/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 23 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) AND
    (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) TransformationProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) YES
    (24) QDP
        (25) TransformationProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (32) YES
    (33) QDP
        (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (35) 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;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;

elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by _ _ [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] _ = [];
nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
 | otherwise = y : nubBy' ys (y : xs);
 };

union :: Eq a => [a]  ->  [a]  ->  [a];
union = unionBy (==);

unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;

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

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

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

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

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

elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by _ _ [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] _ = [];
nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
 | otherwise = y : nubBy' ys (y : xs);
 };

union :: Eq a => [a]  ->  [a]  ->  [a];
union = unionBy (==);

unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;

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

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

elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] wu = [];
nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
 | otherwise = y : nubBy' ys (y : xs);
 };

union :: Eq a => [a]  ->  [a]  ->  [a];
union = unionBy (==);

unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;

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

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"nubBy' [] wu = [];
nubBy' (y : ys) xs|elem_by eq y xsnubBy' ys xs|otherwisey : nubBy' ys (y : xs);
"
is transformed to
"nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
"
"nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
"
"nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
"
"nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
"
"nubBy'3 [] wu = [];
nubBy'3 wz xu = nubBy'2 wz xu;
"

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

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

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

elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubBy' l [] where { 
nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
nubBy'3 [] wu = [];
nubBy'3 wz xu = nubBy'2 wz xu;
 };

union :: Eq a => [a]  ->  [a]  ->  [a];
union = unionBy (==);

unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;

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

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"nubBy' l [] where {
nubBy' [] wu = nubBy'3 [] wu;
nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
;
nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
;
nubBy'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
;
nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
;
nubBy'3 [] wu = [];
nubBy'3 wz xu = nubBy'2 wz xu;
} 
"
are unpacked to the following functions on top level
"nubByNubBy'3 xv [] wu = [];
nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
"
"nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);
"
"nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);
"
"nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;
"
"nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;
"

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

(8)
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;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy wv ww [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

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

elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
elem_by vy vz [] = False;
elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;

nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
nubBy eq l = nubByNubBy' eq l [];

nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;

nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);

nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;

nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);

nubByNubBy'3 xv [] wu = [];
nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;

union :: Eq a => [a]  ->  [a]  ->  [a];
union = unionBy (==);

unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;

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

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.union",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.union xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.union xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.unionBy (==) xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="xw3 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) xw3",fontsize=16,color="burlywood",shape="box"];2440[label="xw3/xw30 : xw31",fontsize=10,color="white",style="solid",shape="box"];6 -> 2440[label="",style="solid", color="burlywood", weight=9];
2440 -> 7[label="",style="solid", color="burlywood", weight=3];
2441[label="xw3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 2441[label="",style="solid", color="burlywood", weight=9];
2441 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="(xw30 : xw31) ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="[] ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="xw30 : xw31 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
10[label="foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 106[label="",style="dashed", color="red", weight=0];
11[label="xw31 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="magenta"];11 -> 107[label="",style="dashed", color="magenta", weight=3];
11 -> 108[label="",style="dashed", color="magenta", weight=3];
11 -> 109[label="",style="dashed", color="magenta", weight=3];
11 -> 110[label="",style="dashed", color="magenta", weight=3];
12[label="List.nubBy (==) xw4",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
107 -> 12[label="",style="dashed", color="red", weight=0];
107[label="List.nubBy (==) xw4",fontsize=16,color="magenta"];108[label="xw31",fontsize=16,color="green",shape="box"];109[label="xw31",fontsize=16,color="green",shape="box"];110[label="xw30",fontsize=16,color="green",shape="box"];106[label="xw8 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="burlywood",shape="triangle"];2442[label="xw8/xw80 : xw81",fontsize=10,color="white",style="solid",shape="box"];106 -> 2442[label="",style="solid", color="burlywood", weight=9];
2442 -> 135[label="",style="solid", color="burlywood", weight=3];
2443[label="xw8/[]",fontsize=10,color="white",style="solid",shape="box"];106 -> 2443[label="",style="solid", color="burlywood", weight=9];
2443 -> 136[label="",style="solid", color="burlywood", weight=3];
15[label="List.nubByNubBy' (==) xw4 []",fontsize=16,color="burlywood",shape="box"];2444[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];15 -> 2444[label="",style="solid", color="burlywood", weight=9];
2444 -> 18[label="",style="solid", color="burlywood", weight=3];
2445[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 2445[label="",style="solid", color="burlywood", weight=9];
2445 -> 19[label="",style="solid", color="burlywood", weight=3];
135[label="(xw80 : xw81) ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];135 -> 138[label="",style="solid", color="black", weight=3];
136[label="[] ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];136 -> 139[label="",style="solid", color="black", weight=3];
18[label="List.nubByNubBy' (==) (xw40 : xw41) []",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
19[label="List.nubByNubBy' (==) [] []",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
138[label="xw80 : xw81 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="green",shape="box"];138 -> 142[label="",style="dashed", color="green", weight=3];
139[label="foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];139 -> 143[label="",style="solid", color="black", weight=3];
23[label="List.nubByNubBy'2 (==) (xw40 : xw41) []",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="List.nubByNubBy'3 (==) [] []",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
142 -> 106[label="",style="dashed", color="red", weight=0];
142[label="xw81 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="magenta"];142 -> 148[label="",style="dashed", color="magenta", weight=3];
143[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) xw11",fontsize=16,color="burlywood",shape="triangle"];2446[label="xw11/xw110 : xw111",fontsize=10,color="white",style="solid",shape="box"];143 -> 2446[label="",style="solid", color="burlywood", weight=9];
2446 -> 149[label="",style="solid", color="burlywood", weight=3];
2447[label="xw11/[]",fontsize=10,color="white",style="solid",shape="box"];143 -> 2447[label="",style="solid", color="burlywood", weight=9];
2447 -> 150[label="",style="solid", color="burlywood", weight=3];
28[label="List.nubByNubBy'1 (==) xw40 xw41 [] (List.elem_by (==) xw40 [])",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3];
29[label="[]",fontsize=16,color="green",shape="box"];148[label="xw81",fontsize=16,color="green",shape="box"];149[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) (xw110 : xw111)",fontsize=16,color="black",shape="box"];149 -> 155[label="",style="solid", color="black", weight=3];
150[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) []",fontsize=16,color="black",shape="box"];150 -> 156[label="",style="solid", color="black", weight=3];
33[label="List.nubByNubBy'1 (==) xw40 xw41 [] False",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
155 -> 143[label="",style="dashed", color="red", weight=0];
155[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) (flip (List.deleteBy (==)) xw9 xw10) xw110) xw111",fontsize=16,color="magenta"];155 -> 162[label="",style="dashed", color="magenta", weight=3];
155 -> 163[label="",style="dashed", color="magenta", weight=3];
155 -> 164[label="",style="dashed", color="magenta", weight=3];
156[label="flip (List.deleteBy (==)) xw9 xw10",fontsize=16,color="black",shape="triangle"];156 -> 165[label="",style="solid", color="black", weight=3];
37[label="List.nubByNubBy'0 (==) xw40 xw41 [] otherwise",fontsize=16,color="black",shape="box"];37 -> 42[label="",style="solid", color="black", weight=3];
162 -> 156[label="",style="dashed", color="red", weight=0];
162[label="flip (List.deleteBy (==)) xw9 xw10",fontsize=16,color="magenta"];163[label="xw111",fontsize=16,color="green",shape="box"];164[label="xw110",fontsize=16,color="green",shape="box"];165[label="List.deleteBy (==) xw10 xw9",fontsize=16,color="burlywood",shape="triangle"];2448[label="xw9/xw90 : xw91",fontsize=10,color="white",style="solid",shape="box"];165 -> 2448[label="",style="solid", color="burlywood", weight=9];
2448 -> 173[label="",style="solid", color="burlywood", weight=3];
2449[label="xw9/[]",fontsize=10,color="white",style="solid",shape="box"];165 -> 2449[label="",style="solid", color="burlywood", weight=9];
2449 -> 174[label="",style="solid", color="burlywood", weight=3];
42[label="List.nubByNubBy'0 (==) xw40 xw41 [] True",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
173[label="List.deleteBy (==) xw10 (xw90 : xw91)",fontsize=16,color="black",shape="box"];173 -> 183[label="",style="solid", color="black", weight=3];
174[label="List.deleteBy (==) xw10 []",fontsize=16,color="black",shape="box"];174 -> 184[label="",style="solid", color="black", weight=3];
49[label="xw40 : List.nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="green",shape="box"];49 -> 54[label="",style="dashed", color="green", weight=3];
183 -> 194[label="",style="dashed", color="red", weight=0];
183[label="List.deleteBy0 xw91 xw90 (==) xw10 ((==) xw10 xw90)",fontsize=16,color="magenta"];183 -> 195[label="",style="dashed", color="magenta", weight=3];
183 -> 196[label="",style="dashed", color="magenta", weight=3];
183 -> 197[label="",style="dashed", color="magenta", weight=3];
183 -> 198[label="",style="dashed", color="magenta", weight=3];
184[label="[]",fontsize=16,color="green",shape="box"];54[label="List.nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="burlywood",shape="box"];2450[label="xw41/xw410 : xw411",fontsize=10,color="white",style="solid",shape="box"];54 -> 2450[label="",style="solid", color="burlywood", weight=9];
2450 -> 58[label="",style="solid", color="burlywood", weight=3];
2451[label="xw41/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 2451[label="",style="solid", color="burlywood", weight=9];
2451 -> 59[label="",style="solid", color="burlywood", weight=3];
195[label="xw91",fontsize=16,color="green",shape="box"];196[label="xw90",fontsize=16,color="green",shape="box"];197[label="xw10",fontsize=16,color="green",shape="box"];198[label="(==) xw10 xw90",fontsize=16,color="blue",shape="box"];2452[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2452[label="",style="solid", color="blue", weight=9];
2452 -> 199[label="",style="solid", color="blue", weight=3];
2453[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2453[label="",style="solid", color="blue", weight=9];
2453 -> 200[label="",style="solid", color="blue", weight=3];
2454[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2454[label="",style="solid", color="blue", weight=9];
2454 -> 201[label="",style="solid", color="blue", weight=3];
2455[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2455[label="",style="solid", color="blue", weight=9];
2455 -> 202[label="",style="solid", color="blue", weight=3];
2456[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2456[label="",style="solid", color="blue", weight=9];
2456 -> 203[label="",style="solid", color="blue", weight=3];
2457[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2457[label="",style="solid", color="blue", weight=9];
2457 -> 204[label="",style="solid", color="blue", weight=3];
2458[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2458[label="",style="solid", color="blue", weight=9];
2458 -> 205[label="",style="solid", color="blue", weight=3];
2459[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2459[label="",style="solid", color="blue", weight=9];
2459 -> 206[label="",style="solid", color="blue", weight=3];
2460[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2460[label="",style="solid", color="blue", weight=9];
2460 -> 207[label="",style="solid", color="blue", weight=3];
2461[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2461[label="",style="solid", color="blue", weight=9];
2461 -> 208[label="",style="solid", color="blue", weight=3];
2462[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2462[label="",style="solid", color="blue", weight=9];
2462 -> 209[label="",style="solid", color="blue", weight=3];
2463[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2463[label="",style="solid", color="blue", weight=9];
2463 -> 210[label="",style="solid", color="blue", weight=3];
2464[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2464[label="",style="solid", color="blue", weight=9];
2464 -> 211[label="",style="solid", color="blue", weight=3];
2465[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2465[label="",style="solid", color="blue", weight=9];
2465 -> 212[label="",style="solid", color="blue", weight=3];
194[label="List.deleteBy0 xw17 xw18 (==) xw19 xw20",fontsize=16,color="burlywood",shape="triangle"];2466[label="xw20/False",fontsize=10,color="white",style="solid",shape="box"];194 -> 2466[label="",style="solid", color="burlywood", weight=9];
2466 -> 213[label="",style="solid", color="burlywood", weight=3];
2467[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];194 -> 2467[label="",style="solid", color="burlywood", weight=9];
2467 -> 214[label="",style="solid", color="burlywood", weight=3];
58[label="List.nubByNubBy' (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3];
59[label="List.nubByNubBy' (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3];
199[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];199 -> 226[label="",style="solid", color="black", weight=3];
200[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];200 -> 227[label="",style="solid", color="black", weight=3];
201[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];201 -> 228[label="",style="solid", color="black", weight=3];
202[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];202 -> 229[label="",style="solid", color="black", weight=3];
203[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];203 -> 230[label="",style="solid", color="black", weight=3];
204[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];204 -> 231[label="",style="solid", color="black", weight=3];
205[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];205 -> 232[label="",style="solid", color="black", weight=3];
206[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];206 -> 233[label="",style="solid", color="black", weight=3];
207[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];207 -> 234[label="",style="solid", color="black", weight=3];
208[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];208 -> 235[label="",style="solid", color="black", weight=3];
209[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];209 -> 236[label="",style="solid", color="black", weight=3];
210[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];210 -> 237[label="",style="solid", color="black", weight=3];
211[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];211 -> 238[label="",style="solid", color="black", weight=3];
212[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];212 -> 239[label="",style="solid", color="black", weight=3];
213[label="List.deleteBy0 xw17 xw18 (==) xw19 False",fontsize=16,color="black",shape="box"];213 -> 240[label="",style="solid", color="black", weight=3];
214[label="List.deleteBy0 xw17 xw18 (==) xw19 True",fontsize=16,color="black",shape="box"];214 -> 241[label="",style="solid", color="black", weight=3];
66[label="List.nubByNubBy'2 (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3];
67[label="List.nubByNubBy'3 (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];67 -> 73[label="",style="solid", color="black", weight=3];
226[label="primEqChar xw10 xw90",fontsize=16,color="burlywood",shape="triangle"];2468[label="xw10/Char xw100",fontsize=10,color="white",style="solid",shape="box"];226 -> 2468[label="",style="solid", color="burlywood", weight=9];
2468 -> 253[label="",style="solid", color="burlywood", weight=3];
227[label="error []",fontsize=16,color="red",shape="box"];228[label="error []",fontsize=16,color="red",shape="box"];229[label="error []",fontsize=16,color="red",shape="box"];230[label="error []",fontsize=16,color="red",shape="box"];231[label="error []",fontsize=16,color="red",shape="box"];232[label="error []",fontsize=16,color="red",shape="box"];233[label="error []",fontsize=16,color="red",shape="box"];234[label="error []",fontsize=16,color="red",shape="box"];235[label="error []",fontsize=16,color="red",shape="box"];236[label="error []",fontsize=16,color="red",shape="box"];237[label="error []",fontsize=16,color="red",shape="box"];238[label="error []",fontsize=16,color="red",shape="box"];239[label="error []",fontsize=16,color="red",shape="box"];240[label="xw18 : List.deleteBy (==) xw19 xw17",fontsize=16,color="green",shape="box"];240 -> 254[label="",style="dashed", color="green", weight=3];
241[label="xw17",fontsize=16,color="green",shape="box"];72[label="List.nubByNubBy'1 (==) xw410 xw411 (xw40 : []) (List.elem_by (==) xw410 (xw40 : []))",fontsize=16,color="black",shape="box"];72 -> 78[label="",style="solid", color="black", weight=3];
73[label="[]",fontsize=16,color="green",shape="box"];253[label="primEqChar (Char xw100) xw90",fontsize=16,color="burlywood",shape="box"];2469[label="xw90/Char xw900",fontsize=10,color="white",style="solid",shape="box"];253 -> 2469[label="",style="solid", color="burlywood", weight=9];
2469 -> 271[label="",style="solid", color="burlywood", weight=3];
254 -> 165[label="",style="dashed", color="red", weight=0];
254[label="List.deleteBy (==) xw19 xw17",fontsize=16,color="magenta"];254 -> 272[label="",style="dashed", color="magenta", weight=3];
254 -> 273[label="",style="dashed", color="magenta", weight=3];
78[label="List.nubByNubBy'1 (==) xw410 xw411 (xw40 : []) ((==) xw40 xw410 || List.elem_by (==) xw410 [])",fontsize=16,color="black",shape="box"];78 -> 89[label="",style="solid", color="black", weight=3];
271[label="primEqChar (Char xw100) (Char xw900)",fontsize=16,color="black",shape="box"];271 -> 289[label="",style="solid", color="black", weight=3];
272[label="xw17",fontsize=16,color="green",shape="box"];273[label="xw19",fontsize=16,color="green",shape="box"];89 -> 2402[label="",style="dashed", color="red", weight=0];
89[label="List.nubByNubBy'1 primEqChar xw410 xw411 (xw40 : []) (primEqChar xw40 xw410 || List.elem_by primEqChar xw410 [])",fontsize=16,color="magenta"];89 -> 2403[label="",style="dashed", color="magenta", weight=3];
89 -> 2404[label="",style="dashed", color="magenta", weight=3];
89 -> 2405[label="",style="dashed", color="magenta", weight=3];
89 -> 2406[label="",style="dashed", color="magenta", weight=3];
89 -> 2407[label="",style="dashed", color="magenta", weight=3];
89 -> 2408[label="",style="dashed", color="magenta", weight=3];
289[label="primEqNat xw100 xw900",fontsize=16,color="burlywood",shape="triangle"];2470[label="xw100/Succ xw1000",fontsize=10,color="white",style="solid",shape="box"];289 -> 2470[label="",style="solid", color="burlywood", weight=9];
2470 -> 305[label="",style="solid", color="burlywood", weight=3];
2471[label="xw100/Zero",fontsize=10,color="white",style="solid",shape="box"];289 -> 2471[label="",style="solid", color="burlywood", weight=9];
2471 -> 306[label="",style="solid", color="burlywood", weight=3];
2403[label="[]",fontsize=16,color="green",shape="box"];2404[label="xw40",fontsize=16,color="green",shape="box"];2405[label="[]",fontsize=16,color="green",shape="box"];2406[label="xw411",fontsize=16,color="green",shape="box"];2407 -> 226[label="",style="dashed", color="red", weight=0];
2407[label="primEqChar xw40 xw410",fontsize=16,color="magenta"];2407 -> 2410[label="",style="dashed", color="magenta", weight=3];
2407 -> 2411[label="",style="dashed", color="magenta", weight=3];
2408[label="xw410",fontsize=16,color="green",shape="box"];2402[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (xw161 || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2472[label="xw161/False",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2472[label="",style="solid", color="burlywood", weight=9];
2472 -> 2412[label="",style="solid", color="burlywood", weight=3];
2473[label="xw161/True",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2473[label="",style="solid", color="burlywood", weight=9];
2473 -> 2413[label="",style="solid", color="burlywood", weight=3];
305[label="primEqNat (Succ xw1000) xw900",fontsize=16,color="burlywood",shape="box"];2474[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];305 -> 2474[label="",style="solid", color="burlywood", weight=9];
2474 -> 309[label="",style="solid", color="burlywood", weight=3];
2475[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];305 -> 2475[label="",style="solid", color="burlywood", weight=9];
2475 -> 310[label="",style="solid", color="burlywood", weight=3];
306[label="primEqNat Zero xw900",fontsize=16,color="burlywood",shape="box"];2476[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];306 -> 2476[label="",style="solid", color="burlywood", weight=9];
2476 -> 311[label="",style="solid", color="burlywood", weight=3];
2477[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];306 -> 2477[label="",style="solid", color="burlywood", weight=9];
2477 -> 312[label="",style="solid", color="burlywood", weight=3];
2410[label="xw410",fontsize=16,color="green",shape="box"];2411[label="xw40",fontsize=16,color="green",shape="box"];2412[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (False || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2412 -> 2414[label="",style="solid", color="black", weight=3];
2413[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (True || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2413 -> 2415[label="",style="solid", color="black", weight=3];
309[label="primEqNat (Succ xw1000) (Succ xw9000)",fontsize=16,color="black",shape="box"];309 -> 333[label="",style="solid", color="black", weight=3];
310[label="primEqNat (Succ xw1000) Zero",fontsize=16,color="black",shape="box"];310 -> 334[label="",style="solid", color="black", weight=3];
311[label="primEqNat Zero (Succ xw9000)",fontsize=16,color="black",shape="box"];311 -> 335[label="",style="solid", color="black", weight=3];
312[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];312 -> 336[label="",style="solid", color="black", weight=3];
2414[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2478[label="xw160/xw1600 : xw1601",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2478[label="",style="solid", color="burlywood", weight=9];
2478 -> 2416[label="",style="solid", color="burlywood", weight=3];
2479[label="xw160/[]",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2479[label="",style="solid", color="burlywood", weight=9];
2479 -> 2417[label="",style="solid", color="burlywood", weight=3];
2415[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2415 -> 2418[label="",style="solid", color="black", weight=3];
333 -> 289[label="",style="dashed", color="red", weight=0];
333[label="primEqNat xw1000 xw9000",fontsize=16,color="magenta"];333 -> 350[label="",style="dashed", color="magenta", weight=3];
333 -> 351[label="",style="dashed", color="magenta", weight=3];
334[label="False",fontsize=16,color="green",shape="box"];335[label="False",fontsize=16,color="green",shape="box"];336[label="True",fontsize=16,color="green",shape="box"];2416[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 (xw1600 : xw1601))",fontsize=16,color="black",shape="box"];2416 -> 2419[label="",style="solid", color="black", weight=3];
2417[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 [])",fontsize=16,color="black",shape="box"];2417 -> 2420[label="",style="solid", color="black", weight=3];
2418[label="List.nubByNubBy' primEqChar xw156 (xw157 : xw158)",fontsize=16,color="burlywood",shape="triangle"];2480[label="xw156/xw1560 : xw1561",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2480[label="",style="solid", color="burlywood", weight=9];
2480 -> 2421[label="",style="solid", color="burlywood", weight=3];
2481[label="xw156/[]",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2481[label="",style="solid", color="burlywood", weight=9];
2481 -> 2422[label="",style="solid", color="burlywood", weight=3];
350[label="xw9000",fontsize=16,color="green",shape="box"];351[label="xw1000",fontsize=16,color="green",shape="box"];2419 -> 2402[label="",style="dashed", color="red", weight=0];
2419[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (primEqChar xw1600 xw155 || List.elem_by primEqChar xw155 xw1601)",fontsize=16,color="magenta"];2419 -> 2423[label="",style="dashed", color="magenta", weight=3];
2419 -> 2424[label="",style="dashed", color="magenta", weight=3];
2420[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) False",fontsize=16,color="black",shape="box"];2420 -> 2425[label="",style="solid", color="black", weight=3];
2421[label="List.nubByNubBy' primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2421 -> 2426[label="",style="solid", color="black", weight=3];
2422[label="List.nubByNubBy' primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2422 -> 2427[label="",style="solid", color="black", weight=3];
2423[label="xw1601",fontsize=16,color="green",shape="box"];2424 -> 226[label="",style="dashed", color="red", weight=0];
2424[label="primEqChar xw1600 xw155",fontsize=16,color="magenta"];2424 -> 2428[label="",style="dashed", color="magenta", weight=3];
2424 -> 2429[label="",style="dashed", color="magenta", weight=3];
2425[label="List.nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) otherwise",fontsize=16,color="black",shape="box"];2425 -> 2430[label="",style="solid", color="black", weight=3];
2426[label="List.nubByNubBy'2 primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2426 -> 2431[label="",style="solid", color="black", weight=3];
2427[label="List.nubByNubBy'3 primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2427 -> 2432[label="",style="solid", color="black", weight=3];
2428[label="xw155",fontsize=16,color="green",shape="box"];2429[label="xw1600",fontsize=16,color="green",shape="box"];2430[label="List.nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2430 -> 2433[label="",style="solid", color="black", weight=3];
2431 -> 2414[label="",style="dashed", color="red", weight=0];
2431[label="List.nubByNubBy'1 primEqChar xw1560 xw1561 (xw157 : xw158) (List.elem_by primEqChar xw1560 (xw157 : xw158))",fontsize=16,color="magenta"];2431 -> 2434[label="",style="dashed", color="magenta", weight=3];
2431 -> 2435[label="",style="dashed", color="magenta", weight=3];
2431 -> 2436[label="",style="dashed", color="magenta", weight=3];
2432[label="[]",fontsize=16,color="green",shape="box"];2433[label="xw155 : List.nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="green",shape="box"];2433 -> 2437[label="",style="dashed", color="green", weight=3];
2434[label="xw157 : xw158",fontsize=16,color="green",shape="box"];2435[label="xw1561",fontsize=16,color="green",shape="box"];2436[label="xw1560",fontsize=16,color="green",shape="box"];2437 -> 2418[label="",style="dashed", color="red", weight=0];
2437[label="List.nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="magenta"];2437 -> 2438[label="",style="dashed", color="magenta", weight=3];
2437 -> 2439[label="",style="dashed", color="magenta", weight=3];
2438[label="xw155",fontsize=16,color="green",shape="box"];2439[label="xw157 : xw158",fontsize=16,color="green",shape="box"];}

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

(10)
Complex Obligation (AND)

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

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

   new_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) -> new_psPs(xw81, xw9, xw10, xw11, ba)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) -> new_psPs(xw81, xw9, xw10, xw11, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5


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

(13)
YES

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

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

   new_deleteBy0(xw17, xw18, xw19, False, ba) -> new_deleteBy(xw19, xw17, ba)
   new_deleteBy(xw10, :(xw90, xw91), bb) -> new_deleteBy0(xw91, xw90, xw10, new_esEs(xw10, xw90, bb), bb)

The TRS R consists of the following rules:

   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
   new_esEs(xw10, xw90, app(ty_Ratio, bf)) -> error([])
   new_primEqNat0(Zero, Zero) -> True
   new_esEs(xw10, xw90, app(app(app(ty_@3, cb), cc), cd)) -> error([])
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_esEs(xw10, xw90, app(ty_[], bc)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
   new_esEs(xw10, xw90, app(app(ty_@2, bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, ty_@0) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, bg), bh)) -> error([])
   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])

The set Q consists of the following terms:

   new_esEs(x0, x1, app(ty_[], x2))
   new_esEs(x0, x1, ty_Float)
   new_primEqNat0(Zero, Zero)
   new_primEqNat0(Succ(x0), Succ(x1))
   new_primEqNat0(Succ(x0), Zero)
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, ty_Char)
   new_esEs(x0, x1, ty_Int)
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, ty_Integer)
   new_primEqChar(Char(x0), Char(x1))
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, ty_Double)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Maybe, x2))

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

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

From the DPs we obtained the following set of size-change graphs:
*new_deleteBy(xw10, :(xw90, xw91), bb) -> new_deleteBy0(xw91, xw90, xw10, new_esEs(xw10, xw90, bb), bb)
The graph contains the following edges 2 > 1, 2 > 2, 1 >= 3, 3 >= 5


*new_deleteBy0(xw17, xw18, xw19, False, ba) -> new_deleteBy(xw19, xw17, ba)
The graph contains the following edges 3 >= 1, 1 >= 2, 5 >= 3


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

(16)
YES

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

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

   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
   new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
   new_nubByNubBy'10(xw155, xw156, xw157, xw158, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))

The TRS R consists of the following rules:

   new_primEqNat0(Zero, Zero) -> True
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)

The set Q consists of the following terms:

   new_primEqNat0(Zero, Zero)
   new_primEqChar(Char(x0), Char(x1))
   new_primEqNat0(Succ(x0), Succ(x1))
   new_primEqNat0(Succ(x0), Zero)
   new_primEqNat0(Zero, Succ(x0))

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

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

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

   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
   new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))

The TRS R consists of the following rules:

   new_primEqNat0(Zero, Zero) -> True
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)

The set Q consists of the following terms:

   new_primEqNat0(Zero, Zero)
   new_primEqChar(Char(x0), Char(x1))
   new_primEqNat0(Succ(x0), Succ(x1))
   new_primEqNat0(Succ(x0), Zero)
   new_primEqNat0(Zero, Succ(x0))

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

(20) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601) we obtained the following new rules [LPAR04]:

   (new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3),new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3))


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

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

   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
   new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3)

The TRS R consists of the following rules:

   new_primEqNat0(Zero, Zero) -> True
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)

The set Q consists of the following terms:

   new_primEqNat0(Zero, Zero)
   new_primEqChar(Char(x0), Char(x1))
   new_primEqNat0(Succ(x0), Succ(x1))
   new_primEqNat0(Succ(x0), Zero)
   new_primEqNat0(Zero, Succ(x0))

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

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

From the DPs we obtained the following set of size-change graphs:
*new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 4


*new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 5 > 3, 4 >= 4, 5 > 4, 4 >= 6, 5 > 6


*new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 6


*new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
The graph contains the following edges 2 >= 1, 1 >= 2


*new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
The graph contains the following edges 2 > 1, 2 > 2, 3 >= 3, 4 >= 4


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

(23)
YES

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

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

   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba)

The TRS R consists of the following rules:

   new_primEqNat0(Zero, Zero) -> True
   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
   new_esEs(xw10, xw90, ty_@0) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))

The set Q consists of the following terms:

   new_esEs(x0, x1, ty_Float)
   new_primEqNat0(Zero, Zero)
   new_deleteBy00(x0, x1, x2, False, x3)
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, ty_Char)
   new_esEs(x0, x1, ty_Int)
   new_deleteBy1(x0, :(x1, x2), x3)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_[], x2))
   new_deleteBy1(x0, [], x1)
   new_esEs(x0, x1, ty_Integer)
   new_primEqChar(Char(x0), Char(x1))
   new_flip(x0, x1, x2)
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, ty_Double)
   new_deleteBy00(x0, x1, x2, True, x3)
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))

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

(25) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba),new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba))


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

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

   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)

The TRS R consists of the following rules:

   new_primEqNat0(Zero, Zero) -> True
   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
   new_esEs(xw10, xw90, ty_@0) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))

The set Q consists of the following terms:

   new_esEs(x0, x1, ty_Float)
   new_primEqNat0(Zero, Zero)
   new_deleteBy00(x0, x1, x2, False, x3)
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, ty_Char)
   new_esEs(x0, x1, ty_Int)
   new_deleteBy1(x0, :(x1, x2), x3)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_[], x2))
   new_deleteBy1(x0, [], x1)
   new_esEs(x0, x1, ty_Integer)
   new_primEqChar(Char(x0), Char(x1))
   new_flip(x0, x1, x2)
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, ty_Double)
   new_deleteBy00(x0, x1, x2, True, x3)
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))

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

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

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

   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)

The TRS R consists of the following rules:

   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
   new_esEs(xw10, xw90, ty_@0) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_primEqNat0(Zero, Zero) -> True
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)

The set Q consists of the following terms:

   new_esEs(x0, x1, ty_Float)
   new_primEqNat0(Zero, Zero)
   new_deleteBy00(x0, x1, x2, False, x3)
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, ty_Char)
   new_esEs(x0, x1, ty_Int)
   new_deleteBy1(x0, :(x1, x2), x3)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_[], x2))
   new_deleteBy1(x0, [], x1)
   new_esEs(x0, x1, ty_Integer)
   new_primEqChar(Char(x0), Char(x1))
   new_flip(x0, x1, x2)
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, ty_Double)
   new_deleteBy00(x0, x1, x2, True, x3)
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))

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

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

   new_flip(x0, x1, x2)


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

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

   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)

The TRS R consists of the following rules:

   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
   new_esEs(xw10, xw90, ty_@0) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
   new_primEqNat0(Zero, Zero) -> True
   new_primEqNat0(Succ(xw1000), Zero) -> False
   new_primEqNat0(Zero, Succ(xw9000)) -> False
   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)

The set Q consists of the following terms:

   new_esEs(x0, x1, ty_Float)
   new_primEqNat0(Zero, Zero)
   new_deleteBy00(x0, x1, x2, False, x3)
   new_primEqNat0(Succ(x0), Succ(x1))
   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_primEqNat0(Succ(x0), Zero)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_primEqNat0(Zero, Succ(x0))
   new_esEs(x0, x1, ty_Char)
   new_esEs(x0, x1, ty_Int)
   new_deleteBy1(x0, :(x1, x2), x3)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, ty_Ordering)
   new_esEs(x0, x1, app(ty_[], x2))
   new_deleteBy1(x0, [], x1)
   new_esEs(x0, x1, ty_Integer)
   new_primEqChar(Char(x0), Char(x1))
   new_esEs(x0, x1, ty_@0)
   new_esEs(x0, x1, ty_Double)
   new_deleteBy00(x0, x1, x2, True, x3)
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Maybe, x2))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))

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

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

From the DPs we obtained the following set of size-change graphs:
*new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)
The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4


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

(32)
YES

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

(33)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primEqNat(Succ(xw1000), Succ(xw9000)) -> new_primEqNat(xw1000, xw9000)

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

(34) 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_primEqNat(Succ(xw1000), Succ(xw9000)) -> new_primEqNat(xw1000, xw9000)
The graph contains the following edges 1 > 1, 2 > 2


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

(35)
YES