300)) ww410)",fontsize=16,color="magenta"];95 -> 113[label="",style="dashed", color="magenta", weight=3];
95 -> 114[label="",style="dashed", color="magenta", weight=3];
95 -> 115[label="",style="dashed", color="magenta", weight=3];
96[label="[]",fontsize=16,color="green",shape="box"];585[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) (Succ ww550))",fontsize=16,color="black",shape="box"];585 -> 593[label="",style="solid", color="black", weight=3];
586[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat (Succ ww540) Zero)",fontsize=16,color="black",shape="box"];586 -> 594[label="",style="solid", color="black", weight=3];
587[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero (Succ ww550))",fontsize=16,color="black",shape="box"];587 -> 595[label="",style="solid", color="black", weight=3];
588[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];588 -> 596[label="",style="solid", color="black", weight=3];
101[label="List.deleteBy primEqInt (Neg Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];101 -> 121[label="",style="solid", color="black", weight=3];
102[label="List.deleteBy primEqInt (Neg Zero) []",fontsize=16,color="black",shape="box"];102 -> 122[label="",style="solid", color="black", weight=3];
589 -> 475[label="",style="dashed", color="red", weight=0];
589[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) (primEqNat ww480 ww490)",fontsize=16,color="magenta"];589 -> 597[label="",style="dashed", color="magenta", weight=3];
589 -> 598[label="",style="dashed", color="magenta", weight=3];
590[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="black",shape="triangle"];590 -> 599[label="",style="solid", color="black", weight=3];
591 -> 590[label="",style="dashed", color="red", weight=0];
591[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) False",fontsize=16,color="magenta"];592[label="List.deleteBy0 ww45 (Pos (Succ ww46)) primEqInt (Pos (Succ ww47)) True",fontsize=16,color="black",shape="box"];592 -> 600[label="",style="solid", color="black", weight=3];
108[label="ww411",fontsize=16,color="green",shape="box"];109[label="Pos (Succ ww300)",fontsize=16,color="green",shape="box"];110[label="ww410",fontsize=16,color="green",shape="box"];111 -> 10[label="",style="dashed", color="red", weight=0];
111[label="List.deleteBy0 ww411 ww410 primEqInt (Pos Zero) (primEqInt (Pos Zero) ww410)",fontsize=16,color="magenta"];111 -> 129[label="",style="dashed", color="magenta", weight=3];
111 -> 130[label="",style="dashed", color="magenta", weight=3];
111 -> 131[label="",style="dashed", color="magenta", weight=3];
112[label="[]",fontsize=16,color="green",shape="box"];113[label="ww411",fontsize=16,color="green",shape="box"];114[label="Neg (Succ ww300)",fontsize=16,color="green",shape="box"];115[label="ww410",fontsize=16,color="green",shape="box"];593 -> 528[label="",style="dashed", color="red", weight=0];
593[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) (primEqNat ww540 ww550)",fontsize=16,color="magenta"];593 -> 601[label="",style="dashed", color="magenta", weight=3];
593 -> 602[label="",style="dashed", color="magenta", weight=3];
594[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="black",shape="triangle"];594 -> 603[label="",style="solid", color="black", weight=3];
595 -> 594[label="",style="dashed", color="red", weight=0];
595[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) False",fontsize=16,color="magenta"];596[label="List.deleteBy0 ww51 (Neg (Succ ww52)) primEqInt (Neg (Succ ww53)) True",fontsize=16,color="black",shape="box"];596 -> 604[label="",style="solid", color="black", weight=3];
121 -> 10[label="",style="dashed", color="red", weight=0];
121[label="List.deleteBy0 ww411 ww410 primEqInt (Neg Zero) (primEqInt (Neg Zero) ww410)",fontsize=16,color="magenta"];121 -> 138[label="",style="dashed", color="magenta", weight=3];
121 -> 139[label="",style="dashed", color="magenta", weight=3];
121 -> 140[label="",style="dashed", color="magenta", weight=3];
122[label="[]",fontsize=16,color="green",shape="box"];597[label="ww490",fontsize=16,color="green",shape="box"];598[label="ww480",fontsize=16,color="green",shape="box"];599[label="Pos (Succ ww46) : List.deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="green",shape="box"];599 -> 605[label="",style="dashed", color="green", weight=3];
600[label="ww45",fontsize=16,color="green",shape="box"];129[label="ww411",fontsize=16,color="green",shape="box"];130[label="Pos Zero",fontsize=16,color="green",shape="box"];131[label="ww410",fontsize=16,color="green",shape="box"];601[label="ww540",fontsize=16,color="green",shape="box"];602[label="ww550",fontsize=16,color="green",shape="box"];603[label="Neg (Succ ww52) : List.deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="green",shape="box"];603 -> 606[label="",style="dashed", color="green", weight=3];
604[label="ww51",fontsize=16,color="green",shape="box"];138[label="ww411",fontsize=16,color="green",shape="box"];139[label="Neg Zero",fontsize=16,color="green",shape="box"];140[label="ww410",fontsize=16,color="green",shape="box"];605 -> 56[label="",style="dashed", color="red", weight=0];
605[label="List.deleteBy primEqInt (Pos (Succ ww47)) ww45",fontsize=16,color="magenta"];605 -> 607[label="",style="dashed", color="magenta", weight=3];
605 -> 608[label="",style="dashed", color="magenta", weight=3];
606 -> 61[label="",style="dashed", color="red", weight=0];
606[label="List.deleteBy primEqInt (Neg (Succ ww53)) ww51",fontsize=16,color="magenta"];606 -> 609[label="",style="dashed", color="magenta", weight=3];
606 -> 610[label="",style="dashed", color="magenta", weight=3];
607[label="ww45",fontsize=16,color="green",shape="box"];608[label="ww47",fontsize=16,color="green",shape="box"];609[label="ww53",fontsize=16,color="green",shape="box"];610[label="ww51",fontsize=16,color="green",shape="box"];}

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

(8)
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(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
   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(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) -> new_deleteBy01(ww411, ww410, Pos(Zero))
   new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) -> new_deleteBy01(ww411, ww410, Neg(Zero))
   new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) -> new_deleteBy3(ww41)
   new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) -> new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
   new_deleteBy01(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
   new_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) -> new_deleteBy1(ww41)
   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(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) -> new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
   new_deleteBy2(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
   new_deleteBy3(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Zero))
   new_deleteBy0(ww45, ww46, ww47, Succ(ww480), Succ(ww490)) -> new_deleteBy0(ww45, ww46, ww47, ww480, ww490)
   new_deleteBy1(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Zero))
   new_deleteBy01(ww41, Neg(Zero), Neg(Succ(ww300))) -> new_deleteBy2(ww300, ww41)
   new_deleteBy01(ww41, Pos(Zero), Pos(Succ(ww300))) -> new_deleteBy(ww300, ww41)
   new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))

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

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

(10)
Complex Obligation (AND)

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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) -> new_deleteBy3(ww41)
   new_deleteBy3(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Zero))
   new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) -> new_deleteBy01(ww411, ww410, Neg(Zero))

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

(12) 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(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2


*new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Neg(Zero)) -> new_deleteBy01(ww411, ww410, Neg(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Zero)) -> new_deleteBy3(ww41)
The graph contains the following edges 1 >= 1


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

(13)
YES

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

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

   new_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) -> new_deleteBy1(ww41)
   new_deleteBy1(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Zero))
   new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) -> new_deleteBy01(ww411, ww410, Pos(Zero))

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

(15) 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(:(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2


*new_deleteBy01(:(ww410, ww411), Pos(Succ(ww4000)), Pos(Zero)) -> new_deleteBy01(ww411, ww410, Pos(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww41, Neg(Succ(ww4000)), Pos(Zero)) -> new_deleteBy1(ww41)
The graph contains the following edges 1 >= 1


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

(16)
YES

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

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

   new_deleteBy2(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
   new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) -> new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
   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(ww41, Neg(Zero), Neg(Succ(ww300))) -> new_deleteBy2(ww300, ww41)
   new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))

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

(18) 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(ww41, Neg(Zero), Neg(Succ(ww300))) -> new_deleteBy2(ww300, ww41)
The graph contains the following edges 3 > 1, 1 >= 2


*new_deleteBy01(:(ww410, ww411), Pos(ww400), Neg(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww41, Neg(Succ(ww4000)), Neg(Succ(ww300))) -> new_deleteBy02(ww41, ww4000, ww300, ww300, ww4000)
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(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Neg(Succ(ww300)))
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


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(19)
YES

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(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
   new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) -> new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
   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(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
   new_deleteBy01(ww41, Pos(Zero), Pos(Succ(ww300))) -> new_deleteBy(ww300, ww41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(21) 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(ww41, Pos(Zero), Pos(Succ(ww300))) -> new_deleteBy(ww300, ww41)
The graph contains the following edges 3 > 1, 1 >= 2


*new_deleteBy01(:(ww410, ww411), Neg(ww400), Pos(Succ(ww300))) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(ww41, Pos(Succ(ww4000)), Pos(Succ(ww300))) -> new_deleteBy0(ww41, ww4000, ww300, ww300, ww4000)
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(ww300, :(ww410, ww411)) -> new_deleteBy01(ww411, ww410, Pos(Succ(ww300)))
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


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(22)
YES