/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 19 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) DependencyGraphProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 40 ms]
        (16) YES
    (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) TransformationProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) DependencyGraphProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) UsableRulesProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) QReductionProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) TransformationProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QReductionProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) TransformationProof [EQUIVALENT, 0 ms]
        (36) QDP
        (37) TransformationProof [EQUIVALENT, 0 ms]
        (38) QDP
        (39) TransformationProof [EQUIVALENT, 0 ms]
        (40) QDP
        (41) DependencyGraphProof [EQUIVALENT, 0 ms]
        (42) AND
            (43) QDP
                (44) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (45) YES
            (46) QDP
                (47) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (48) YES
            (49) QDP
                (50) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (51) YES
            (52) QDP
                (53) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (54) YES
            (55) QDP
                (56) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (57) 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;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = case x `cmp` y of { 
GT-> y : merge cmp (x : xs) ys; 
_-> x : merge cmp xs (y : ys); 
 } ;

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
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 : merge cmp (x : xs) ys;
_  -> x : merge cmp xs (y : ys)}
"
is transformed to
"merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys _ = x : merge cmp xs (y : 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;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys _ = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

}
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;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

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

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"compare x y|x == yEQ|x <= yLT|otherwiseGT;
"
is transformed to
"compare x y = compare3 x y;
"
"compare1 x y True = LT;
compare1 x y False = compare0 x y otherwise;
"
"compare0 x y True = GT;
"
"compare2 x y True = EQ;
compare2 x y False = compare1 x y (x <= y);
"
"compare3 x y = compare2 x y (x == y);
"
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;
merge :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a]  ->  [a];
merge cmp xs [] = xs;
merge cmp [] ys = ys;
merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y);

merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys;
merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys);

merge_pairs :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [[a]];
merge_pairs cmp [] = [];
merge_pairs cmp (xs : []) = xs : [];
merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss;

mergesort :: (a  ->  a  ->  Ordering)  ->  [a]  ->  [a];
mergesort cmp = mergesort' cmp . map wrap;

mergesort' :: (a  ->  a  ->  Ordering)  ->  [[a]]  ->  [a];
mergesort' cmp [] = [];
mergesort' cmp (xs : []) = xs;
mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss);

sort :: Ord a => [a]  ->  [a];
sort l = mergesort compare l;

wrap :: a  ->  [a];
wrap x = x : [];

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

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

(7) Narrow (SOUND)
Haskell To QDPs

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480 -> 17[label="",style="dashed", color="red", weight=0];
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861 -> 488[label="",style="solid", color="burlywood", weight=3];
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862 -> 491[label="",style="solid", color="burlywood", weight=3];
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509 -> 513[label="",style="dashed", color="magenta", weight=3];
510 -> 324[label="",style="dashed", color="red", weight=0];
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510 -> 515[label="",style="dashed", color="magenta", weight=3];
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869 -> 519[label="",style="solid", color="burlywood", weight=3];
870[label="vz3510/[]",fontsize=10,color="white",style="solid",shape="box"];512 -> 870[label="",style="solid", color="burlywood", weight=9];
870 -> 520[label="",style="solid", color="burlywood", weight=3];
513[label="List.merge_pairs compare vz3511",fontsize=16,color="burlywood",shape="triangle"];871[label="vz3511/vz35110 : vz35111",fontsize=10,color="white",style="solid",shape="box"];513 -> 871[label="",style="solid", color="burlywood", weight=9];
871 -> 521[label="",style="solid", color="burlywood", weight=3];
872[label="vz3511/[]",fontsize=10,color="white",style="solid",shape="box"];513 -> 872[label="",style="solid", color="burlywood", weight=9];
872 -> 522[label="",style="solid", color="burlywood", weight=3];
514[label="List.merge compare vz34 vz350",fontsize=16,color="burlywood",shape="triangle"];873[label="vz350/vz3500 : vz3501",fontsize=10,color="white",style="solid",shape="box"];514 -> 873[label="",style="solid", color="burlywood", weight=9];
873 -> 523[label="",style="solid", color="burlywood", weight=3];
874[label="vz350/[]",fontsize=10,color="white",style="solid",shape="box"];514 -> 874[label="",style="solid", color="burlywood", weight=9];
874 -> 524[label="",style="solid", color="burlywood", weight=3];
515[label="[]",fontsize=16,color="green",shape="box"];722[label="compare2 vz370 vz360 (vz370 == vz360)",fontsize=16,color="burlywood",shape="box"];875[label="vz370/LT",fontsize=10,color="white",style="solid",shape="box"];722 -> 875[label="",style="solid", color="burlywood", weight=9];
875 -> 769[label="",style="solid", color="burlywood", weight=3];
876[label="vz370/EQ",fontsize=10,color="white",style="solid",shape="box"];722 -> 876[label="",style="solid", color="burlywood", weight=9];
876 -> 770[label="",style="solid", color="burlywood", weight=3];
877[label="vz370/GT",fontsize=10,color="white",style="solid",shape="box"];722 -> 877[label="",style="solid", color="burlywood", weight=9];
877 -> 771[label="",style="solid", color="burlywood", weight=3];
723[label="vz45 : List.merge compare vz46 (vz44 : vz47)",fontsize=16,color="green",shape="box"];723 -> 772[label="",style="dashed", color="green", weight=3];
724[label="vz45 : List.merge compare vz46 (vz44 : vz47)",fontsize=16,color="green",shape="box"];724 -> 773[label="",style="dashed", color="green", weight=3];
725[label="vz44 : List.merge compare (vz45 : vz46) vz47",fontsize=16,color="green",shape="box"];725 -> 774[label="",style="dashed", color="green", weight=3];
519[label="List.merge compare vz34 (List.merge compare vz350 (vz35100 : vz35101))",fontsize=16,color="burlywood",shape="box"];878[label="vz350/vz3500 : vz3501",fontsize=10,color="white",style="solid",shape="box"];519 -> 878[label="",style="solid", color="burlywood", weight=9];
878 -> 534[label="",style="solid", color="burlywood", weight=3];
879[label="vz350/[]",fontsize=10,color="white",style="solid",shape="box"];519 -> 879[label="",style="solid", color="burlywood", weight=9];
879 -> 535[label="",style="solid", color="burlywood", weight=3];
520[label="List.merge compare vz34 (List.merge compare vz350 [])",fontsize=16,color="black",shape="box"];520 -> 536[label="",style="solid", color="black", weight=3];
521[label="List.merge_pairs compare (vz35110 : vz35111)",fontsize=16,color="burlywood",shape="box"];880[label="vz35111/vz351110 : vz351111",fontsize=10,color="white",style="solid",shape="box"];521 -> 880[label="",style="solid", color="burlywood", weight=9];
880 -> 537[label="",style="solid", color="burlywood", weight=3];
881[label="vz35111/[]",fontsize=10,color="white",style="solid",shape="box"];521 -> 881[label="",style="solid", color="burlywood", weight=9];
881 -> 538[label="",style="solid", color="burlywood", weight=3];
522[label="List.merge_pairs compare []",fontsize=16,color="black",shape="box"];522 -> 539[label="",style="solid", color="black", weight=3];
523[label="List.merge compare vz34 (vz3500 : vz3501)",fontsize=16,color="burlywood",shape="box"];882[label="vz34/vz340 : vz341",fontsize=10,color="white",style="solid",shape="box"];523 -> 882[label="",style="solid", color="burlywood", weight=9];
882 -> 540[label="",style="solid", color="burlywood", weight=3];
883[label="vz34/[]",fontsize=10,color="white",style="solid",shape="box"];523 -> 883[label="",style="solid", color="burlywood", weight=9];
883 -> 541[label="",style="solid", color="burlywood", weight=3];
524[label="List.merge compare vz34 []",fontsize=16,color="black",shape="box"];524 -> 542[label="",style="solid", color="black", weight=3];
769[label="compare2 LT vz360 (LT == vz360)",fontsize=16,color="burlywood",shape="box"];884[label="vz360/LT",fontsize=10,color="white",style="solid",shape="box"];769 -> 884[label="",style="solid", color="burlywood", weight=9];
884 -> 788[label="",style="solid", color="burlywood", weight=3];
885[label="vz360/EQ",fontsize=10,color="white",style="solid",shape="box"];769 -> 885[label="",style="solid", color="burlywood", weight=9];
885 -> 789[label="",style="solid", color="burlywood", weight=3];
886[label="vz360/GT",fontsize=10,color="white",style="solid",shape="box"];769 -> 886[label="",style="solid", color="burlywood", weight=9];
886 -> 790[label="",style="solid", color="burlywood", weight=3];
770[label="compare2 EQ vz360 (EQ == vz360)",fontsize=16,color="burlywood",shape="box"];887[label="vz360/LT",fontsize=10,color="white",style="solid",shape="box"];770 -> 887[label="",style="solid", color="burlywood", weight=9];
887 -> 791[label="",style="solid", color="burlywood", weight=3];
888[label="vz360/EQ",fontsize=10,color="white",style="solid",shape="box"];770 -> 888[label="",style="solid", color="burlywood", weight=9];
888 -> 792[label="",style="solid", color="burlywood", weight=3];
889[label="vz360/GT",fontsize=10,color="white",style="solid",shape="box"];770 -> 889[label="",style="solid", color="burlywood", weight=9];
889 -> 793[label="",style="solid", color="burlywood", weight=3];
771[label="compare2 GT vz360 (GT == vz360)",fontsize=16,color="burlywood",shape="box"];890[label="vz360/LT",fontsize=10,color="white",style="solid",shape="box"];771 -> 890[label="",style="solid", color="burlywood", weight=9];
890 -> 794[label="",style="solid", color="burlywood", weight=3];
891[label="vz360/EQ",fontsize=10,color="white",style="solid",shape="box"];771 -> 891[label="",style="solid", color="burlywood", weight=9];
891 -> 795[label="",style="solid", color="burlywood", weight=3];
892[label="vz360/GT",fontsize=10,color="white",style="solid",shape="box"];771 -> 892[label="",style="solid", color="burlywood", weight=9];
892 -> 796[label="",style="solid", color="burlywood", weight=3];
772 -> 514[label="",style="dashed", color="red", weight=0];
772[label="List.merge compare vz46 (vz44 : vz47)",fontsize=16,color="magenta"];772 -> 797[label="",style="dashed", color="magenta", weight=3];
772 -> 798[label="",style="dashed", color="magenta", weight=3];
773 -> 514[label="",style="dashed", color="red", weight=0];
773[label="List.merge compare vz46 (vz44 : vz47)",fontsize=16,color="magenta"];773 -> 799[label="",style="dashed", color="magenta", weight=3];
773 -> 800[label="",style="dashed", color="magenta", weight=3];
774 -> 514[label="",style="dashed", color="red", weight=0];
774[label="List.merge compare (vz45 : vz46) vz47",fontsize=16,color="magenta"];774 -> 801[label="",style="dashed", color="magenta", weight=3];
774 -> 802[label="",style="dashed", color="magenta", weight=3];
534[label="List.merge compare vz34 (List.merge compare (vz3500 : vz3501) (vz35100 : vz35101))",fontsize=16,color="black",shape="box"];534 -> 552[label="",style="solid", color="black", weight=3];
535[label="List.merge compare vz34 (List.merge compare [] (vz35100 : vz35101))",fontsize=16,color="black",shape="box"];535 -> 553[label="",style="solid", color="black", weight=3];
536 -> 514[label="",style="dashed", color="red", weight=0];
536[label="List.merge compare vz34 vz350",fontsize=16,color="magenta"];537[label="List.merge_pairs compare (vz35110 : vz351110 : vz351111)",fontsize=16,color="black",shape="box"];537 -> 554[label="",style="solid", color="black", weight=3];
538[label="List.merge_pairs compare (vz35110 : [])",fontsize=16,color="black",shape="box"];538 -> 555[label="",style="solid", color="black", weight=3];
539[label="[]",fontsize=16,color="green",shape="box"];540[label="List.merge compare (vz340 : vz341) (vz3500 : vz3501)",fontsize=16,color="black",shape="box"];540 -> 556[label="",style="solid", color="black", weight=3];
541[label="List.merge compare [] (vz3500 : vz3501)",fontsize=16,color="black",shape="box"];541 -> 557[label="",style="solid", color="black", weight=3];
542[label="vz34",fontsize=16,color="green",shape="box"];788[label="compare2 LT LT (LT == LT)",fontsize=16,color="black",shape="box"];788 -> 816[label="",style="solid", color="black", weight=3];
789[label="compare2 LT EQ (LT == EQ)",fontsize=16,color="black",shape="box"];789 -> 817[label="",style="solid", color="black", weight=3];
790[label="compare2 LT GT (LT == GT)",fontsize=16,color="black",shape="box"];790 -> 818[label="",style="solid", color="black", weight=3];
791[label="compare2 EQ LT (EQ == LT)",fontsize=16,color="black",shape="box"];791 -> 819[label="",style="solid", color="black", weight=3];
792[label="compare2 EQ EQ (EQ == EQ)",fontsize=16,color="black",shape="box"];792 -> 820[label="",style="solid", color="black", weight=3];
793[label="compare2 EQ GT (EQ == GT)",fontsize=16,color="black",shape="box"];793 -> 821[label="",style="solid", color="black", weight=3];
794[label="compare2 GT LT (GT == LT)",fontsize=16,color="black",shape="box"];794 -> 822[label="",style="solid", color="black", weight=3];
795[label="compare2 GT EQ (GT == EQ)",fontsize=16,color="black",shape="box"];795 -> 823[label="",style="solid", color="black", weight=3];
796[label="compare2 GT GT (GT == GT)",fontsize=16,color="black",shape="box"];796 -> 824[label="",style="solid", color="black", weight=3];
797[label="vz46",fontsize=16,color="green",shape="box"];798[label="vz44 : vz47",fontsize=16,color="green",shape="box"];799[label="vz46",fontsize=16,color="green",shape="box"];800[label="vz44 : vz47",fontsize=16,color="green",shape="box"];801[label="vz45 : vz46",fontsize=16,color="green",shape="box"];802[label="vz47",fontsize=16,color="green",shape="box"];552 -> 514[label="",style="dashed", color="red", weight=0];
552[label="List.merge compare vz34 (List.merge0 vz35100 compare vz3500 vz3501 vz35101 (compare vz3500 vz35100))",fontsize=16,color="magenta"];552 -> 567[label="",style="dashed", color="magenta", weight=3];
553 -> 514[label="",style="dashed", color="red", weight=0];
553[label="List.merge compare vz34 (vz35100 : vz35101)",fontsize=16,color="magenta"];553 -> 568[label="",style="dashed", color="magenta", weight=3];
554[label="List.merge compare vz35110 vz351110 : List.merge_pairs compare vz351111",fontsize=16,color="green",shape="box"];554 -> 569[label="",style="dashed", color="green", weight=3];
554 -> 570[label="",style="dashed", color="green", weight=3];
555[label="vz35110 : []",fontsize=16,color="green",shape="box"];556 -> 571[label="",style="dashed", color="red", weight=0];
556[label="List.merge0 vz3500 compare vz340 vz341 vz3501 (compare vz340 vz3500)",fontsize=16,color="magenta"];556 -> 632[label="",style="dashed", color="magenta", weight=3];
556 -> 633[label="",style="dashed", color="magenta", weight=3];
556 -> 634[label="",style="dashed", color="magenta", weight=3];
556 -> 635[label="",style="dashed", color="magenta", weight=3];
556 -> 636[label="",style="dashed", color="magenta", weight=3];
557[label="vz3500 : vz3501",fontsize=16,color="green",shape="box"];816[label="compare2 LT LT True",fontsize=16,color="black",shape="box"];816 -> 825[label="",style="solid", color="black", weight=3];
817[label="compare2 LT EQ False",fontsize=16,color="black",shape="box"];817 -> 826[label="",style="solid", color="black", weight=3];
818[label="compare2 LT GT False",fontsize=16,color="black",shape="box"];818 -> 827[label="",style="solid", color="black", weight=3];
819[label="compare2 EQ LT False",fontsize=16,color="black",shape="box"];819 -> 828[label="",style="solid", color="black", weight=3];
820[label="compare2 EQ EQ True",fontsize=16,color="black",shape="box"];820 -> 829[label="",style="solid", color="black", weight=3];
821[label="compare2 EQ GT False",fontsize=16,color="black",shape="box"];821 -> 830[label="",style="solid", color="black", weight=3];
822[label="compare2 GT LT False",fontsize=16,color="black",shape="box"];822 -> 831[label="",style="solid", color="black", weight=3];
823[label="compare2 GT EQ False",fontsize=16,color="black",shape="box"];823 -> 832[label="",style="solid", color="black", weight=3];
824[label="compare2 GT GT True",fontsize=16,color="black",shape="box"];824 -> 833[label="",style="solid", color="black", weight=3];
567 -> 571[label="",style="dashed", color="red", weight=0];
567[label="List.merge0 vz35100 compare vz3500 vz3501 vz35101 (compare vz3500 vz35100)",fontsize=16,color="magenta"];567 -> 682[label="",style="dashed", color="magenta", weight=3];
567 -> 683[label="",style="dashed", color="magenta", weight=3];
567 -> 684[label="",style="dashed", color="magenta", weight=3];
567 -> 685[label="",style="dashed", color="magenta", weight=3];
567 -> 686[label="",style="dashed", color="magenta", weight=3];
568[label="vz35100 : vz35101",fontsize=16,color="green",shape="box"];569 -> 514[label="",style="dashed", color="red", weight=0];
569[label="List.merge compare vz35110 vz351110",fontsize=16,color="magenta"];569 -> 691[label="",style="dashed", color="magenta", weight=3];
569 -> 692[label="",style="dashed", color="magenta", weight=3];
570 -> 513[label="",style="dashed", color="red", weight=0];
570[label="List.merge_pairs compare vz351111",fontsize=16,color="magenta"];570 -> 693[label="",style="dashed", color="magenta", weight=3];
632[label="compare vz340 vz3500",fontsize=16,color="blue",shape="box"];893[label="compare :: ([] a) -> ([] a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 893[label="",style="solid", color="blue", weight=9];
893 -> 694[label="",style="solid", color="blue", weight=3];
894[label="compare :: ((@2) a b) -> ((@2) a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 894[label="",style="solid", color="blue", weight=9];
894 -> 695[label="",style="solid", color="blue", weight=3];
895[label="compare :: Char -> Char -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 895[label="",style="solid", color="blue", weight=9];
895 -> 696[label="",style="solid", color="blue", weight=3];
896[label="compare :: ((@3) a b c) -> ((@3) a b c) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 896[label="",style="solid", color="blue", weight=9];
896 -> 697[label="",style="solid", color="blue", weight=3];
897[label="compare :: Ordering -> Ordering -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 897[label="",style="solid", color="blue", weight=9];
897 -> 698[label="",style="solid", color="blue", weight=3];
898[label="compare :: Bool -> Bool -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 898[label="",style="solid", color="blue", weight=9];
898 -> 699[label="",style="solid", color="blue", weight=3];
899[label="compare :: Integer -> Integer -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 899[label="",style="solid", color="blue", weight=9];
899 -> 700[label="",style="solid", color="blue", weight=3];
900[label="compare :: (Maybe a) -> (Maybe a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 900[label="",style="solid", color="blue", weight=9];
900 -> 701[label="",style="solid", color="blue", weight=3];
901[label="compare :: () -> () -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 901[label="",style="solid", color="blue", weight=9];
901 -> 702[label="",style="solid", color="blue", weight=3];
902[label="compare :: Int -> Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 902[label="",style="solid", color="blue", weight=9];
902 -> 703[label="",style="solid", color="blue", weight=3];
903[label="compare :: (Ratio a) -> (Ratio a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 903[label="",style="solid", color="blue", weight=9];
903 -> 704[label="",style="solid", color="blue", weight=3];
904[label="compare :: Float -> Float -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 904[label="",style="solid", color="blue", weight=9];
904 -> 705[label="",style="solid", color="blue", weight=3];
905[label="compare :: Double -> Double -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 905[label="",style="solid", color="blue", weight=9];
905 -> 706[label="",style="solid", color="blue", weight=3];
906[label="compare :: (Either a b) -> (Either a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];632 -> 906[label="",style="solid", color="blue", weight=9];
906 -> 707[label="",style="solid", color="blue", weight=3];
633[label="vz3501",fontsize=16,color="green",shape="box"];634[label="vz340",fontsize=16,color="green",shape="box"];635[label="vz3500",fontsize=16,color="green",shape="box"];636[label="vz341",fontsize=16,color="green",shape="box"];825[label="EQ",fontsize=16,color="green",shape="box"];826[label="compare1 LT EQ (LT <= EQ)",fontsize=16,color="black",shape="box"];826 -> 834[label="",style="solid", color="black", weight=3];
827[label="compare1 LT GT (LT <= GT)",fontsize=16,color="black",shape="box"];827 -> 835[label="",style="solid", color="black", weight=3];
828[label="compare1 EQ LT (EQ <= LT)",fontsize=16,color="black",shape="box"];828 -> 836[label="",style="solid", color="black", weight=3];
829[label="EQ",fontsize=16,color="green",shape="box"];830[label="compare1 EQ GT (EQ <= GT)",fontsize=16,color="black",shape="box"];830 -> 837[label="",style="solid", color="black", weight=3];
831[label="compare1 GT LT (GT <= LT)",fontsize=16,color="black",shape="box"];831 -> 838[label="",style="solid", color="black", weight=3];
832[label="compare1 GT EQ (GT <= EQ)",fontsize=16,color="black",shape="box"];832 -> 839[label="",style="solid", color="black", weight=3];
833[label="EQ",fontsize=16,color="green",shape="box"];682[label="compare vz3500 vz35100",fontsize=16,color="blue",shape="box"];907[label="compare :: ([] a) -> ([] a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 907[label="",style="solid", color="blue", weight=9];
907 -> 708[label="",style="solid", color="blue", weight=3];
908[label="compare :: ((@2) a b) -> ((@2) a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 908[label="",style="solid", color="blue", weight=9];
908 -> 709[label="",style="solid", color="blue", weight=3];
909[label="compare :: Char -> Char -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 909[label="",style="solid", color="blue", weight=9];
909 -> 710[label="",style="solid", color="blue", weight=3];
910[label="compare :: ((@3) a b c) -> ((@3) a b c) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 910[label="",style="solid", color="blue", weight=9];
910 -> 711[label="",style="solid", color="blue", weight=3];
911[label="compare :: Ordering -> Ordering -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 911[label="",style="solid", color="blue", weight=9];
911 -> 712[label="",style="solid", color="blue", weight=3];
912[label="compare :: Bool -> Bool -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 912[label="",style="solid", color="blue", weight=9];
912 -> 713[label="",style="solid", color="blue", weight=3];
913[label="compare :: Integer -> Integer -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 913[label="",style="solid", color="blue", weight=9];
913 -> 714[label="",style="solid", color="blue", weight=3];
914[label="compare :: (Maybe a) -> (Maybe a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 914[label="",style="solid", color="blue", weight=9];
914 -> 715[label="",style="solid", color="blue", weight=3];
915[label="compare :: () -> () -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 915[label="",style="solid", color="blue", weight=9];
915 -> 716[label="",style="solid", color="blue", weight=3];
916[label="compare :: Int -> Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 916[label="",style="solid", color="blue", weight=9];
916 -> 717[label="",style="solid", color="blue", weight=3];
917[label="compare :: (Ratio a) -> (Ratio a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 917[label="",style="solid", color="blue", weight=9];
917 -> 718[label="",style="solid", color="blue", weight=3];
918[label="compare :: Float -> Float -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 918[label="",style="solid", color="blue", weight=9];
918 -> 719[label="",style="solid", color="blue", weight=3];
919[label="compare :: Double -> Double -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 919[label="",style="solid", color="blue", weight=9];
919 -> 720[label="",style="solid", color="blue", weight=3];
920[label="compare :: (Either a b) -> (Either a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];682 -> 920[label="",style="solid", color="blue", weight=9];
920 -> 721[label="",style="solid", color="blue", weight=3];
683[label="vz35101",fontsize=16,color="green",shape="box"];684[label="vz3500",fontsize=16,color="green",shape="box"];685[label="vz35100",fontsize=16,color="green",shape="box"];686[label="vz3501",fontsize=16,color="green",shape="box"];691[label="vz35110",fontsize=16,color="green",shape="box"];692[label="vz351110",fontsize=16,color="green",shape="box"];693[label="vz351111",fontsize=16,color="green",shape="box"];694[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];694 -> 726[label="",style="solid", color="black", weight=3];
695[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];695 -> 727[label="",style="solid", color="black", weight=3];
696[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];696 -> 728[label="",style="solid", color="black", weight=3];
697[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];697 -> 729[label="",style="solid", color="black", weight=3];
698 -> 572[label="",style="dashed", color="red", weight=0];
698[label="compare vz340 vz3500",fontsize=16,color="magenta"];698 -> 730[label="",style="dashed", color="magenta", weight=3];
698 -> 731[label="",style="dashed", color="magenta", weight=3];
699[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];699 -> 732[label="",style="solid", color="black", weight=3];
700[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];700 -> 733[label="",style="solid", color="black", weight=3];
701[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];701 -> 734[label="",style="solid", color="black", weight=3];
702[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];702 -> 735[label="",style="solid", color="black", weight=3];
703[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];703 -> 736[label="",style="solid", color="black", weight=3];
704[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];704 -> 737[label="",style="solid", color="black", weight=3];
705[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];705 -> 738[label="",style="solid", color="black", weight=3];
706[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];706 -> 739[label="",style="solid", color="black", weight=3];
707[label="compare vz340 vz3500",fontsize=16,color="black",shape="triangle"];707 -> 740[label="",style="solid", color="black", weight=3];
834[label="compare1 LT EQ True",fontsize=16,color="black",shape="box"];834 -> 840[label="",style="solid", color="black", weight=3];
835[label="compare1 LT GT True",fontsize=16,color="black",shape="box"];835 -> 841[label="",style="solid", color="black", weight=3];
836[label="compare1 EQ LT False",fontsize=16,color="black",shape="box"];836 -> 842[label="",style="solid", color="black", weight=3];
837[label="compare1 EQ GT True",fontsize=16,color="black",shape="box"];837 -> 843[label="",style="solid", color="black", weight=3];
838[label="compare1 GT LT False",fontsize=16,color="black",shape="box"];838 -> 844[label="",style="solid", color="black", weight=3];
839[label="compare1 GT EQ False",fontsize=16,color="black",shape="box"];839 -> 845[label="",style="solid", color="black", weight=3];
708 -> 694[label="",style="dashed", color="red", weight=0];
708[label="compare vz3500 vz35100",fontsize=16,color="magenta"];708 -> 741[label="",style="dashed", color="magenta", weight=3];
708 -> 742[label="",style="dashed", color="magenta", weight=3];
709 -> 695[label="",style="dashed", color="red", weight=0];
709[label="compare vz3500 vz35100",fontsize=16,color="magenta"];709 -> 743[label="",style="dashed", color="magenta", weight=3];
709 -> 744[label="",style="dashed", color="magenta", weight=3];
710 -> 696[label="",style="dashed", color="red", weight=0];
710[label="compare vz3500 vz35100",fontsize=16,color="magenta"];710 -> 745[label="",style="dashed", color="magenta", weight=3];
710 -> 746[label="",style="dashed", color="magenta", weight=3];
711 -> 697[label="",style="dashed", color="red", weight=0];
711[label="compare vz3500 vz35100",fontsize=16,color="magenta"];711 -> 747[label="",style="dashed", color="magenta", weight=3];
711 -> 748[label="",style="dashed", color="magenta", weight=3];
712 -> 572[label="",style="dashed", color="red", weight=0];
712[label="compare vz3500 vz35100",fontsize=16,color="magenta"];712 -> 749[label="",style="dashed", color="magenta", weight=3];
712 -> 750[label="",style="dashed", color="magenta", weight=3];
713 -> 699[label="",style="dashed", color="red", weight=0];
713[label="compare vz3500 vz35100",fontsize=16,color="magenta"];713 -> 751[label="",style="dashed", color="magenta", weight=3];
713 -> 752[label="",style="dashed", color="magenta", weight=3];
714 -> 700[label="",style="dashed", color="red", weight=0];
714[label="compare vz3500 vz35100",fontsize=16,color="magenta"];714 -> 753[label="",style="dashed", color="magenta", weight=3];
714 -> 754[label="",style="dashed", color="magenta", weight=3];
715 -> 701[label="",style="dashed", color="red", weight=0];
715[label="compare vz3500 vz35100",fontsize=16,color="magenta"];715 -> 755[label="",style="dashed", color="magenta", weight=3];
715 -> 756[label="",style="dashed", color="magenta", weight=3];
716 -> 702[label="",style="dashed", color="red", weight=0];
716[label="compare vz3500 vz35100",fontsize=16,color="magenta"];716 -> 757[label="",style="dashed", color="magenta", weight=3];
716 -> 758[label="",style="dashed", color="magenta", weight=3];
717 -> 703[label="",style="dashed", color="red", weight=0];
717[label="compare vz3500 vz35100",fontsize=16,color="magenta"];717 -> 759[label="",style="dashed", color="magenta", weight=3];
717 -> 760[label="",style="dashed", color="magenta", weight=3];
718 -> 704[label="",style="dashed", color="red", weight=0];
718[label="compare vz3500 vz35100",fontsize=16,color="magenta"];718 -> 761[label="",style="dashed", color="magenta", weight=3];
718 -> 762[label="",style="dashed", color="magenta", weight=3];
719 -> 705[label="",style="dashed", color="red", weight=0];
719[label="compare vz3500 vz35100",fontsize=16,color="magenta"];719 -> 763[label="",style="dashed", color="magenta", weight=3];
719 -> 764[label="",style="dashed", color="magenta", weight=3];
720 -> 706[label="",style="dashed", color="red", weight=0];
720[label="compare vz3500 vz35100",fontsize=16,color="magenta"];720 -> 765[label="",style="dashed", color="magenta", weight=3];
720 -> 766[label="",style="dashed", color="magenta", weight=3];
721 -> 707[label="",style="dashed", color="red", weight=0];
721[label="compare vz3500 vz35100",fontsize=16,color="magenta"];721 -> 767[label="",style="dashed", color="magenta", weight=3];
721 -> 768[label="",style="dashed", color="magenta", weight=3];
726[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];726 -> 775[label="",style="solid", color="black", weight=3];
727[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];727 -> 776[label="",style="solid", color="black", weight=3];
728[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];728 -> 777[label="",style="solid", color="black", weight=3];
729[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];729 -> 778[label="",style="solid", color="black", weight=3];
730[label="vz3500",fontsize=16,color="green",shape="box"];731[label="vz340",fontsize=16,color="green",shape="box"];732[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];732 -> 779[label="",style="solid", color="black", weight=3];
733[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];733 -> 780[label="",style="solid", color="black", weight=3];
734[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];734 -> 781[label="",style="solid", color="black", weight=3];
735[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];735 -> 782[label="",style="solid", color="black", weight=3];
736[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];736 -> 783[label="",style="solid", color="black", weight=3];
737[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];737 -> 784[label="",style="solid", color="black", weight=3];
738[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];738 -> 785[label="",style="solid", color="black", weight=3];
739[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];739 -> 786[label="",style="solid", color="black", weight=3];
740[label="compare3 vz340 vz3500",fontsize=16,color="black",shape="box"];740 -> 787[label="",style="solid", color="black", weight=3];
840[label="LT",fontsize=16,color="green",shape="box"];841[label="LT",fontsize=16,color="green",shape="box"];842[label="compare0 EQ LT otherwise",fontsize=16,color="black",shape="box"];842 -> 846[label="",style="solid", color="black", weight=3];
843[label="LT",fontsize=16,color="green",shape="box"];844[label="compare0 GT LT otherwise",fontsize=16,color="black",shape="box"];844 -> 847[label="",style="solid", color="black", weight=3];
845[label="compare0 GT EQ otherwise",fontsize=16,color="black",shape="box"];845 -> 848[label="",style="solid", color="black", weight=3];
741[label="vz3500",fontsize=16,color="green",shape="box"];742[label="vz35100",fontsize=16,color="green",shape="box"];743[label="vz3500",fontsize=16,color="green",shape="box"];744[label="vz35100",fontsize=16,color="green",shape="box"];745[label="vz3500",fontsize=16,color="green",shape="box"];746[label="vz35100",fontsize=16,color="green",shape="box"];747[label="vz3500",fontsize=16,color="green",shape="box"];748[label="vz35100",fontsize=16,color="green",shape="box"];749[label="vz35100",fontsize=16,color="green",shape="box"];750[label="vz3500",fontsize=16,color="green",shape="box"];751[label="vz3500",fontsize=16,color="green",shape="box"];752[label="vz35100",fontsize=16,color="green",shape="box"];753[label="vz3500",fontsize=16,color="green",shape="box"];754[label="vz35100",fontsize=16,color="green",shape="box"];755[label="vz3500",fontsize=16,color="green",shape="box"];756[label="vz35100",fontsize=16,color="green",shape="box"];757[label="vz3500",fontsize=16,color="green",shape="box"];758[label="vz35100",fontsize=16,color="green",shape="box"];759[label="vz3500",fontsize=16,color="green",shape="box"];760[label="vz35100",fontsize=16,color="green",shape="box"];761[label="vz3500",fontsize=16,color="green",shape="box"];762[label="vz35100",fontsize=16,color="green",shape="box"];763[label="vz3500",fontsize=16,color="green",shape="box"];764[label="vz35100",fontsize=16,color="green",shape="box"];765[label="vz3500",fontsize=16,color="green",shape="box"];766[label="vz35100",fontsize=16,color="green",shape="box"];767[label="vz3500",fontsize=16,color="green",shape="box"];768[label="vz35100",fontsize=16,color="green",shape="box"];775[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];775 -> 803[label="",style="solid", color="black", weight=3];
776[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];776 -> 804[label="",style="solid", color="black", weight=3];
777[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];777 -> 805[label="",style="solid", color="black", weight=3];
778[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];778 -> 806[label="",style="solid", color="black", weight=3];
779[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];779 -> 807[label="",style="solid", color="black", weight=3];
780[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];780 -> 808[label="",style="solid", color="black", weight=3];
781[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];781 -> 809[label="",style="solid", color="black", weight=3];
782[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];782 -> 810[label="",style="solid", color="black", weight=3];
783[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];783 -> 811[label="",style="solid", color="black", weight=3];
784[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];784 -> 812[label="",style="solid", color="black", weight=3];
785[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];785 -> 813[label="",style="solid", color="black", weight=3];
786[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];786 -> 814[label="",style="solid", color="black", weight=3];
787[label="compare2 vz340 vz3500 (vz340 == vz3500)",fontsize=16,color="black",shape="box"];787 -> 815[label="",style="solid", color="black", weight=3];
846[label="compare0 EQ LT True",fontsize=16,color="black",shape="box"];846 -> 849[label="",style="solid", color="black", weight=3];
847[label="compare0 GT LT True",fontsize=16,color="black",shape="box"];847 -> 850[label="",style="solid", color="black", weight=3];
848[label="compare0 GT EQ True",fontsize=16,color="black",shape="box"];848 -> 851[label="",style="solid", color="black", weight=3];
803[label="error []",fontsize=16,color="red",shape="box"];804[label="error []",fontsize=16,color="red",shape="box"];805[label="error []",fontsize=16,color="red",shape="box"];806[label="error []",fontsize=16,color="red",shape="box"];807[label="error []",fontsize=16,color="red",shape="box"];808[label="error []",fontsize=16,color="red",shape="box"];809[label="error []",fontsize=16,color="red",shape="box"];810[label="error []",fontsize=16,color="red",shape="box"];811[label="error []",fontsize=16,color="red",shape="box"];812[label="error []",fontsize=16,color="red",shape="box"];813[label="error []",fontsize=16,color="red",shape="box"];814[label="error []",fontsize=16,color="red",shape="box"];815[label="error []",fontsize=16,color="red",shape="box"];849[label="GT",fontsize=16,color="green",shape="box"];850[label="GT",fontsize=16,color="green",shape="box"];851[label="GT",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_merge_pairs(:(vz35110, :(vz351110, vz351111)), ba) -> new_merge_pairs(vz351111, ba)

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

(10) 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_merge_pairs(:(vz35110, :(vz351110, vz351111)), ba) -> new_merge_pairs(vz351111, ba)
The graph contains the following edges 1 > 1, 2 >= 2


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

(11)
YES

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

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

   new_mergesort'(vz34, :(vz350, []), ba) -> new_mergesort'(new_merge2(vz34, vz350, ba), [], ba)
   new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba)

The TRS R consists of the following rules:

   new_compare9(vz340, vz3500, ca) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare6(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Double) -> new_compare12(vz3500, vz35100)
   new_compare4(vz340, vz3500) -> error([])
   new_compare11(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, ty_@0) -> new_compare10(vz3500, vz35100)
   new_compare13(vz340, vz3500, cb, cc) -> error([])
   new_compare6(GT, GT) -> EQ
   new_compare(vz340, vz3500, app(ty_[], bh)) -> new_compare5(vz340, vz3500, bh)
   new_compare(vz340, vz3500, app(app(ty_@2, bb), bc)) -> new_compare0(vz340, vz3500, bb, bc)
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_compare14(vz3500, vz35100, app(ty_Ratio, bg)) -> new_compare3(vz3500, vz35100, bg)
   new_compare(vz340, vz3500, app(ty_Maybe, ca)) -> new_compare9(vz340, vz3500, ca)
   new_merge1(vz34, :(vz3500, vz3501), :(vz35100, vz35101), ba) -> new_merge2(vz34, new_merge00(vz35100, vz3500, vz3501, vz35101, new_compare14(vz3500, vz35100, ba), ba), ba)
   new_compare14(vz3500, vz35100, ty_Bool) -> new_compare7(vz3500, vz35100)
   new_compare8(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bh) -> error([])
   new_compare6(LT, EQ) -> LT
   new_compare12(vz340, vz3500) -> error([])
   new_merge2([], :(vz3500, vz3501), ba) -> :(vz3500, vz3501)
   new_compare0(vz340, vz3500, bb, bc) -> error([])
   new_compare14(vz3500, vz35100, ty_Float) -> new_compare11(vz3500, vz35100)
   new_compare14(vz3500, vz35100, ty_Ordering) -> new_compare6(vz3500, vz35100)
   new_compare6(LT, LT) -> EQ
   new_compare(vz340, vz3500, ty_Integer) -> new_compare8(vz340, vz3500)
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))
   new_compare(vz340, vz3500, ty_Int) -> new_compare2(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Char) -> new_compare4(vz340, vz3500)
   new_merge00(vz44, vz45, vz46, vz47, LT, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare7(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, app(app(ty_@2, bb), bc)) -> new_compare0(vz3500, vz35100, bb, bc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare7(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Int) -> new_compare2(vz3500, vz35100)
   new_merge2(:(vz340, vz341), :(vz3500, vz3501), ba) -> new_merge00(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, ba), ba)
   new_compare14(vz3500, vz35100, ty_Integer) -> new_compare8(vz3500, vz35100)
   new_compare6(GT, LT) -> GT
   new_compare6(GT, EQ) -> GT
   new_merge2(vz34, [], ba) -> vz34
   new_compare6(EQ, LT) -> GT
   new_compare(vz340, vz3500, ty_@0) -> new_compare10(vz340, vz3500)
   new_merge_pairs0([], ba) -> []
   new_merge1(vz34, [], :(vz35100, vz35101), ba) -> new_merge2(vz34, :(vz35100, vz35101), ba)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bd, be, bf) -> error([])
   new_compare3(vz340, vz3500, bg) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, GT, cd) -> :(vz44, new_merge2(:(vz45, vz46), vz47, cd))
   new_compare(vz340, vz3500, app(app(ty_Either, cb), cc)) -> new_compare13(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bd), be), bf)) -> new_compare1(vz340, vz3500, bd, be, bf)
   new_compare14(vz3500, vz35100, ty_Char) -> new_compare4(vz3500, vz35100)
   new_merge00(vz44, vz45, vz46, vz47, EQ, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare10(vz340, vz3500) -> error([])
   new_compare6(LT, GT) -> LT
   new_compare(vz340, vz3500, ty_Float) -> new_compare11(vz340, vz3500)
   new_compare14(vz3500, vz35100, app(ty_[], bh)) -> new_compare5(vz3500, vz35100, bh)
   new_compare(vz340, vz3500, ty_Double) -> new_compare12(vz340, vz3500)
   new_compare14(vz3500, vz35100, app(app(app(ty_@3, bd), be), bf)) -> new_compare1(vz3500, vz35100, bd, be, bf)
   new_compare14(vz3500, vz35100, app(ty_Maybe, ca)) -> new_compare9(vz3500, vz35100, ca)
   new_compare14(vz3500, vz35100, app(app(ty_Either, cb), cc)) -> new_compare13(vz3500, vz35100, cb, cc)
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare3(vz340, vz3500, bg)
   new_merge1(vz34, vz350, [], ba) -> new_merge2(vz34, vz350, ba)

The set Q consists of the following terms:

   new_compare0(x0, x1, x2, x3)
   new_compare14(x0, x1, ty_Double)
   new_merge1(x0, :(x1, x2), :(x3, x4), x5)
   new_compare13(x0, x1, x2, x3)
   new_compare10(x0, x1)
   new_compare14(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare14(x0, x1, ty_Bool)
   new_merge2([], :(x0, x1), x2)
   new_compare11(x0, x1)
   new_compare14(x0, x1, ty_@0)
   new_merge_pairs0(:(x0, :(x1, x2)), x3)
   new_compare(x0, x1, ty_Bool)
   new_compare14(x0, x1, ty_Char)
   new_merge00(x0, x1, x2, x3, EQ, x4)
   new_compare9(x0, x1, x2)
   new_compare14(x0, x1, app(ty_Ratio, x2))
   new_merge1(x0, [], :(x1, x2), x3)
   new_compare14(x0, x1, ty_Ordering)
   new_compare7(x0, x1)
   new_merge2(x0, [], x1)
   new_compare(x0, x1, ty_Int)
   new_merge00(x0, x1, x2, x3, GT, x4)
   new_merge1(x0, x1, [], x2)
   new_compare3(x0, x1, x2)
   new_compare14(x0, x1, app(app(ty_@2, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare14(x0, x1, ty_Int)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)
   new_compare6(EQ, GT)
   new_compare6(GT, EQ)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare12(x0, x1)
   new_compare14(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(LT, LT)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare6(LT, EQ)
   new_compare6(EQ, LT)
   new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare8(x0, x1)
   new_compare6(GT, GT)
   new_compare(x0, x1, ty_Char)
   new_merge_pairs0(:(x0, []), x1)
   new_compare14(x0, x1, ty_Float)
   new_merge_pairs0([], x0)
   new_merge2(:(x0, x1), :(x2, x3), x4)
   new_compare(x0, x1, ty_Double)
   new_compare14(x0, x1, ty_Integer)
   new_merge00(x0, x1, x2, x3, LT, x4)
   new_compare14(x0, x1, app(ty_[], x2))

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

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

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

   new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba)

The TRS R consists of the following rules:

   new_compare9(vz340, vz3500, ca) -> error([])
   new_compare(vz340, vz3500, ty_Ordering) -> new_compare6(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Double) -> new_compare12(vz3500, vz35100)
   new_compare4(vz340, vz3500) -> error([])
   new_compare11(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, ty_@0) -> new_compare10(vz3500, vz35100)
   new_compare13(vz340, vz3500, cb, cc) -> error([])
   new_compare6(GT, GT) -> EQ
   new_compare(vz340, vz3500, app(ty_[], bh)) -> new_compare5(vz340, vz3500, bh)
   new_compare(vz340, vz3500, app(app(ty_@2, bb), bc)) -> new_compare0(vz340, vz3500, bb, bc)
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_compare14(vz3500, vz35100, app(ty_Ratio, bg)) -> new_compare3(vz3500, vz35100, bg)
   new_compare(vz340, vz3500, app(ty_Maybe, ca)) -> new_compare9(vz340, vz3500, ca)
   new_merge1(vz34, :(vz3500, vz3501), :(vz35100, vz35101), ba) -> new_merge2(vz34, new_merge00(vz35100, vz3500, vz3501, vz35101, new_compare14(vz3500, vz35100, ba), ba), ba)
   new_compare14(vz3500, vz35100, ty_Bool) -> new_compare7(vz3500, vz35100)
   new_compare8(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, bh) -> error([])
   new_compare6(LT, EQ) -> LT
   new_compare12(vz340, vz3500) -> error([])
   new_merge2([], :(vz3500, vz3501), ba) -> :(vz3500, vz3501)
   new_compare0(vz340, vz3500, bb, bc) -> error([])
   new_compare14(vz3500, vz35100, ty_Float) -> new_compare11(vz3500, vz35100)
   new_compare14(vz3500, vz35100, ty_Ordering) -> new_compare6(vz3500, vz35100)
   new_compare6(LT, LT) -> EQ
   new_compare(vz340, vz3500, ty_Integer) -> new_compare8(vz340, vz3500)
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))
   new_compare(vz340, vz3500, ty_Int) -> new_compare2(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Char) -> new_compare4(vz340, vz3500)
   new_merge00(vz44, vz45, vz46, vz47, LT, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare7(vz340, vz3500) -> error([])
   new_compare14(vz3500, vz35100, app(app(ty_@2, bb), bc)) -> new_compare0(vz3500, vz35100, bb, bc)
   new_compare(vz340, vz3500, ty_Bool) -> new_compare7(vz340, vz3500)
   new_compare14(vz3500, vz35100, ty_Int) -> new_compare2(vz3500, vz35100)
   new_merge2(:(vz340, vz341), :(vz3500, vz3501), ba) -> new_merge00(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, ba), ba)
   new_compare14(vz3500, vz35100, ty_Integer) -> new_compare8(vz3500, vz35100)
   new_compare6(GT, LT) -> GT
   new_compare6(GT, EQ) -> GT
   new_merge2(vz34, [], ba) -> vz34
   new_compare6(EQ, LT) -> GT
   new_compare(vz340, vz3500, ty_@0) -> new_compare10(vz340, vz3500)
   new_merge_pairs0([], ba) -> []
   new_merge1(vz34, [], :(vz35100, vz35101), ba) -> new_merge2(vz34, :(vz35100, vz35101), ba)
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, bd, be, bf) -> error([])
   new_compare3(vz340, vz3500, bg) -> error([])
   new_merge00(vz44, vz45, vz46, vz47, GT, cd) -> :(vz44, new_merge2(:(vz45, vz46), vz47, cd))
   new_compare(vz340, vz3500, app(app(ty_Either, cb), cc)) -> new_compare13(vz340, vz3500, cb, cc)
   new_compare(vz340, vz3500, app(app(app(ty_@3, bd), be), bf)) -> new_compare1(vz340, vz3500, bd, be, bf)
   new_compare14(vz3500, vz35100, ty_Char) -> new_compare4(vz3500, vz35100)
   new_merge00(vz44, vz45, vz46, vz47, EQ, cd) -> :(vz45, new_merge2(vz46, :(vz44, vz47), cd))
   new_compare10(vz340, vz3500) -> error([])
   new_compare6(LT, GT) -> LT
   new_compare(vz340, vz3500, ty_Float) -> new_compare11(vz340, vz3500)
   new_compare14(vz3500, vz35100, app(ty_[], bh)) -> new_compare5(vz3500, vz35100, bh)
   new_compare(vz340, vz3500, ty_Double) -> new_compare12(vz340, vz3500)
   new_compare14(vz3500, vz35100, app(app(app(ty_@3, bd), be), bf)) -> new_compare1(vz3500, vz35100, bd, be, bf)
   new_compare14(vz3500, vz35100, app(ty_Maybe, ca)) -> new_compare9(vz3500, vz35100, ca)
   new_compare14(vz3500, vz35100, app(app(ty_Either, cb), cc)) -> new_compare13(vz3500, vz35100, cb, cc)
   new_compare(vz340, vz3500, app(ty_Ratio, bg)) -> new_compare3(vz340, vz3500, bg)
   new_merge1(vz34, vz350, [], ba) -> new_merge2(vz34, vz350, ba)

The set Q consists of the following terms:

   new_compare0(x0, x1, x2, x3)
   new_compare14(x0, x1, ty_Double)
   new_merge1(x0, :(x1, x2), :(x3, x4), x5)
   new_compare13(x0, x1, x2, x3)
   new_compare10(x0, x1)
   new_compare14(x0, x1, app(ty_Maybe, x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare14(x0, x1, ty_Bool)
   new_merge2([], :(x0, x1), x2)
   new_compare11(x0, x1)
   new_compare14(x0, x1, ty_@0)
   new_merge_pairs0(:(x0, :(x1, x2)), x3)
   new_compare(x0, x1, ty_Bool)
   new_compare14(x0, x1, ty_Char)
   new_merge00(x0, x1, x2, x3, EQ, x4)
   new_compare9(x0, x1, x2)
   new_compare14(x0, x1, app(ty_Ratio, x2))
   new_merge1(x0, [], :(x1, x2), x3)
   new_compare14(x0, x1, ty_Ordering)
   new_compare7(x0, x1)
   new_merge2(x0, [], x1)
   new_compare(x0, x1, ty_Int)
   new_merge00(x0, x1, x2, x3, GT, x4)
   new_merge1(x0, x1, [], x2)
   new_compare3(x0, x1, x2)
   new_compare14(x0, x1, app(app(ty_@2, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare14(x0, x1, ty_Int)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)
   new_compare6(EQ, GT)
   new_compare6(GT, EQ)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare12(x0, x1)
   new_compare14(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(LT, LT)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare6(LT, EQ)
   new_compare6(EQ, LT)
   new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare8(x0, x1)
   new_compare6(GT, GT)
   new_compare(x0, x1, ty_Char)
   new_merge_pairs0(:(x0, []), x1)
   new_compare14(x0, x1, ty_Float)
   new_merge_pairs0([], x0)
   new_merge2(:(x0, x1), :(x2, x3), x4)
   new_compare(x0, x1, ty_Double)
   new_compare14(x0, x1, ty_Integer)
   new_merge00(x0, x1, x2, x3, LT, x4)
   new_compare14(x0, x1, app(ty_[], x2))

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

(15) QDPSizeChangeProof (EQUIVALENT)
We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 

Order:Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_2
   POL([]) = 0
   POL(new_merge2(x_1, x_2, x_3)) = 0
   POL(new_merge_pairs0(x_1, x_2)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_mergesort'(vz34, :(vz350, :(vz3510, vz3511)), ba) -> new_mergesort'(new_merge1(vz34, vz350, vz3510, ba), new_merge_pairs0(vz3511, ba), ba) (allowed arguments on rhs = {2, 3})
The graph contains the following edges 2 > 2, 3 >= 3



We oriented the following set of usable rules [AAECC05,FROCOS05].

   new_merge_pairs0([], ba) -> []
   new_merge_pairs0(:(vz35110, []), ba) -> :(vz35110, [])
   new_merge_pairs0(:(vz35110, :(vz351110, vz351111)), ba) -> :(new_merge2(vz35110, vz351110, ba), new_merge_pairs0(vz351111, ba))

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

(16)
YES

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

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

   new_map(:(vz3110, vz3111)) -> new_map(vz3111)

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_map(:(vz3110, vz3111)) -> new_map(vz3111)
The graph contains the following edges 1 > 1


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

(19)
YES

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

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

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(vz340, vz341), :(vz3500, vz3501), bb) -> new_merge0(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, bb), bb)

The TRS R consists of the following rules:

   new_compare(vz340, vz3500, ty_Ordering) -> new_compare6(vz340, vz3500)
   new_compare9(vz340, vz3500, cb) -> error([])
   new_compare(vz340, vz3500, ty_Bool) -> new_compare7(vz340, vz3500)
   new_compare4(vz340, vz3500) -> error([])
   new_compare11(vz340, vz3500) -> error([])
   new_compare6(GT, LT) -> GT
   new_compare13(vz340, vz3500, cc, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_[], ca)) -> new_compare5(vz340, vz3500, ca)
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare(vz340, vz3500, app(app(ty_@2, bc), bd)) -> new_compare0(vz340, vz3500, bc, bd)
   new_compare(vz340, vz3500, ty_@0) -> new_compare10(vz340, vz3500)
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, be, bf, bg) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, cb)) -> new_compare9(vz340, vz3500, cb)
   new_compare3(vz340, vz3500, bh) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, cc), cd)) -> new_compare13(vz340, vz3500, cc, cd)
   new_compare8(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, ca) -> error([])
   new_compare6(LT, EQ) -> LT
   new_compare(vz340, vz3500, app(app(app(ty_@3, be), bf), bg)) -> new_compare1(vz340, vz3500, be, bf, bg)
   new_compare12(vz340, vz3500) -> error([])
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500, bc, bd) -> error([])
   new_compare(vz340, vz3500, ty_Float) -> new_compare11(vz340, vz3500)
   new_compare6(LT, GT) -> LT
   new_compare(vz340, vz3500, ty_Double) -> new_compare12(vz340, vz3500)
   new_compare6(LT, LT) -> EQ
   new_compare(vz340, vz3500, ty_Integer) -> new_compare8(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare2(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Char) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_Ratio, bh)) -> new_compare3(vz340, vz3500, bh)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare3(x0, x1, x2)
   new_compare9(x0, x1, x2)
   new_compare11(x0, x1)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare13(x0, x1, x2, x3)
   new_compare12(x0, x1)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare7(x0, x1)
   new_compare0(x0, x1, x2, x3)
   new_compare8(x0, x1)
   new_compare(x0, x1, ty_Int)
   new_compare6(GT, GT)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

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

(21) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_merge(:(vz340, vz341), :(vz3500, vz3501), bb) -> new_merge0(vz3500, vz340, vz341, vz3501, new_compare(vz340, vz3500, bb), bb) at position [4] we obtained the following new rules [LPAR04]:

   (new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering),new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering))
   (new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Bool),new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Bool))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_[], x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_[], x2)))
   (new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1, x2, x3), app(app(ty_@2, x2), x3)),new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1, x2, x3), app(app(ty_@2, x2), x3)))
   (new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_@0),new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_@0))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2), app(ty_Maybe, x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2), app(ty_Maybe, x2)))
   (new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2, x3), app(app(ty_Either, x2), x3)),new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2, x3), app(app(ty_Either, x2), x3)))
   (new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4)),new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4)))
   (new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Float),new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Float))
   (new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1), ty_Double),new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1), ty_Double))
   (new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1), ty_Integer),new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1), ty_Integer))
   (new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Int),new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Int))
   (new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_Char),new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_Char))
   (new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1, x2), app(ty_Ratio, x2)),new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1, x2), app(ty_Ratio, x2)))


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

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

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering)
   new_merge(:(x0, y1), :(x1, y3), ty_Bool) -> new_merge0(x1, x0, y1, y3, new_compare7(x0, x1), ty_Bool)
   new_merge(:(x0, y1), :(x1, y3), app(ty_[], x2)) -> new_merge0(x1, x0, y1, y3, new_compare5(x0, x1, x2), app(ty_[], x2))
   new_merge(:(x0, y1), :(x1, y3), app(app(ty_@2, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare0(x0, x1, x2, x3), app(app(ty_@2, x2), x3))
   new_merge(:(x0, y1), :(x1, y3), ty_@0) -> new_merge0(x1, x0, y1, y3, new_compare10(x0, x1), ty_@0)
   new_merge(:(x0, y1), :(x1, y3), app(ty_Maybe, x2)) -> new_merge0(x1, x0, y1, y3, new_compare9(x0, x1, x2), app(ty_Maybe, x2))
   new_merge(:(x0, y1), :(x1, y3), app(app(ty_Either, x2), x3)) -> new_merge0(x1, x0, y1, y3, new_compare13(x0, x1, x2, x3), app(app(ty_Either, x2), x3))
   new_merge(:(x0, y1), :(x1, y3), app(app(app(ty_@3, x2), x3), x4)) -> new_merge0(x1, x0, y1, y3, new_compare1(x0, x1, x2, x3, x4), app(app(app(ty_@3, x2), x3), x4))
   new_merge(:(x0, y1), :(x1, y3), ty_Float) -> new_merge0(x1, x0, y1, y3, new_compare11(x0, x1), ty_Float)
   new_merge(:(x0, y1), :(x1, y3), ty_Double) -> new_merge0(x1, x0, y1, y3, new_compare12(x0, x1), ty_Double)
   new_merge(:(x0, y1), :(x1, y3), ty_Integer) -> new_merge0(x1, x0, y1, y3, new_compare8(x0, x1), ty_Integer)
   new_merge(:(x0, y1), :(x1, y3), ty_Int) -> new_merge0(x1, x0, y1, y3, new_compare2(x0, x1), ty_Int)
   new_merge(:(x0, y1), :(x1, y3), ty_Char) -> new_merge0(x1, x0, y1, y3, new_compare4(x0, x1), ty_Char)
   new_merge(:(x0, y1), :(x1, y3), app(ty_Ratio, x2)) -> new_merge0(x1, x0, y1, y3, new_compare3(x0, x1, x2), app(ty_Ratio, x2))

The TRS R consists of the following rules:

   new_compare(vz340, vz3500, ty_Ordering) -> new_compare6(vz340, vz3500)
   new_compare9(vz340, vz3500, cb) -> error([])
   new_compare(vz340, vz3500, ty_Bool) -> new_compare7(vz340, vz3500)
   new_compare4(vz340, vz3500) -> error([])
   new_compare11(vz340, vz3500) -> error([])
   new_compare6(GT, LT) -> GT
   new_compare13(vz340, vz3500, cc, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_[], ca)) -> new_compare5(vz340, vz3500, ca)
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare(vz340, vz3500, app(app(ty_@2, bc), bd)) -> new_compare0(vz340, vz3500, bc, bd)
   new_compare(vz340, vz3500, ty_@0) -> new_compare10(vz340, vz3500)
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, be, bf, bg) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, cb)) -> new_compare9(vz340, vz3500, cb)
   new_compare3(vz340, vz3500, bh) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, cc), cd)) -> new_compare13(vz340, vz3500, cc, cd)
   new_compare8(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, ca) -> error([])
   new_compare6(LT, EQ) -> LT
   new_compare(vz340, vz3500, app(app(app(ty_@3, be), bf), bg)) -> new_compare1(vz340, vz3500, be, bf, bg)
   new_compare12(vz340, vz3500) -> error([])
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500, bc, bd) -> error([])
   new_compare(vz340, vz3500, ty_Float) -> new_compare11(vz340, vz3500)
   new_compare6(LT, GT) -> LT
   new_compare(vz340, vz3500, ty_Double) -> new_compare12(vz340, vz3500)
   new_compare6(LT, LT) -> EQ
   new_compare(vz340, vz3500, ty_Integer) -> new_compare8(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare2(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Char) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_Ratio, bh)) -> new_compare3(vz340, vz3500, bh)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare3(x0, x1, x2)
   new_compare9(x0, x1, x2)
   new_compare11(x0, x1)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare13(x0, x1, x2, x3)
   new_compare12(x0, x1)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare7(x0, x1)
   new_compare0(x0, x1, x2, x3)
   new_compare8(x0, x1)
   new_compare(x0, x1, ty_Int)
   new_compare6(GT, GT)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

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

(23) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 13 less nodes.
----------------------------------------

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

   new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare(vz340, vz3500, ty_Ordering) -> new_compare6(vz340, vz3500)
   new_compare9(vz340, vz3500, cb) -> error([])
   new_compare(vz340, vz3500, ty_Bool) -> new_compare7(vz340, vz3500)
   new_compare4(vz340, vz3500) -> error([])
   new_compare11(vz340, vz3500) -> error([])
   new_compare6(GT, LT) -> GT
   new_compare13(vz340, vz3500, cc, cd) -> error([])
   new_compare(vz340, vz3500, app(ty_[], ca)) -> new_compare5(vz340, vz3500, ca)
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare(vz340, vz3500, app(app(ty_@2, bc), bd)) -> new_compare0(vz340, vz3500, bc, bd)
   new_compare(vz340, vz3500, ty_@0) -> new_compare10(vz340, vz3500)
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare2(vz340, vz3500) -> error([])
   new_compare1(vz340, vz3500, be, bf, bg) -> error([])
   new_compare(vz340, vz3500, app(ty_Maybe, cb)) -> new_compare9(vz340, vz3500, cb)
   new_compare3(vz340, vz3500, bh) -> error([])
   new_compare(vz340, vz3500, app(app(ty_Either, cc), cd)) -> new_compare13(vz340, vz3500, cc, cd)
   new_compare8(vz340, vz3500) -> error([])
   new_compare5(vz340, vz3500, ca) -> error([])
   new_compare6(LT, EQ) -> LT
   new_compare(vz340, vz3500, app(app(app(ty_@3, be), bf), bg)) -> new_compare1(vz340, vz3500, be, bf, bg)
   new_compare12(vz340, vz3500) -> error([])
   new_compare10(vz340, vz3500) -> error([])
   new_compare0(vz340, vz3500, bc, bd) -> error([])
   new_compare(vz340, vz3500, ty_Float) -> new_compare11(vz340, vz3500)
   new_compare6(LT, GT) -> LT
   new_compare(vz340, vz3500, ty_Double) -> new_compare12(vz340, vz3500)
   new_compare6(LT, LT) -> EQ
   new_compare(vz340, vz3500, ty_Integer) -> new_compare8(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Int) -> new_compare2(vz340, vz3500)
   new_compare(vz340, vz3500, ty_Char) -> new_compare4(vz340, vz3500)
   new_compare(vz340, vz3500, app(ty_Ratio, bh)) -> new_compare3(vz340, vz3500, bh)
   new_compare7(vz340, vz3500) -> error([])

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare3(x0, x1, x2)
   new_compare9(x0, x1, x2)
   new_compare11(x0, x1)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare13(x0, x1, x2, x3)
   new_compare12(x0, x1)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare7(x0, x1)
   new_compare0(x0, x1, x2, x3)
   new_compare8(x0, x1)
   new_compare(x0, x1, ty_Int)
   new_compare6(GT, GT)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

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

(25) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare6(GT, LT) -> GT
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare6(LT, EQ) -> LT
   new_compare6(LT, GT) -> LT
   new_compare6(LT, LT) -> EQ

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare10(x0, x1)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare3(x0, x1, x2)
   new_compare9(x0, x1, x2)
   new_compare11(x0, x1)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, ty_Integer)
   new_compare6(EQ, EQ)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare13(x0, x1, x2, x3)
   new_compare12(x0, x1)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare7(x0, x1)
   new_compare0(x0, x1, x2, x3)
   new_compare8(x0, x1)
   new_compare(x0, x1, ty_Int)
   new_compare6(GT, GT)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare6(GT, LT)
   new_compare6(LT, GT)
   new_compare4(x0, x1)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)

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

(27) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_compare10(x0, x1)
   new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_compare3(x0, x1, x2)
   new_compare9(x0, x1, x2)
   new_compare11(x0, x1)
   new_compare(x0, x1, app(ty_Ratio, x2))
   new_compare(x0, x1, ty_Integer)
   new_compare(x0, x1, ty_Bool)
   new_compare2(x0, x1)
   new_compare(x0, x1, ty_Float)
   new_compare(x0, x1, app(ty_Maybe, x2))
   new_compare13(x0, x1, x2, x3)
   new_compare12(x0, x1)
   new_compare7(x0, x1)
   new_compare0(x0, x1, x2, x3)
   new_compare8(x0, x1)
   new_compare(x0, x1, ty_Int)
   new_compare5(x0, x1, x2)
   new_compare(x0, x1, ty_Char)
   new_compare(x0, x1, app(ty_[], x2))
   new_compare(x0, x1, app(app(ty_@2, x2), x3))
   new_compare(x0, x1, app(app(ty_Either, x2), x3))
   new_compare4(x0, x1)
   new_compare1(x0, x1, x2, x3, x4)
   new_compare(x0, x1, ty_Double)
   new_compare(x0, x1, ty_@0)
   new_compare(x0, x1, ty_Ordering)


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

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

   new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering)
   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)

The TRS R consists of the following rules:

   new_compare6(GT, LT) -> GT
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare6(LT, EQ) -> LT
   new_compare6(LT, GT) -> LT
   new_compare6(LT, LT) -> EQ

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare6(EQ, EQ)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare6(GT, GT)
   new_compare6(GT, LT)
   new_compare6(LT, GT)

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

(29) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_merge(:(x0, y1), :(x1, y3), ty_Ordering) -> new_merge0(x1, x0, y1, y3, new_compare6(x0, x1), ty_Ordering) at position [4] we obtained the following new rules [LPAR04]:

   (new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering),new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering))
   (new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering),new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering))
   (new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering),new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering))
   (new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering),new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering))
   (new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering),new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering))
   (new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering),new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering))
   (new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering),new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering))
   (new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering),new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering))
   (new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering),new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering))


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

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

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)

The TRS R consists of the following rules:

   new_compare6(GT, LT) -> GT
   new_compare6(GT, EQ) -> GT
   new_compare6(GT, GT) -> EQ
   new_compare6(EQ, LT) -> GT
   new_compare6(EQ, GT) -> LT
   new_compare6(EQ, EQ) -> EQ
   new_compare6(LT, EQ) -> LT
   new_compare6(LT, GT) -> LT
   new_compare6(LT, LT) -> EQ

The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare6(EQ, EQ)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare6(GT, GT)
   new_compare6(GT, LT)
   new_compare6(LT, GT)

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

(31) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)

R is empty.
The set Q consists of the following terms:

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare6(EQ, EQ)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare6(GT, GT)
   new_compare6(GT, LT)
   new_compare6(LT, GT)

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

(33) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_compare6(GT, EQ)
   new_compare6(EQ, GT)
   new_compare6(EQ, EQ)
   new_compare6(LT, LT)
   new_compare6(EQ, LT)
   new_compare6(LT, EQ)
   new_compare6(GT, GT)
   new_compare6(GT, LT)
   new_compare6(LT, GT)


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

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

   new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)

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

(35) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_merge0(vz44, vz45, vz46, vz47, EQ, ba) -> new_merge(vz46, :(vz44, vz47), ba) we obtained the following new rules [LPAR04]:

   (new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering),new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering))
   (new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering),new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering))
   (new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering),new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering))


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

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

   new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba)
   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)
   new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
   new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering)

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

(37) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_merge0(vz44, vz45, vz46, vz47, GT, ba) -> new_merge(:(vz45, vz46), vz47, ba) we obtained the following new rules [LPAR04]:

   (new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering),new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering))
   (new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering),new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering))
   (new_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering),new_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering))


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

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

   new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)
   new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
   new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering)
   new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
   new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
   new_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering)

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

(39) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_merge0(vz44, vz45, vz46, vz47, LT, ba) -> new_merge(vz46, :(vz44, vz47), ba) we obtained the following new rules [LPAR04]:

   (new_merge0(GT, EQ, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering),new_merge0(GT, EQ, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering))
   (new_merge0(EQ, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering),new_merge0(EQ, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering))
   (new_merge0(GT, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering),new_merge0(GT, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering))


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

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

   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)
   new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
   new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering)
   new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
   new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
   new_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering)
   new_merge0(GT, EQ, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge0(EQ, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
   new_merge0(GT, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)

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

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs.
----------------------------------------

(42)
Complex Obligation (AND)

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

(43)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
   new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
   new_merge0(GT, EQ, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
   new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
   new_merge0(GT, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)

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

(44) 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_merge(:(GT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, GT, y1, y3, EQ, ty_Ordering)
The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 2, 1 > 3, 2 > 4, 3 >= 6


*new_merge0(GT, GT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge0(GT, EQ, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge0(GT, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(GT, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge(:(EQ, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, EQ, y1, y3, LT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 3 >= 6


*new_merge(:(LT, y1), :(GT, y3), ty_Ordering) -> new_merge0(GT, LT, y1, y3, LT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 1 > 5, 3 >= 6


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

(45)
YES

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

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

   new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
   new_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
   new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
   new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)

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

(47) 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_merge(:(GT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, GT, y1, y3, GT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 1 > 5, 3 >= 6


*new_merge(:(GT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, GT, y1, y3, GT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 1 > 5, 3 >= 6


*new_merge0(LT, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
The graph contains the following edges 4 >= 2, 6 >= 3


*new_merge0(EQ, GT, z0, z1, GT, ty_Ordering) -> new_merge(:(GT, z0), z1, ty_Ordering)
The graph contains the following edges 4 >= 2, 6 >= 3


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

(48)
YES

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

(49)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
   new_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
   new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
   new_merge0(EQ, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)

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

(50) 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_merge0(EQ, EQ, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge0(EQ, LT, z0, z1, LT, ty_Ordering) -> new_merge(z0, :(EQ, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge(:(EQ, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, EQ, y1, y3, EQ, ty_Ordering)
The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 2, 1 > 3, 2 > 4, 1 > 5, 2 > 5, 3 >= 6


*new_merge(:(LT, y1), :(EQ, y3), ty_Ordering) -> new_merge0(EQ, LT, y1, y3, LT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 1 > 5, 3 >= 6


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(51)
YES

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

   new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
   new_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(53) 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_merge0(LT, EQ, z0, z1, GT, ty_Ordering) -> new_merge(:(EQ, z0), z1, ty_Ordering)
The graph contains the following edges 4 >= 2, 6 >= 3


*new_merge(:(EQ, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, EQ, y1, y3, GT, ty_Ordering)
The graph contains the following edges 2 > 1, 1 > 2, 1 > 3, 2 > 4, 3 >= 6


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(54)
YES

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

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

   new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)
   new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering)

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

(56) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_merge0(LT, LT, z0, z1, EQ, ty_Ordering) -> new_merge(z0, :(LT, z1), ty_Ordering)
The graph contains the following edges 3 >= 1, 6 >= 3


*new_merge(:(LT, y1), :(LT, y3), ty_Ordering) -> new_merge0(LT, LT, y1, y3, EQ, ty_Ordering)
The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 2, 1 > 3, 2 > 4, 3 >= 6


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

(57)
YES