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


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


MAYBE
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 not be shown:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 27 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) NonTerminationLoopProof [COMPLETE, 0 ms]
        (16) NO
(17) Narrow [COMPLETE, 0 ms]
(18) QDP
    (19) PisEmptyProof [EQUIVALENT, 0 ms]
    (20) 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;
groupBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [[a]];
groupBy _ [] = [];
groupBy eq (x : xs) = (x : ys) : groupBy eq zs where { 
vv10 = span (eq x) xs;
ys = (\(ys,_) ->ys) vv10;
zs = (\(_,zs) ->zs) vv10;
 };

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

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(_,zs)->zs"
is transformed to
"zs0 (_,zs) = zs;
"
The following Lambda expression
"\(ys,_)->ys"
is transformed to
"ys0 (ys,_) = ys;
"
The following Lambda expression
"\(_,zs)->zs"
is transformed to
"zs1 (_,zs) = zs;
"
The following Lambda expression
"\(ys,_)->ys"
is transformed to
"ys1 (ys,_) = 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;
groupBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [[a]];
groupBy _ [] = [];
groupBy eq (x : xs) = (x : ys) : groupBy eq zs where { 
vv10 = span (eq x) xs;
ys = ys1 vv10;
ys1 (ys,_) = ys;
zs = zs1 vv10;
zs1 (_,zs) = zs;
 };

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

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

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.

Binding Reductions:
The bind variable of the following binding Pattern
"xs@(vx : vy)"
is replaced by the following term
"vx : vy"

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

(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;
groupBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [[a]];
groupBy ww [] = [];
groupBy eq (x : xs) = (x : ys) : groupBy eq zs where { 
vv10 = span (eq x) xs;
ys = ys1 vv10;
ys1 (ys,wx) = ys;
zs = zs1 vv10;
zs1 (wy,zs) = zs;
 };

}
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
"span p [] = ([],[]);
span p (vx : vy)|p vx(vx : ys,zs)|otherwise([],vx : vy) where {
vu43  = span p vy;
;
ys  = ys0 vu43;
;
ys0 (ys,wu) = ys;
;
zs  = zs0 vu43;
;
zs0 (vz,zs) = zs;
} 
;
"
is transformed to
"span p [] = span3 p [];
span p (vx : vy) = span2 p (vx : vy);
"
"span2 p (vx : vy) = span1 p vx vy (p vx) where {
span0 p vx vy True = ([],vx : vy);
;
span1 p vx vy True = (vx : ys,zs);
span1 p vx vy False = span0 p vx vy otherwise;
;
vu43  = span p vy;
;
ys  = ys0 vu43;
;
ys0 (ys,wu) = ys;
;
zs  = zs0 vu43;
;
zs0 (vz,zs) = zs;
} 
;
"
"span3 p [] = ([],[]);
span3 xv xw = span2 xv xw;
"

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

(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;
groupBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [[a]];
groupBy ww [] = [];
groupBy eq (x : xs) = (x : ys) : groupBy eq zs where { 
vv10 = span (eq x) xs;
ys = ys1 vv10;
ys1 (ys,wx) = ys;
zs = zs1 vv10;
zs1 (wy,zs) = zs;
 };

}
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
"span1 p vx vy (p vx) where {
span0 p vx vy True = ([],vx : vy);
;
span1 p vx vy True = (vx : ys,zs);
span1 p vx vy False = span0 p vx vy otherwise;
;
vu43  = span p vy;
;
ys  = ys0 vu43;
;
ys0 (ys,wu) = ys;
;
zs  = zs0 vu43;
;
zs0 (vz,zs) = zs;
} 
"
are unpacked to the following functions on top level
"span2Vu43 xx xy = span xx xy;
"
"span2Ys xx xy = span2Ys0 xx xy (span2Vu43 xx xy);
"
"span2Ys0 xx xy (ys,wu) = ys;
"
"span2Span1 xx xy p vx vy True = (vx : span2Ys xx xy,span2Zs xx xy);
span2Span1 xx xy p vx vy False = span2Span0 xx xy p vx vy otherwise;
"
"span2Zs0 xx xy (vz,zs) = zs;
"
"span2Span0 xx xy p vx vy True = ([],vx : vy);
"
"span2Zs xx xy = span2Zs0 xx xy (span2Vu43 xx xy);
"
The bindings of the following Let/Where expression
"(x : ys) : groupBy eq zs where {
vv10  = span (eq x) xs;
;
ys  = ys1 vv10;
;
ys1 (ys,wx) = ys;
;
zs  = zs1 vv10;
;
zs1 (wy,zs) = zs;
} 
"
are unpacked to the following functions on top level
"groupByZs xz yu yv = groupByZs1 xz yu yv (groupByVv10 xz yu yv);
"
"groupByYs1 xz yu yv (ys,wx) = ys;
"
"groupByZs1 xz yu yv (wy,zs) = zs;
"
"groupByYs xz yu yv = groupByYs1 xz yu yv (groupByVv10 xz yu yv);
"
"groupByVv10 xz yu yv = span (xz yu) yv;
"

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

(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;
groupBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [[a]];
groupBy ww [] = [];
groupBy eq (x : xs) = (x : groupByYs eq x xs) : groupBy eq (groupByZs eq x xs);

groupByVv10 xz yu yv = span (xz yu) yv;

groupByYs xz yu yv = groupByYs1 xz yu yv (groupByVv10 xz yu yv);

groupByYs1 xz yu yv (ys,wx) = ys;

groupByZs xz yu yv = groupByZs1 xz yu yv (groupByVv10 xz yu yv);

groupByZs1 xz yu yv (wy,zs) = zs;

}
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.groupBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.groupBy yw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
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113 -> 5[label="",style="solid", color="burlywood", weight=3];
114[label="yw4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 6[label="",style="solid", color="burlywood", weight=3];
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6[label="List.groupBy yw3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(yw40 : List.groupByYs yw3 yw40 yw41) : List.groupBy yw3 (List.groupByZs yw3 yw40 yw41)",fontsize=16,color="green",shape="box"];7 -> 9[label="",style="dashed", color="green", weight=3];
7 -> 10[label="",style="dashed", color="green", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="List.groupByYs yw3 yw40 yw41",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="List.groupBy yw3 (List.groupByZs yw3 yw40 yw41)",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
11[label="List.groupByYs1 yw3 yw40 yw41 (List.groupByVv10 yw3 yw40 yw41)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="List.groupByZs yw3 yw40 yw41",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="List.groupByYs1 yw3 yw40 yw41 (span (yw3 yw40) yw41)",fontsize=16,color="burlywood",shape="box"];115[label="yw41/yw410 : yw411",fontsize=10,color="white",style="solid",shape="box"];13 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 15[label="",style="solid", color="burlywood", weight=3];
116[label="yw41/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="List.groupByZs1 yw3 yw40 yw41 (List.groupByVv10 yw3 yw40 yw41)",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="List.groupByYs1 yw3 yw40 [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="List.groupByZs1 yw3 yw40 yw41 (span (yw3 yw40) yw41)",fontsize=16,color="burlywood",shape="box"];117[label="yw41/yw410 : yw411",fontsize=10,color="white",style="solid",shape="box"];17 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 20[label="",style="solid", color="burlywood", weight=3];
118[label="yw41/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2 (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.groupByYs1 yw3 yw40 [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.groupByZs1 yw3 yw40 [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22 -> 26[label="",style="dashed", color="red", weight=0];
22[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 (yw3 yw40 yw410))",fontsize=16,color="magenta"];22 -> 27[label="",style="dashed", color="magenta", weight=3];
23[label="List.groupByYs1 yw3 yw40 [] ([],[])",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2 (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
25[label="List.groupByZs1 yw3 yw40 [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
27[label="yw3 yw40 yw410",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
27 -> 36[label="",style="dashed", color="green", weight=3];
26[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 yw5)",fontsize=16,color="burlywood",shape="triangle"];119[label="yw5/False",fontsize=10,color="white",style="solid",shape="box"];26 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 33[label="",style="solid", color="burlywood", weight=3];
120[label="yw5/True",fontsize=10,color="white",style="solid",shape="box"];26 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 34[label="",style="solid", color="burlywood", weight=3];
28[label="[]",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="red", weight=0];
29[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 (yw3 yw40 yw410))",fontsize=16,color="magenta"];29 -> 38[label="",style="dashed", color="magenta", weight=3];
30[label="List.groupByZs1 yw3 yw40 [] ([],[])",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
35[label="yw40",fontsize=16,color="green",shape="box"];36[label="yw410",fontsize=16,color="green",shape="box"];33[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 False)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
34[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3];
38[label="yw3 yw40 yw410",fontsize=16,color="green",shape="box"];38 -> 46[label="",style="dashed", color="green", weight=3];
38 -> 47[label="",style="dashed", color="green", weight=3];
37[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 yw6)",fontsize=16,color="burlywood",shape="triangle"];121[label="yw6/False",fontsize=10,color="white",style="solid",shape="box"];37 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 44[label="",style="solid", color="burlywood", weight=3];
122[label="yw6/True",fontsize=10,color="white",style="solid",shape="box"];37 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 45[label="",style="solid", color="burlywood", weight=3];
39[label="[]",fontsize=16,color="green",shape="box"];40[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 otherwise)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (yw410 : span2Ys (yw3 yw40) yw411,span2Zs (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
46[label="yw40",fontsize=16,color="green",shape="box"];47[label="yw410",fontsize=16,color="green",shape="box"];44[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 False)",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
48[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
49[label="yw410 : span2Ys (yw3 yw40) yw411",fontsize=16,color="green",shape="box"];49 -> 53[label="",style="dashed", color="green", weight=3];
50[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 otherwise)",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (yw410 : span2Ys (yw3 yw40) yw411,span2Zs (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) ([],yw410 : yw411)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="span2Ys (yw3 yw40) yw411",fontsize=16,color="black",shape="triangle"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="span2Zs (yw3 yw40) yw411",fontsize=16,color="black",shape="triangle"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="[]",fontsize=16,color="green",shape="box"];57[label="span2Ys0 (yw3 yw40) yw411 (span2Vu43 (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
58[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) ([],yw410 : yw411)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="span2Zs0 (yw3 yw40) yw411 (span2Vu43 (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="span2Ys0 (yw3 yw40) yw411 (span (yw3 yw40) yw411)",fontsize=16,color="burlywood",shape="box"];123[label="yw411/yw4110 : yw4111",fontsize=10,color="white",style="solid",shape="box"];60 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 63[label="",style="solid", color="burlywood", weight=3];
124[label="yw411/[]",fontsize=10,color="white",style="solid",shape="box"];60 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 64[label="",style="solid", color="burlywood", weight=3];
61[label="yw410 : yw411",fontsize=16,color="green",shape="box"];62[label="span2Zs0 (yw3 yw40) yw411 (span (yw3 yw40) yw411)",fontsize=16,color="burlywood",shape="box"];125[label="yw411/yw4110 : yw4111",fontsize=10,color="white",style="solid",shape="box"];62 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 65[label="",style="solid", color="burlywood", weight=3];
126[label="yw411/[]",fontsize=10,color="white",style="solid",shape="box"];62 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 66[label="",style="solid", color="burlywood", weight=3];
63[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64[label="span2Ys0 (yw3 yw40) [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
65[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66[label="span2Zs0 (yw3 yw40) [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2 (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="span2Ys0 (yw3 yw40) [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2 (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
70[label="span2Zs0 (yw3 yw40) [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];70 -> 74[label="",style="solid", color="black", weight=3];
71 -> 75[label="",style="dashed", color="red", weight=0];
71[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 (yw3 yw40 yw4110))",fontsize=16,color="magenta"];71 -> 76[label="",style="dashed", color="magenta", weight=3];
72[label="span2Ys0 (yw3 yw40) [] ([],[])",fontsize=16,color="black",shape="box"];72 -> 77[label="",style="solid", color="black", weight=3];
73 -> 78[label="",style="dashed", color="red", weight=0];
73[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 (yw3 yw40 yw4110))",fontsize=16,color="magenta"];73 -> 79[label="",style="dashed", color="magenta", weight=3];
74[label="span2Zs0 (yw3 yw40) [] ([],[])",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
76[label="yw3 yw40 yw4110",fontsize=16,color="green",shape="box"];76 -> 81[label="",style="dashed", color="green", weight=3];
76 -> 82[label="",style="dashed", color="green", weight=3];
75[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 yw7)",fontsize=16,color="burlywood",shape="triangle"];127[label="yw7/False",fontsize=10,color="white",style="solid",shape="box"];75 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 83[label="",style="solid", color="burlywood", weight=3];
128[label="yw7/True",fontsize=10,color="white",style="solid",shape="box"];75 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 84[label="",style="solid", color="burlywood", weight=3];
77[label="[]",fontsize=16,color="green",shape="box"];79[label="yw3 yw40 yw4110",fontsize=16,color="green",shape="box"];79 -> 89[label="",style="dashed", color="green", weight=3];
79 -> 90[label="",style="dashed", color="green", weight=3];
78[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 yw8)",fontsize=16,color="burlywood",shape="triangle"];129[label="yw8/False",fontsize=10,color="white",style="solid",shape="box"];78 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 87[label="",style="solid", color="burlywood", weight=3];
130[label="yw8/True",fontsize=10,color="white",style="solid",shape="box"];78 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 88[label="",style="solid", color="burlywood", weight=3];
80[label="[]",fontsize=16,color="green",shape="box"];81[label="yw40",fontsize=16,color="green",shape="box"];82[label="yw4110",fontsize=16,color="green",shape="box"];83[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 False)",fontsize=16,color="black",shape="box"];83 -> 91[label="",style="solid", color="black", weight=3];
84[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3];
89[label="yw40",fontsize=16,color="green",shape="box"];90[label="yw4110",fontsize=16,color="green",shape="box"];87[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 False)",fontsize=16,color="black",shape="box"];87 -> 93[label="",style="solid", color="black", weight=3];
88[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];88 -> 94[label="",style="solid", color="black", weight=3];
91[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 otherwise)",fontsize=16,color="black",shape="box"];91 -> 95[label="",style="solid", color="black", weight=3];
92 -> 96[label="",style="dashed", color="red", weight=0];
92[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : span2Ys (yw3 yw40) yw4111,span2Zs (yw3 yw40) yw4111)",fontsize=16,color="magenta"];92 -> 97[label="",style="dashed", color="magenta", weight=3];
92 -> 98[label="",style="dashed", color="magenta", weight=3];
93[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 otherwise)",fontsize=16,color="black",shape="box"];93 -> 99[label="",style="solid", color="black", weight=3];
94 -> 100[label="",style="dashed", color="red", weight=0];
94[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : span2Ys (yw3 yw40) yw4111,span2Zs (yw3 yw40) yw4111)",fontsize=16,color="magenta"];94 -> 101[label="",style="dashed", color="magenta", weight=3];
94 -> 102[label="",style="dashed", color="magenta", weight=3];
95[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];95 -> 103[label="",style="solid", color="black", weight=3];
97 -> 55[label="",style="dashed", color="red", weight=0];
97[label="span2Zs (yw3 yw40) yw4111",fontsize=16,color="magenta"];97 -> 104[label="",style="dashed", color="magenta", weight=3];
98 -> 53[label="",style="dashed", color="red", weight=0];
98[label="span2Ys (yw3 yw40) yw4111",fontsize=16,color="magenta"];98 -> 105[label="",style="dashed", color="magenta", weight=3];
96[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : yw10,yw9)",fontsize=16,color="black",shape="triangle"];96 -> 106[label="",style="solid", color="black", weight=3];
99[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];99 -> 107[label="",style="solid", color="black", weight=3];
101 -> 55[label="",style="dashed", color="red", weight=0];
101[label="span2Zs (yw3 yw40) yw4111",fontsize=16,color="magenta"];101 -> 108[label="",style="dashed", color="magenta", weight=3];
102 -> 53[label="",style="dashed", color="red", weight=0];
102[label="span2Ys (yw3 yw40) yw4111",fontsize=16,color="magenta"];102 -> 109[label="",style="dashed", color="magenta", weight=3];
100[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : yw12,yw11)",fontsize=16,color="black",shape="triangle"];100 -> 110[label="",style="solid", color="black", weight=3];
103[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) ([],yw4110 : yw4111)",fontsize=16,color="black",shape="box"];103 -> 111[label="",style="solid", color="black", weight=3];
104[label="yw4111",fontsize=16,color="green",shape="box"];105[label="yw4111",fontsize=16,color="green",shape="box"];106[label="yw4110 : yw10",fontsize=16,color="green",shape="box"];107[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) ([],yw4110 : yw4111)",fontsize=16,color="black",shape="box"];107 -> 112[label="",style="solid", color="black", weight=3];
108[label="yw4111",fontsize=16,color="green",shape="box"];109[label="yw4111",fontsize=16,color="green",shape="box"];110[label="yw11",fontsize=16,color="green",shape="box"];111[label="[]",fontsize=16,color="green",shape="box"];112[label="yw4110 : yw4111",fontsize=16,color="green",shape="box"];}

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

(10)
Complex Obligation (AND)

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

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

   new_span2Zs0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Ys(yw3, yw40, yw4111, ba)
   new_span2Ys0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Ys(yw3, yw40, yw4111, ba)
   new_span2Zs(yw3, yw40, :(yw4110, yw4111), ba) -> new_span2Zs0(yw3, yw40, yw4110, yw4111, ba)
   new_span2Zs0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Zs(yw3, yw40, yw4111, ba)
   new_span2Ys0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Zs(yw3, yw40, yw4111, ba)
   new_span2Ys(yw3, yw40, :(yw4110, yw4111), ba) -> new_span2Ys0(yw3, yw40, yw4110, yw4111, 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_span2Ys(yw3, yw40, :(yw4110, yw4111), ba) -> new_span2Ys0(yw3, yw40, yw4110, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5


*new_span2Zs(yw3, yw40, :(yw4110, yw4111), ba) -> new_span2Zs0(yw3, yw40, yw4110, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5


*new_span2Zs0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Zs(yw3, yw40, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 5 >= 4


*new_span2Zs0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Ys(yw3, yw40, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 5 >= 4


*new_span2Ys0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Zs(yw3, yw40, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 5 >= 4


*new_span2Ys0(yw3, yw40, yw4110, yw4111, ba) -> new_span2Ys(yw3, yw40, yw4111, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 5 >= 4


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

(13)
YES

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

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

   new_groupBy(yw3, ba) -> new_groupBy(yw3, ba)

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

(15) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = new_groupBy(yw3, ba) evaluates to  t =new_groupBy(yw3, ba)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_groupBy(yw3, ba) to new_groupBy(yw3, ba).




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

(16)
NO

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

(17) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.groupBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.groupBy yw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.groupBy yw3 yw4",fontsize=16,color="burlywood",shape="triangle"];113[label="yw4/yw40 : yw41",fontsize=10,color="white",style="solid",shape="box"];4 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 5[label="",style="solid", color="burlywood", weight=3];
114[label="yw4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.groupBy yw3 (yw40 : yw41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="List.groupBy yw3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(yw40 : List.groupByYs yw3 yw40 yw41) : List.groupBy yw3 (List.groupByZs yw3 yw40 yw41)",fontsize=16,color="green",shape="box"];7 -> 9[label="",style="dashed", color="green", weight=3];
7 -> 10[label="",style="dashed", color="green", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];9[label="List.groupByYs yw3 yw40 yw41",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="List.groupBy yw3 (List.groupByZs yw3 yw40 yw41)",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
11[label="List.groupByYs1 yw3 yw40 yw41 (List.groupByVv10 yw3 yw40 yw41)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="List.groupByZs yw3 yw40 yw41",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="List.groupByYs1 yw3 yw40 yw41 (span (yw3 yw40) yw41)",fontsize=16,color="burlywood",shape="box"];115[label="yw41/yw410 : yw411",fontsize=10,color="white",style="solid",shape="box"];13 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 15[label="",style="solid", color="burlywood", weight=3];
116[label="yw41/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="List.groupByZs1 yw3 yw40 yw41 (List.groupByVv10 yw3 yw40 yw41)",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="List.groupByYs1 yw3 yw40 [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="List.groupByZs1 yw3 yw40 yw41 (span (yw3 yw40) yw41)",fontsize=16,color="burlywood",shape="box"];117[label="yw41/yw410 : yw411",fontsize=10,color="white",style="solid",shape="box"];17 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 20[label="",style="solid", color="burlywood", weight=3];
118[label="yw41/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2 (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.groupByYs1 yw3 yw40 [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.groupByZs1 yw3 yw40 [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22 -> 26[label="",style="dashed", color="red", weight=0];
22[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 (yw3 yw40 yw410))",fontsize=16,color="magenta"];22 -> 27[label="",style="dashed", color="magenta", weight=3];
23[label="List.groupByYs1 yw3 yw40 [] ([],[])",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2 (yw3 yw40) (yw410 : yw411))",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
25[label="List.groupByZs1 yw3 yw40 [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
27[label="yw3 yw40 yw410",fontsize=16,color="green",shape="box"];27 -> 35[label="",style="dashed", color="green", weight=3];
27 -> 36[label="",style="dashed", color="green", weight=3];
26[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 yw5)",fontsize=16,color="burlywood",shape="triangle"];119[label="yw5/False",fontsize=10,color="white",style="solid",shape="box"];26 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 33[label="",style="solid", color="burlywood", weight=3];
120[label="yw5/True",fontsize=10,color="white",style="solid",shape="box"];26 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 34[label="",style="solid", color="burlywood", weight=3];
28[label="[]",fontsize=16,color="green",shape="box"];29 -> 37[label="",style="dashed", color="red", weight=0];
29[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 (yw3 yw40 yw410))",fontsize=16,color="magenta"];29 -> 38[label="",style="dashed", color="magenta", weight=3];
30[label="List.groupByZs1 yw3 yw40 [] ([],[])",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
35[label="yw40",fontsize=16,color="green",shape="box"];36[label="yw410",fontsize=16,color="green",shape="box"];33[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 False)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
34[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3];
38[label="yw3 yw40 yw410",fontsize=16,color="green",shape="box"];38 -> 46[label="",style="dashed", color="green", weight=3];
38 -> 47[label="",style="dashed", color="green", weight=3];
37[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 yw6)",fontsize=16,color="burlywood",shape="triangle"];121[label="yw6/False",fontsize=10,color="white",style="solid",shape="box"];37 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 44[label="",style="solid", color="burlywood", weight=3];
122[label="yw6/True",fontsize=10,color="white",style="solid",shape="box"];37 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 45[label="",style="solid", color="burlywood", weight=3];
39[label="[]",fontsize=16,color="green",shape="box"];40[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 otherwise)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (yw410 : span2Ys (yw3 yw40) yw411,span2Zs (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
46[label="yw40",fontsize=16,color="green",shape="box"];47[label="yw410",fontsize=16,color="green",shape="box"];44[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 False)",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span1 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
48[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
49[label="yw410 : span2Ys (yw3 yw40) yw411",fontsize=16,color="green",shape="box"];49 -> 53[label="",style="dashed", color="green", weight=3];
50[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 otherwise)",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (yw410 : span2Ys (yw3 yw40) yw411,span2Zs (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="List.groupByYs1 yw3 yw40 (yw410 : yw411) ([],yw410 : yw411)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="span2Ys (yw3 yw40) yw411",fontsize=16,color="black",shape="triangle"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) (span2Span0 (yw3 yw40) yw411 (yw3 yw40) yw410 yw411 True)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="span2Zs (yw3 yw40) yw411",fontsize=16,color="black",shape="triangle"];55 -> 59[label="",style="solid", color="black", weight=3];
56[label="[]",fontsize=16,color="green",shape="box"];57[label="span2Ys0 (yw3 yw40) yw411 (span2Vu43 (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
58[label="List.groupByZs1 yw3 yw40 (yw410 : yw411) ([],yw410 : yw411)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="span2Zs0 (yw3 yw40) yw411 (span2Vu43 (yw3 yw40) yw411)",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="span2Ys0 (yw3 yw40) yw411 (span (yw3 yw40) yw411)",fontsize=16,color="burlywood",shape="box"];123[label="yw411/yw4110 : yw4111",fontsize=10,color="white",style="solid",shape="box"];60 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 63[label="",style="solid", color="burlywood", weight=3];
124[label="yw411/[]",fontsize=10,color="white",style="solid",shape="box"];60 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 64[label="",style="solid", color="burlywood", weight=3];
61[label="yw410 : yw411",fontsize=16,color="green",shape="box"];62[label="span2Zs0 (yw3 yw40) yw411 (span (yw3 yw40) yw411)",fontsize=16,color="burlywood",shape="box"];125[label="yw411/yw4110 : yw4111",fontsize=10,color="white",style="solid",shape="box"];62 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 65[label="",style="solid", color="burlywood", weight=3];
126[label="yw411/[]",fontsize=10,color="white",style="solid",shape="box"];62 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 66[label="",style="solid", color="burlywood", weight=3];
63[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64[label="span2Ys0 (yw3 yw40) [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
65[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66[label="span2Zs0 (yw3 yw40) [] (span (yw3 yw40) [])",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2 (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="span2Ys0 (yw3 yw40) [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2 (yw3 yw40) (yw4110 : yw4111))",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
70[label="span2Zs0 (yw3 yw40) [] (span3 (yw3 yw40) [])",fontsize=16,color="black",shape="box"];70 -> 74[label="",style="solid", color="black", weight=3];
71 -> 75[label="",style="dashed", color="red", weight=0];
71[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 (yw3 yw40 yw4110))",fontsize=16,color="magenta"];71 -> 76[label="",style="dashed", color="magenta", weight=3];
72[label="span2Ys0 (yw3 yw40) [] ([],[])",fontsize=16,color="black",shape="box"];72 -> 77[label="",style="solid", color="black", weight=3];
73 -> 78[label="",style="dashed", color="red", weight=0];
73[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 (yw3 yw40 yw4110))",fontsize=16,color="magenta"];73 -> 79[label="",style="dashed", color="magenta", weight=3];
74[label="span2Zs0 (yw3 yw40) [] ([],[])",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
76[label="yw3 yw40 yw4110",fontsize=16,color="green",shape="box"];76 -> 81[label="",style="dashed", color="green", weight=3];
76 -> 82[label="",style="dashed", color="green", weight=3];
75[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 yw7)",fontsize=16,color="burlywood",shape="triangle"];127[label="yw7/False",fontsize=10,color="white",style="solid",shape="box"];75 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 83[label="",style="solid", color="burlywood", weight=3];
128[label="yw7/True",fontsize=10,color="white",style="solid",shape="box"];75 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 84[label="",style="solid", color="burlywood", weight=3];
77[label="[]",fontsize=16,color="green",shape="box"];79[label="yw3 yw40 yw4110",fontsize=16,color="green",shape="box"];79 -> 89[label="",style="dashed", color="green", weight=3];
79 -> 90[label="",style="dashed", color="green", weight=3];
78[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 yw8)",fontsize=16,color="burlywood",shape="triangle"];129[label="yw8/False",fontsize=10,color="white",style="solid",shape="box"];78 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 87[label="",style="solid", color="burlywood", weight=3];
130[label="yw8/True",fontsize=10,color="white",style="solid",shape="box"];78 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 88[label="",style="solid", color="burlywood", weight=3];
80[label="[]",fontsize=16,color="green",shape="box"];81[label="yw40",fontsize=16,color="green",shape="box"];82[label="yw4110",fontsize=16,color="green",shape="box"];83[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 False)",fontsize=16,color="black",shape="box"];83 -> 91[label="",style="solid", color="black", weight=3];
84[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3];
89[label="yw40",fontsize=16,color="green",shape="box"];90[label="yw4110",fontsize=16,color="green",shape="box"];87[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 False)",fontsize=16,color="black",shape="box"];87 -> 93[label="",style="solid", color="black", weight=3];
88[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span1 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];88 -> 94[label="",style="solid", color="black", weight=3];
91[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 otherwise)",fontsize=16,color="black",shape="box"];91 -> 95[label="",style="solid", color="black", weight=3];
92 -> 96[label="",style="dashed", color="red", weight=0];
92[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : span2Ys (yw3 yw40) yw4111,span2Zs (yw3 yw40) yw4111)",fontsize=16,color="magenta"];92 -> 97[label="",style="dashed", color="magenta", weight=3];
92 -> 98[label="",style="dashed", color="magenta", weight=3];
93[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 otherwise)",fontsize=16,color="black",shape="box"];93 -> 99[label="",style="solid", color="black", weight=3];
94 -> 100[label="",style="dashed", color="red", weight=0];
94[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : span2Ys (yw3 yw40) yw4111,span2Zs (yw3 yw40) yw4111)",fontsize=16,color="magenta"];94 -> 101[label="",style="dashed", color="magenta", weight=3];
94 -> 102[label="",style="dashed", color="magenta", weight=3];
95[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];95 -> 103[label="",style="solid", color="black", weight=3];
97 -> 55[label="",style="dashed", color="red", weight=0];
97[label="span2Zs (yw3 yw40) yw4111",fontsize=16,color="magenta"];97 -> 104[label="",style="dashed", color="magenta", weight=3];
98 -> 53[label="",style="dashed", color="red", weight=0];
98[label="span2Ys (yw3 yw40) yw4111",fontsize=16,color="magenta"];98 -> 105[label="",style="dashed", color="magenta", weight=3];
96[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : yw10,yw9)",fontsize=16,color="black",shape="triangle"];96 -> 106[label="",style="solid", color="black", weight=3];
99[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (span2Span0 (yw3 yw40) yw4111 (yw3 yw40) yw4110 yw4111 True)",fontsize=16,color="black",shape="box"];99 -> 107[label="",style="solid", color="black", weight=3];
101 -> 55[label="",style="dashed", color="red", weight=0];
101[label="span2Zs (yw3 yw40) yw4111",fontsize=16,color="magenta"];101 -> 108[label="",style="dashed", color="magenta", weight=3];
102 -> 53[label="",style="dashed", color="red", weight=0];
102[label="span2Ys (yw3 yw40) yw4111",fontsize=16,color="magenta"];102 -> 109[label="",style="dashed", color="magenta", weight=3];
100[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) (yw4110 : yw12,yw11)",fontsize=16,color="black",shape="triangle"];100 -> 110[label="",style="solid", color="black", weight=3];
103[label="span2Ys0 (yw3 yw40) (yw4110 : yw4111) ([],yw4110 : yw4111)",fontsize=16,color="black",shape="box"];103 -> 111[label="",style="solid", color="black", weight=3];
104[label="yw4111",fontsize=16,color="green",shape="box"];105[label="yw4111",fontsize=16,color="green",shape="box"];106[label="yw4110 : yw10",fontsize=16,color="green",shape="box"];107[label="span2Zs0 (yw3 yw40) (yw4110 : yw4111) ([],yw4110 : yw4111)",fontsize=16,color="black",shape="box"];107 -> 112[label="",style="solid", color="black", weight=3];
108[label="yw4111",fontsize=16,color="green",shape="box"];109[label="yw4111",fontsize=16,color="green",shape="box"];110[label="yw11",fontsize=16,color="green",shape="box"];111[label="[]",fontsize=16,color="green",shape="box"];112[label="yw4110 : yw4111",fontsize=16,color="green",shape="box"];}

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

(18)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(19) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(20)
YES