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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) HASKELL
(1) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 22 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) AND
    (11) QDP
        (12) TransformationProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) UsableRulesProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QReductionProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) DependencyGraphProof [EQUIVALENT, 0 ms]
        (25) TRUE
    (26) QDP
        (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (28) 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'1 y ys xs True = nubBy' ys xs;
nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
"
"nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
"
"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' 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'3 xv [] wu = [];
nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
"
"nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv 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;
"

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

(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"];185[label="xw3/xw30 : xw31",fontsize=10,color="white",style="solid",shape="box"];6 -> 185[label="",style="solid", color="burlywood", weight=9];
185 -> 7[label="",style="solid", color="burlywood", weight=3];
186[label="xw3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 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 -> 91[label="",style="dashed", color="red", weight=0];
11[label="xw31 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="magenta"];11 -> 92[label="",style="dashed", color="magenta", weight=3];
11 -> 93[label="",style="dashed", color="magenta", weight=3];
11 -> 94[label="",style="dashed", color="magenta", weight=3];
11 -> 95[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];
92[label="xw31",fontsize=16,color="green",shape="box"];93[label="xw31",fontsize=16,color="green",shape="box"];94[label="xw30",fontsize=16,color="green",shape="box"];95 -> 12[label="",style="dashed", color="red", weight=0];
95[label="List.nubBy (==) xw4",fontsize=16,color="magenta"];91[label="xw8 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="burlywood",shape="triangle"];187[label="xw8/xw80 : xw81",fontsize=10,color="white",style="solid",shape="box"];91 -> 187[label="",style="solid", color="burlywood", weight=9];
187 -> 120[label="",style="solid", color="burlywood", weight=3];
188[label="xw8/[]",fontsize=10,color="white",style="solid",shape="box"];91 -> 188[label="",style="solid", color="burlywood", weight=9];
188 -> 121[label="",style="solid", color="burlywood", weight=3];
15[label="List.nubByNubBy' (==) xw4 []",fontsize=16,color="burlywood",shape="box"];189[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];15 -> 189[label="",style="solid", color="burlywood", weight=9];
189 -> 18[label="",style="solid", color="burlywood", weight=3];
190[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 190[label="",style="solid", color="burlywood", weight=9];
190 -> 19[label="",style="solid", color="burlywood", weight=3];
120[label="(xw80 : xw81) ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];120 -> 123[label="",style="solid", color="black", weight=3];
121[label="[] ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];121 -> 124[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];
123[label="xw80 : xw81 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="green",shape="box"];123 -> 126[label="",style="dashed", color="green", weight=3];
124[label="foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="black",shape="box"];124 -> 127[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];
126 -> 91[label="",style="dashed", color="red", weight=0];
126[label="xw81 ++ foldl (flip (List.deleteBy (==))) xw9 (xw10 : xw11)",fontsize=16,color="magenta"];126 -> 130[label="",style="dashed", color="magenta", weight=3];
127[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) xw11",fontsize=16,color="burlywood",shape="triangle"];191[label="xw11/xw110 : xw111",fontsize=10,color="white",style="solid",shape="box"];127 -> 191[label="",style="solid", color="burlywood", weight=9];
191 -> 131[label="",style="solid", color="burlywood", weight=3];
192[label="xw11/[]",fontsize=10,color="white",style="solid",shape="box"];127 -> 192[label="",style="solid", color="burlywood", weight=9];
192 -> 132[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"];130[label="xw81",fontsize=16,color="green",shape="box"];131[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) (xw110 : xw111)",fontsize=16,color="black",shape="box"];131 -> 133[label="",style="solid", color="black", weight=3];
132[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) xw9 xw10) []",fontsize=16,color="black",shape="box"];132 -> 134[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];
133 -> 127[label="",style="dashed", color="red", weight=0];
133[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) (flip (List.deleteBy (==)) xw9 xw10) xw110) xw111",fontsize=16,color="magenta"];133 -> 135[label="",style="dashed", color="magenta", weight=3];
133 -> 136[label="",style="dashed", color="magenta", weight=3];
133 -> 137[label="",style="dashed", color="magenta", weight=3];
134[label="flip (List.deleteBy (==)) xw9 xw10",fontsize=16,color="black",shape="triangle"];134 -> 138[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];
135[label="xw111",fontsize=16,color="green",shape="box"];136[label="xw110",fontsize=16,color="green",shape="box"];137 -> 134[label="",style="dashed", color="red", weight=0];
137[label="flip (List.deleteBy (==)) xw9 xw10",fontsize=16,color="magenta"];138[label="List.deleteBy (==) xw10 xw9",fontsize=16,color="burlywood",shape="triangle"];193[label="xw9/xw90 : xw91",fontsize=10,color="white",style="solid",shape="box"];138 -> 193[label="",style="solid", color="burlywood", weight=9];
193 -> 139[label="",style="solid", color="burlywood", weight=3];
194[label="xw9/[]",fontsize=10,color="white",style="solid",shape="box"];138 -> 194[label="",style="solid", color="burlywood", weight=9];
194 -> 140[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];
139[label="List.deleteBy (==) xw10 (xw90 : xw91)",fontsize=16,color="black",shape="box"];139 -> 141[label="",style="solid", color="black", weight=3];
140[label="List.deleteBy (==) xw10 []",fontsize=16,color="black",shape="box"];140 -> 142[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];
141 -> 143[label="",style="dashed", color="red", weight=0];
141[label="List.deleteBy0 xw91 xw90 (==) xw10 ((==) xw10 xw90)",fontsize=16,color="magenta"];141 -> 144[label="",style="dashed", color="magenta", weight=3];
141 -> 145[label="",style="dashed", color="magenta", weight=3];
141 -> 146[label="",style="dashed", color="magenta", weight=3];
141 -> 147[label="",style="dashed", color="magenta", weight=3];
142[label="[]",fontsize=16,color="green",shape="box"];54[label="List.nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="burlywood",shape="triangle"];195[label="xw41/xw410 : xw411",fontsize=10,color="white",style="solid",shape="box"];54 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 58[label="",style="solid", color="burlywood", weight=3];
196[label="xw41/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 59[label="",style="solid", color="burlywood", weight=3];
144[label="xw91",fontsize=16,color="green",shape="box"];145[label="xw10",fontsize=16,color="green",shape="box"];146[label="(==) xw10 xw90",fontsize=16,color="blue",shape="box"];197[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 197[label="",style="solid", color="blue", weight=9];
197 -> 148[label="",style="solid", color="blue", weight=3];
198[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 198[label="",style="solid", color="blue", weight=9];
198 -> 149[label="",style="solid", color="blue", weight=3];
199[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 199[label="",style="solid", color="blue", weight=9];
199 -> 150[label="",style="solid", color="blue", weight=3];
200[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 200[label="",style="solid", color="blue", weight=9];
200 -> 151[label="",style="solid", color="blue", weight=3];
201[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 201[label="",style="solid", color="blue", weight=9];
201 -> 152[label="",style="solid", color="blue", weight=3];
202[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 202[label="",style="solid", color="blue", weight=9];
202 -> 153[label="",style="solid", color="blue", weight=3];
203[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 203[label="",style="solid", color="blue", weight=9];
203 -> 154[label="",style="solid", color="blue", weight=3];
204[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 204[label="",style="solid", color="blue", weight=9];
204 -> 155[label="",style="solid", color="blue", weight=3];
205[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 205[label="",style="solid", color="blue", weight=9];
205 -> 156[label="",style="solid", color="blue", weight=3];
206[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 206[label="",style="solid", color="blue", weight=9];
206 -> 157[label="",style="solid", color="blue", weight=3];
207[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 207[label="",style="solid", color="blue", weight=9];
207 -> 158[label="",style="solid", color="blue", weight=3];
208[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 208[label="",style="solid", color="blue", weight=9];
208 -> 159[label="",style="solid", color="blue", weight=3];
209[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 209[label="",style="solid", color="blue", weight=9];
209 -> 160[label="",style="solid", color="blue", weight=3];
210[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];146 -> 210[label="",style="solid", color="blue", weight=9];
210 -> 161[label="",style="solid", color="blue", weight=3];
147[label="xw90",fontsize=16,color="green",shape="box"];143[label="List.deleteBy0 xw17 xw18 (==) xw19 xw20",fontsize=16,color="burlywood",shape="triangle"];211[label="xw20/False",fontsize=10,color="white",style="solid",shape="box"];143 -> 211[label="",style="solid", color="burlywood", weight=9];
211 -> 162[label="",style="solid", color="burlywood", weight=3];
212[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];143 -> 212[label="",style="solid", color="burlywood", weight=9];
212 -> 163[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];
148[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];148 -> 164[label="",style="solid", color="black", weight=3];
149[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];149 -> 165[label="",style="solid", color="black", weight=3];
150[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];150 -> 166[label="",style="solid", color="black", weight=3];
151[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];151 -> 167[label="",style="solid", color="black", weight=3];
152[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];152 -> 168[label="",style="solid", color="black", weight=3];
153[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];153 -> 169[label="",style="solid", color="black", weight=3];
154[label="(==) xw10 xw90",fontsize=16,color="burlywood",shape="box"];213[label="xw10/()",fontsize=10,color="white",style="solid",shape="box"];154 -> 213[label="",style="solid", color="burlywood", weight=9];
213 -> 170[label="",style="solid", color="burlywood", weight=3];
155[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];155 -> 171[label="",style="solid", color="black", weight=3];
156[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];156 -> 172[label="",style="solid", color="black", weight=3];
157[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];157 -> 173[label="",style="solid", color="black", weight=3];
158[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];158 -> 174[label="",style="solid", color="black", weight=3];
159[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];159 -> 175[label="",style="solid", color="black", weight=3];
160[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];160 -> 176[label="",style="solid", color="black", weight=3];
161[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];161 -> 177[label="",style="solid", color="black", weight=3];
162[label="List.deleteBy0 xw17 xw18 (==) xw19 False",fontsize=16,color="black",shape="box"];162 -> 178[label="",style="solid", color="black", weight=3];
163[label="List.deleteBy0 xw17 xw18 (==) xw19 True",fontsize=16,color="black",shape="box"];163 -> 179[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];
164[label="error []",fontsize=16,color="red",shape="box"];165[label="error []",fontsize=16,color="red",shape="box"];166[label="error []",fontsize=16,color="red",shape="box"];167[label="error []",fontsize=16,color="red",shape="box"];168[label="error []",fontsize=16,color="red",shape="box"];169[label="error []",fontsize=16,color="red",shape="box"];170[label="(==) () xw90",fontsize=16,color="burlywood",shape="box"];214[label="xw90/()",fontsize=10,color="white",style="solid",shape="box"];170 -> 214[label="",style="solid", color="burlywood", weight=9];
214 -> 180[label="",style="solid", color="burlywood", weight=3];
171[label="error []",fontsize=16,color="red",shape="box"];172[label="error []",fontsize=16,color="red",shape="box"];173[label="error []",fontsize=16,color="red",shape="box"];174[label="error []",fontsize=16,color="red",shape="box"];175[label="error []",fontsize=16,color="red",shape="box"];176[label="error []",fontsize=16,color="red",shape="box"];177[label="error []",fontsize=16,color="red",shape="box"];178[label="xw18 : List.deleteBy (==) xw19 xw17",fontsize=16,color="green",shape="box"];178 -> 181[label="",style="dashed", color="green", weight=3];
179[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 -> 76[label="",style="solid", color="black", weight=3];
73[label="[]",fontsize=16,color="green",shape="box"];180[label="(==) () ()",fontsize=16,color="black",shape="box"];180 -> 182[label="",style="solid", color="black", weight=3];
181 -> 138[label="",style="dashed", color="red", weight=0];
181[label="List.deleteBy (==) xw19 xw17",fontsize=16,color="magenta"];181 -> 183[label="",style="dashed", color="magenta", weight=3];
181 -> 184[label="",style="dashed", color="magenta", weight=3];
76[label="List.nubByNubBy'1 (==) xw410 xw411 (xw40 : []) ((==) xw40 xw410 || List.elem_by (==) xw410 [])",fontsize=16,color="burlywood",shape="box"];215[label="xw40/()",fontsize=10,color="white",style="solid",shape="box"];76 -> 215[label="",style="solid", color="burlywood", weight=9];
215 -> 83[label="",style="solid", color="burlywood", weight=3];
182[label="True",fontsize=16,color="green",shape="box"];183[label="xw19",fontsize=16,color="green",shape="box"];184[label="xw17",fontsize=16,color="green",shape="box"];83[label="List.nubByNubBy'1 (==) xw410 xw411 (() : []) ((==) () xw410 || List.elem_by (==) xw410 [])",fontsize=16,color="burlywood",shape="box"];216[label="xw410/()",fontsize=10,color="white",style="solid",shape="box"];83 -> 216[label="",style="solid", color="burlywood", weight=9];
216 -> 87[label="",style="solid", color="burlywood", weight=3];
87[label="List.nubByNubBy'1 (==) () xw411 (() : []) ((==) () () || List.elem_by (==) () [])",fontsize=16,color="black",shape="box"];87 -> 90[label="",style="solid", color="black", weight=3];
90[label="List.nubByNubBy'1 (==) () xw411 (() : []) (True || List.elem_by (==) () [])",fontsize=16,color="black",shape="box"];90 -> 122[label="",style="solid", color="black", weight=3];
122[label="List.nubByNubBy'1 (==) () xw411 (() : []) True",fontsize=16,color="black",shape="box"];122 -> 125[label="",style="solid", color="black", weight=3];
125 -> 54[label="",style="dashed", color="red", weight=0];
125[label="List.nubByNubBy' (==) xw411 (() : [])",fontsize=16,color="magenta"];125 -> 128[label="",style="dashed", color="magenta", weight=3];
125 -> 129[label="",style="dashed", color="magenta", weight=3];
128[label="xw411",fontsize=16,color="green",shape="box"];129[label="()",fontsize=16,color="green",shape="box"];}

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

(10)
Complex Obligation (AND)

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

(11)
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_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
   new_deleteBy00(xw17, xw18, xw19, True, bb) -> xw17
   new_esEs(xw10, xw90, app(ty_Ratio, bh)) -> error([])
   new_esEs(xw10, xw90, app(app(app(ty_@3, bc), bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
   new_esEs(@0, @0, ty_@0) -> True
   new_esEs(xw10, xw90, app(ty_[], cd)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bf), bg)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> error([])
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, False, bb) -> :(xw18, new_deleteBy1(xw19, xw17, bb))
   new_esEs(xw10, xw90, ty_Double) -> error([])

The set Q consists of the following terms:

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

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

(12) 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))


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

(13)
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_deleteBy00(xw17, xw18, xw19, True, bb) -> xw17
   new_esEs(xw10, xw90, app(ty_Ratio, bh)) -> error([])
   new_esEs(xw10, xw90, app(app(app(ty_@3, bc), bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
   new_esEs(@0, @0, ty_@0) -> True
   new_esEs(xw10, xw90, app(ty_[], cd)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bf), bg)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> error([])
   new_deleteBy1(xw10, [], ba) -> []
   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
   new_deleteBy00(xw17, xw18, xw19, False, bb) -> :(xw18, new_deleteBy1(xw19, xw17, bb))
   new_esEs(xw10, xw90, ty_Double) -> error([])

The set Q consists of the following terms:

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

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

(14) 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.
----------------------------------------

(15)
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, bh)) -> error([])
   new_esEs(xw10, xw90, app(app(app(ty_@3, bc), bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_esEs(@0, @0, ty_@0) -> True
   new_esEs(xw10, xw90, app(ty_[], cd)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bf), bg)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> error([])
   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_deleteBy00(xw17, xw18, xw19, True, bb) -> xw17
   new_deleteBy00(xw17, xw18, xw19, False, bb) -> :(xw18, new_deleteBy1(xw19, xw17, bb))

The set Q consists of the following terms:

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

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

(16) 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)


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

(17)
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, bh)) -> error([])
   new_esEs(xw10, xw90, app(app(app(ty_@3, bc), bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_esEs(@0, @0, ty_@0) -> True
   new_esEs(xw10, xw90, app(ty_[], cd)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bf), bg)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> error([])
   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
   new_esEs(xw10, xw90, ty_Double) -> error([])
   new_deleteBy00(xw17, xw18, xw19, True, bb) -> xw17
   new_deleteBy00(xw17, xw18, xw19, False, bb) -> :(xw18, new_deleteBy1(xw19, xw17, bb))

The set Q consists of the following terms:

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

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


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(19)
YES

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(20)
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.
----------------------------------------

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

From the DPs we obtained the following set of size-change graphs:
*new_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


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(22)
YES

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(23)
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, app(ty_Ratio, bh)) -> error([])
   new_esEs(xw10, xw90, app(app(app(ty_@3, bc), bd), be)) -> error([])
   new_esEs(xw10, xw90, ty_Integer) -> error([])
   new_esEs(@0, @0, ty_@0) -> True
   new_esEs(xw10, xw90, app(ty_[], cd)) -> error([])
   new_esEs(xw10, xw90, ty_Ordering) -> error([])
   new_esEs(xw10, xw90, app(app(ty_@2, bf), bg)) -> error([])
   new_esEs(xw10, xw90, ty_Float) -> error([])
   new_esEs(xw10, xw90, ty_Bool) -> error([])
   new_esEs(xw10, xw90, app(app(ty_Either, cb), cc)) -> error([])
   new_esEs(xw10, xw90, ty_Int) -> error([])
   new_esEs(xw10, xw90, ty_Char) -> 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, ty_Float)
   new_esEs(@0, @0, ty_@0)
   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_esEs(x0, x1, app(app(ty_Either, x2), x3))
   new_esEs(x0, x1, ty_Double)
   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
   new_esEs(x0, x1, app(ty_[], x2))
   new_esEs(x0, x1, ty_Bool)
   new_esEs(x0, x1, app(ty_Ratio, x2))
   new_esEs(x0, x1, app(ty_Maybe, x2))

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

(24) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
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(25)
TRUE

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

   new_nubByNubBy'(:(@0, xw411), @0) -> new_nubByNubBy'(xw411, @0)

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

(27) 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'(:(@0, xw411), @0) -> new_nubByNubBy'(xw411, @0)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2


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(28)
YES