/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 22 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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = case cmp x y of { 
GT-> y : insertBy cmp x ys'; 
_-> x : ys; 
 } ;

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

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

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case cmp x y of {
GT  -> y : insertBy cmp x ys';
_  -> x : ys}
"
is transformed to
"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = 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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = 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.

Binding Reductions:
The bind variable of the following binding Pattern
"ys@(wu : wv)"
is replaced by the following term
"wu : wv"

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

(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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = 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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = 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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656 -> 6[label="",style="solid", color="burlywood", weight=3];
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658 -> 11[label="",style="solid", color="burlywood", weight=3];
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660 -> 13[label="",style="solid", color="burlywood", weight=3];
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51 -> 513[label="",style="dashed", color="magenta", weight=3];
51 -> 514[label="",style="dashed", color="magenta", weight=3];
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51 -> 516[label="",style="dashed", color="magenta", weight=3];
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684 -> 67[label="",style="solid", color="burlywood", weight=3];
685[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 685[label="",style="solid", color="burlywood", weight=9];
685 -> 68[label="",style="solid", color="burlywood", weight=3];
54[label="List.insertBy0 (Pos (Succ ww4000)) primCmpInt (Pos Zero) ww41 (Pos (Succ ww4000) : ww41) LT",fontsize=16,color="black",shape="box"];54 -> 69[label="",style="solid", color="black", weight=3];
55[label="Pos Zero : Pos Zero : ww41",fontsize=16,color="green",shape="box"];56[label="Neg (Succ ww4000) : List.insertBy primCmpInt (Pos Zero) ww41",fontsize=16,color="green",shape="box"];56 -> 70[label="",style="dashed", color="green", weight=3];
57[label="Pos Zero : Neg Zero : ww41",fontsize=16,color="green",shape="box"];58 -> 566[label="",style="dashed", color="red", weight=0];
58[label="List.insertBy0 (Neg (Succ ww4000)) primCmpInt (Neg (Succ ww300)) ww41 (Neg (Succ ww4000) : ww41) (primCmpNat ww4000 ww300)",fontsize=16,color="magenta"];58 -> 567[label="",style="dashed", color="magenta", weight=3];
58 -> 568[label="",style="dashed", color="magenta", weight=3];
58 -> 569[label="",style="dashed", color="magenta", weight=3];
58 -> 570[label="",style="dashed", color="magenta", weight=3];
58 -> 571[label="",style="dashed", color="magenta", weight=3];
59[label="List.insertBy0 (Neg Zero) primCmpInt (Neg (Succ ww300)) ww41 (Neg Zero : ww41) LT",fontsize=16,color="black",shape="box"];59 -> 73[label="",style="solid", color="black", weight=3];
60[label="Neg Zero : Pos (Succ ww4000) : ww41",fontsize=16,color="green",shape="box"];61[label="Neg Zero : Pos Zero : ww41",fontsize=16,color="green",shape="box"];62[label="List.insertBy0 (Neg (Succ ww4000)) primCmpInt (Neg Zero) ww41 (Neg (Succ ww4000) : ww41) GT",fontsize=16,color="black",shape="box"];62 -> 74[label="",style="solid", color="black", weight=3];
63[label="Neg Zero : Neg Zero : ww41",fontsize=16,color="green",shape="box"];512[label="ww4000",fontsize=16,color="green",shape="box"];513[label="ww300",fontsize=16,color="green",shape="box"];514[label="ww300",fontsize=16,color="green",shape="box"];515[label="ww4000",fontsize=16,color="green",shape="box"];516[label="ww41",fontsize=16,color="green",shape="box"];511[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat ww62 ww63)",fontsize=16,color="burlywood",shape="triangle"];686[label="ww62/Succ ww620",fontsize=10,color="white",style="solid",shape="box"];511 -> 686[label="",style="solid", color="burlywood", weight=9];
686 -> 562[label="",style="solid", color="burlywood", weight=3];
687[label="ww62/Zero",fontsize=10,color="white",style="solid",shape="box"];511 -> 687[label="",style="solid", color="burlywood", weight=9];
687 -> 563[label="",style="solid", color="burlywood", weight=3];
66[label="Pos Zero : List.insertBy primCmpInt (Pos (Succ ww300)) ww41",fontsize=16,color="green",shape="box"];66 -> 79[label="",style="dashed", color="green", weight=3];
67[label="List.insertBy primCmpInt (Pos (Succ ww300)) (ww410 : ww411)",fontsize=16,color="black",shape="box"];67 -> 80[label="",style="solid", color="black", weight=3];
68[label="List.insertBy primCmpInt (Pos (Succ ww300)) []",fontsize=16,color="black",shape="box"];68 -> 81[label="",style="solid", color="black", weight=3];
69[label="Pos Zero : Pos (Succ ww4000) : ww41",fontsize=16,color="green",shape="box"];70[label="List.insertBy primCmpInt (Pos Zero) ww41",fontsize=16,color="burlywood",shape="box"];688[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];70 -> 688[label="",style="solid", color="burlywood", weight=9];
688 -> 82[label="",style="solid", color="burlywood", weight=3];
689[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];70 -> 689[label="",style="solid", color="burlywood", weight=9];
689 -> 83[label="",style="solid", color="burlywood", weight=3];
567[label="ww300",fontsize=16,color="green",shape="box"];568[label="ww4000",fontsize=16,color="green",shape="box"];569[label="ww41",fontsize=16,color="green",shape="box"];570[label="ww4000",fontsize=16,color="green",shape="box"];571[label="ww300",fontsize=16,color="green",shape="box"];566[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat ww68 ww69)",fontsize=16,color="burlywood",shape="triangle"];690[label="ww68/Succ ww680",fontsize=10,color="white",style="solid",shape="box"];566 -> 690[label="",style="solid", color="burlywood", weight=9];
690 -> 617[label="",style="solid", color="burlywood", weight=3];
691[label="ww68/Zero",fontsize=10,color="white",style="solid",shape="box"];566 -> 691[label="",style="solid", color="burlywood", weight=9];
691 -> 618[label="",style="solid", color="burlywood", weight=3];
73[label="Neg (Succ ww300) : Neg Zero : ww41",fontsize=16,color="green",shape="box"];74[label="Neg (Succ ww4000) : List.insertBy primCmpInt (Neg Zero) ww41",fontsize=16,color="green",shape="box"];74 -> 88[label="",style="dashed", color="green", weight=3];
562[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) ww63)",fontsize=16,color="burlywood",shape="box"];692[label="ww63/Succ ww630",fontsize=10,color="white",style="solid",shape="box"];562 -> 692[label="",style="solid", color="burlywood", weight=9];
692 -> 619[label="",style="solid", color="burlywood", weight=3];
693[label="ww63/Zero",fontsize=10,color="white",style="solid",shape="box"];562 -> 693[label="",style="solid", color="burlywood", weight=9];
693 -> 620[label="",style="solid", color="burlywood", weight=3];
563[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero ww63)",fontsize=16,color="burlywood",shape="box"];694[label="ww63/Succ ww630",fontsize=10,color="white",style="solid",shape="box"];563 -> 694[label="",style="solid", color="burlywood", weight=9];
694 -> 621[label="",style="solid", color="burlywood", weight=3];
695[label="ww63/Zero",fontsize=10,color="white",style="solid",shape="box"];563 -> 695[label="",style="solid", color="burlywood", weight=9];
695 -> 622[label="",style="solid", color="burlywood", weight=3];
79 -> 53[label="",style="dashed", color="red", weight=0];
79[label="List.insertBy primCmpInt (Pos (Succ ww300)) ww41",fontsize=16,color="magenta"];80 -> 10[label="",style="dashed", color="red", weight=0];
80[label="List.insertBy0 ww410 primCmpInt (Pos (Succ ww300)) ww411 (ww410 : ww411) (primCmpInt (Pos (Succ ww300)) ww410)",fontsize=16,color="magenta"];80 -> 93[label="",style="dashed", color="magenta", weight=3];
80 -> 94[label="",style="dashed", color="magenta", weight=3];
80 -> 95[label="",style="dashed", color="magenta", weight=3];
81[label="Pos (Succ ww300) : []",fontsize=16,color="green",shape="box"];82[label="List.insertBy primCmpInt (Pos Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];82 -> 96[label="",style="solid", color="black", weight=3];
83[label="List.insertBy primCmpInt (Pos Zero) []",fontsize=16,color="black",shape="box"];83 -> 97[label="",style="solid", color="black", weight=3];
617[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) ww69)",fontsize=16,color="burlywood",shape="box"];696[label="ww69/Succ ww690",fontsize=10,color="white",style="solid",shape="box"];617 -> 696[label="",style="solid", color="burlywood", weight=9];
696 -> 623[label="",style="solid", color="burlywood", weight=3];
697[label="ww69/Zero",fontsize=10,color="white",style="solid",shape="box"];617 -> 697[label="",style="solid", color="burlywood", weight=9];
697 -> 624[label="",style="solid", color="burlywood", weight=3];
618[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero ww69)",fontsize=16,color="burlywood",shape="box"];698[label="ww69/Succ ww690",fontsize=10,color="white",style="solid",shape="box"];618 -> 698[label="",style="solid", color="burlywood", weight=9];
698 -> 625[label="",style="solid", color="burlywood", weight=3];
699[label="ww69/Zero",fontsize=10,color="white",style="solid",shape="box"];618 -> 699[label="",style="solid", color="burlywood", weight=9];
699 -> 626[label="",style="solid", color="burlywood", weight=3];
88[label="List.insertBy primCmpInt (Neg Zero) ww41",fontsize=16,color="burlywood",shape="box"];700[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];88 -> 700[label="",style="solid", color="burlywood", weight=9];
700 -> 102[label="",style="solid", color="burlywood", weight=3];
701[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];88 -> 701[label="",style="solid", color="burlywood", weight=9];
701 -> 103[label="",style="solid", color="burlywood", weight=3];
619[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) (Succ ww630))",fontsize=16,color="black",shape="box"];619 -> 627[label="",style="solid", color="black", weight=3];
620[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat (Succ ww620) Zero)",fontsize=16,color="black",shape="box"];620 -> 628[label="",style="solid", color="black", weight=3];
621[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero (Succ ww630))",fontsize=16,color="black",shape="box"];621 -> 629[label="",style="solid", color="black", weight=3];
622[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];622 -> 630[label="",style="solid", color="black", weight=3];
93[label="ww411",fontsize=16,color="green",shape="box"];94[label="ww410",fontsize=16,color="green",shape="box"];95[label="Pos (Succ ww300)",fontsize=16,color="green",shape="box"];96 -> 10[label="",style="dashed", color="red", weight=0];
96[label="List.insertBy0 ww410 primCmpInt (Pos Zero) ww411 (ww410 : ww411) (primCmpInt (Pos Zero) ww410)",fontsize=16,color="magenta"];96 -> 109[label="",style="dashed", color="magenta", weight=3];
96 -> 110[label="",style="dashed", color="magenta", weight=3];
96 -> 111[label="",style="dashed", color="magenta", weight=3];
97[label="Pos Zero : []",fontsize=16,color="green",shape="box"];623[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) (Succ ww690))",fontsize=16,color="black",shape="box"];623 -> 631[label="",style="solid", color="black", weight=3];
624[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat (Succ ww680) Zero)",fontsize=16,color="black",shape="box"];624 -> 632[label="",style="solid", color="black", weight=3];
625[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero (Succ ww690))",fontsize=16,color="black",shape="box"];625 -> 633[label="",style="solid", color="black", weight=3];
626[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat Zero Zero)",fontsize=16,color="black",shape="box"];626 -> 634[label="",style="solid", color="black", weight=3];
102[label="List.insertBy primCmpInt (Neg Zero) (ww410 : ww411)",fontsize=16,color="black",shape="box"];102 -> 117[label="",style="solid", color="black", weight=3];
103[label="List.insertBy primCmpInt (Neg Zero) []",fontsize=16,color="black",shape="box"];103 -> 118[label="",style="solid", color="black", weight=3];
627 -> 511[label="",style="dashed", color="red", weight=0];
627[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) (primCmpNat ww620 ww630)",fontsize=16,color="magenta"];627 -> 635[label="",style="dashed", color="magenta", weight=3];
627 -> 636[label="",style="dashed", color="magenta", weight=3];
628[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) GT",fontsize=16,color="black",shape="box"];628 -> 637[label="",style="solid", color="black", weight=3];
629[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) LT",fontsize=16,color="black",shape="box"];629 -> 638[label="",style="solid", color="black", weight=3];
630[label="List.insertBy0 (Pos (Succ ww59)) primCmpInt (Pos (Succ ww60)) ww61 (Pos (Succ ww59) : ww61) EQ",fontsize=16,color="black",shape="box"];630 -> 639[label="",style="solid", color="black", weight=3];
109[label="ww411",fontsize=16,color="green",shape="box"];110[label="ww410",fontsize=16,color="green",shape="box"];111[label="Pos Zero",fontsize=16,color="green",shape="box"];631 -> 566[label="",style="dashed", color="red", weight=0];
631[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) (primCmpNat ww680 ww690)",fontsize=16,color="magenta"];631 -> 640[label="",style="dashed", color="magenta", weight=3];
631 -> 641[label="",style="dashed", color="magenta", weight=3];
632[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) GT",fontsize=16,color="black",shape="box"];632 -> 642[label="",style="solid", color="black", weight=3];
633[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) LT",fontsize=16,color="black",shape="box"];633 -> 643[label="",style="solid", color="black", weight=3];
634[label="List.insertBy0 (Neg (Succ ww65)) primCmpInt (Neg (Succ ww66)) ww67 (Neg (Succ ww65) : ww67) EQ",fontsize=16,color="black",shape="box"];634 -> 644[label="",style="solid", color="black", weight=3];
117 -> 10[label="",style="dashed", color="red", weight=0];
117[label="List.insertBy0 ww410 primCmpInt (Neg Zero) ww411 (ww410 : ww411) (primCmpInt (Neg Zero) ww410)",fontsize=16,color="magenta"];117 -> 129[label="",style="dashed", color="magenta", weight=3];
117 -> 130[label="",style="dashed", color="magenta", weight=3];
117 -> 131[label="",style="dashed", color="magenta", weight=3];
118[label="Neg Zero : []",fontsize=16,color="green",shape="box"];635[label="ww630",fontsize=16,color="green",shape="box"];636[label="ww620",fontsize=16,color="green",shape="box"];637[label="Pos (Succ ww59) : List.insertBy primCmpInt (Pos (Succ ww60)) ww61",fontsize=16,color="green",shape="box"];637 -> 645[label="",style="dashed", color="green", weight=3];
638[label="Pos (Succ ww60) : Pos (Succ ww59) : ww61",fontsize=16,color="green",shape="box"];639[label="Pos (Succ ww60) : Pos (Succ ww59) : ww61",fontsize=16,color="green",shape="box"];640[label="ww680",fontsize=16,color="green",shape="box"];641[label="ww690",fontsize=16,color="green",shape="box"];642[label="Neg (Succ ww65) : List.insertBy primCmpInt (Neg (Succ ww66)) ww67",fontsize=16,color="green",shape="box"];642 -> 646[label="",style="dashed", color="green", weight=3];
643[label="Neg (Succ ww66) : Neg (Succ ww65) : ww67",fontsize=16,color="green",shape="box"];644[label="Neg (Succ ww66) : Neg (Succ ww65) : ww67",fontsize=16,color="green",shape="box"];129[label="ww411",fontsize=16,color="green",shape="box"];130[label="ww410",fontsize=16,color="green",shape="box"];131[label="Neg Zero",fontsize=16,color="green",shape="box"];645 -> 53[label="",style="dashed", color="red", weight=0];
645[label="List.insertBy primCmpInt (Pos (Succ ww60)) ww61",fontsize=16,color="magenta"];645 -> 647[label="",style="dashed", color="magenta", weight=3];
645 -> 648[label="",style="dashed", color="magenta", weight=3];
646[label="List.insertBy primCmpInt (Neg (Succ ww66)) ww67",fontsize=16,color="burlywood",shape="box"];702[label="ww67/ww670 : ww671",fontsize=10,color="white",style="solid",shape="box"];646 -> 702[label="",style="solid", color="burlywood", weight=9];
702 -> 649[label="",style="solid", color="burlywood", weight=3];
703[label="ww67/[]",fontsize=10,color="white",style="solid",shape="box"];646 -> 703[label="",style="solid", color="burlywood", weight=9];
703 -> 650[label="",style="solid", color="burlywood", weight=3];
647[label="ww61",fontsize=16,color="green",shape="box"];648[label="ww60",fontsize=16,color="green",shape="box"];649[label="List.insertBy primCmpInt (Neg (Succ ww66)) (ww670 : ww671)",fontsize=16,color="black",shape="box"];649 -> 651[label="",style="solid", color="black", weight=3];
650[label="List.insertBy primCmpInt (Neg (Succ ww66)) []",fontsize=16,color="black",shape="box"];650 -> 652[label="",style="solid", color="black", weight=3];
651 -> 10[label="",style="dashed", color="red", weight=0];
651[label="List.insertBy0 ww670 primCmpInt (Neg (Succ ww66)) ww671 (ww670 : ww671) (primCmpInt (Neg (Succ ww66)) ww670)",fontsize=16,color="magenta"];651 -> 653[label="",style="dashed", color="magenta", weight=3];
651 -> 654[label="",style="dashed", color="magenta", weight=3];
651 -> 655[label="",style="dashed", color="magenta", weight=3];
652[label="Neg (Succ ww66) : []",fontsize=16,color="green",shape="box"];653[label="ww671",fontsize=16,color="green",shape="box"];654[label="ww670",fontsize=16,color="green",shape="box"];655[label="Neg (Succ ww66)",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) -> new_insertBy(ww300, ww41)
   new_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) -> new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
   new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) -> new_insertBy01(ww65, ww66, ww67, ww680, ww690)
   new_insertBy(ww300, :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
   new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) -> new_insertBy0(ww59, ww60, ww61, ww620, ww630)
   new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
   new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) -> new_insertBy(ww60, ww61)
   new_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Zero), ww411)
   new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) -> new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
   new_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Neg(Zero), ww411)
   new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) -> new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)

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_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Neg(Zero), ww411)

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_insertBy00(Neg(Succ(ww4000)), Neg(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Neg(Zero), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3


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

(13)
YES

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

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

   new_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Zero), ww411)

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_insertBy00(Neg(Succ(ww4000)), Pos(Zero), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Zero), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3


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

(16)
YES

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

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

   new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) -> new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
   new_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) -> new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
   new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) -> new_insertBy01(ww65, ww66, ww67, ww680, ww690)

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_insertBy01(ww65, ww66, :(ww670, ww671), Succ(ww680), Zero) -> new_insertBy00(ww670, Neg(Succ(ww66)), ww671)
The graph contains the following edges 3 > 1, 3 > 3


*new_insertBy01(ww65, ww66, ww67, Succ(ww680), Succ(ww690)) -> new_insertBy01(ww65, ww66, ww67, ww680, ww690)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


*new_insertBy00(Neg(Succ(ww4000)), Neg(Succ(ww300)), ww41) -> new_insertBy01(ww4000, ww300, ww41, ww4000, ww300)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 1 > 4, 2 > 5


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(19)
YES

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

   new_insertBy(ww300, :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
   new_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) -> new_insertBy(ww300, ww41)
   new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
   new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) -> new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
   new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) -> new_insertBy0(ww59, ww60, ww61, ww620, ww630)
   new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) -> new_insertBy(ww60, ww61)

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_insertBy00(Pos(Zero), Pos(Succ(ww300)), ww41) -> new_insertBy(ww300, ww41)
The graph contains the following edges 2 > 1, 3 >= 2


*new_insertBy0(ww59, ww60, ww61, Succ(ww620), Zero) -> new_insertBy(ww60, ww61)
The graph contains the following edges 2 >= 1, 3 >= 2


*new_insertBy00(Neg(ww400), Pos(Succ(ww300)), :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
The graph contains the following edges 3 > 1, 2 >= 2, 3 > 3


*new_insertBy00(Pos(Succ(ww4000)), Pos(Succ(ww300)), ww41) -> new_insertBy0(ww4000, ww300, ww41, ww300, ww4000)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 2 > 4, 1 > 5


*new_insertBy(ww300, :(ww410, ww411)) -> new_insertBy00(ww410, Pos(Succ(ww300)), ww411)
The graph contains the following edges 2 > 1, 2 > 3


*new_insertBy0(ww59, ww60, ww61, Succ(ww620), Succ(ww630)) -> new_insertBy0(ww59, ww60, ww61, ww620, ww630)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


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(22)
YES