/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 18 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) AND
    (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) 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;
group :: Eq a => [a]  ->  [[a]];
group = groupBy (==);

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;
group :: Eq a => [a]  ->  [[a]];
group = groupBy (==);

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;
group :: Eq a => [a]  ->  [[a]];
group = groupBy (==);

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;
group :: Eq a => [a]  ->  [[a]];
group = groupBy (==);

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
"span2Ys0 xx xy (ys,wu) = ys;
"
"span2Vu43 xx xy = span xx xy;
"
"span2Ys xx xy = span2Ys0 xx xy (span2Vu43 xx xy);
"
"span2Span0 xx xy p vx vy True = ([],vx : vy);
"
"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;
"
"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
"groupByYs1 xz yu yv (ys,wx) = ys;
"
"groupByZs1 xz yu yv (wy,zs) = zs;
"
"groupByZs xz yu yv = groupByZs1 xz yu yv (groupByVv10 xz yu yv);
"
"groupByVv10 xz yu yv = span (xz yu) yv;
"
"groupByYs xz yu yv = groupByYs1 xz yu yv (groupByVv10 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;
group :: Eq a => [a]  ->  [[a]];
group = groupBy (==);

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.group",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.group yw3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="List.groupBy (==) yw3",fontsize=16,color="burlywood",shape="triangle"];172[label="yw3/yw30 : yw31",fontsize=10,color="white",style="solid",shape="box"];4 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 5[label="",style="solid", color="burlywood", weight=3];
173[label="yw3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.groupBy (==) (yw30 : yw31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="List.groupBy (==) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(yw30 : List.groupByYs (==) yw30 yw31) : List.groupBy (==) (List.groupByZs (==) yw30 yw31)",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 (==) yw30 yw31",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 (==) (List.groupByZs (==) yw30 yw31)",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
11[label="List.groupByYs1 (==) yw30 yw31 (List.groupByVv10 (==) yw30 yw31)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="List.groupByZs (==) yw30 yw31",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="List.groupByYs1 (==) yw30 yw31 (span ((==) yw30) yw31)",fontsize=16,color="burlywood",shape="box"];174[label="yw31/yw310 : yw311",fontsize=10,color="white",style="solid",shape="box"];13 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 15[label="",style="solid", color="burlywood", weight=3];
175[label="yw31/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 175[label="",style="solid", color="burlywood", weight=9];
175 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="List.groupByZs1 (==) yw30 yw31 (List.groupByVv10 (==) yw30 yw31)",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="List.groupByYs1 (==) yw30 (yw310 : yw311) (span ((==) yw30) (yw310 : yw311))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
16[label="List.groupByYs1 (==) yw30 [] (span ((==) yw30) [])",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="List.groupByZs1 (==) yw30 yw31 (span ((==) yw30) yw31)",fontsize=16,color="burlywood",shape="box"];176[label="yw31/yw310 : yw311",fontsize=10,color="white",style="solid",shape="box"];17 -> 176[label="",style="solid", color="burlywood", weight=9];
176 -> 20[label="",style="solid", color="burlywood", weight=3];
177[label="yw31/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 177[label="",style="solid", color="burlywood", weight=9];
177 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="List.groupByYs1 (==) yw30 (yw310 : yw311) (span2 ((==) yw30) (yw310 : yw311))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.groupByYs1 (==) yw30 [] (span3 ((==) yw30) [])",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="List.groupByZs1 (==) yw30 (yw310 : yw311) (span ((==) yw30) (yw310 : yw311))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.groupByZs1 (==) yw30 [] (span ((==) yw30) [])",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="List.groupByYs1 (==) yw30 (yw310 : yw311) (span2Span1 ((==) yw30) yw311 ((==) yw30) yw310 yw311 ((==) yw30 yw310))",fontsize=16,color="burlywood",shape="box"];178[label="yw30/False",fontsize=10,color="white",style="solid",shape="box"];22 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 26[label="",style="solid", color="burlywood", weight=3];
179[label="yw30/True",fontsize=10,color="white",style="solid",shape="box"];22 -> 179[label="",style="solid", color="burlywood", weight=9];
179 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="List.groupByYs1 (==) yw30 [] ([],[])",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="List.groupByZs1 (==) yw30 (yw310 : yw311) (span2 ((==) yw30) (yw310 : yw311))",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
25[label="List.groupByZs1 (==) yw30 [] (span3 ((==) yw30) [])",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
26[label="List.groupByYs1 (==) False (yw310 : yw311) (span2Span1 ((==) False) yw311 ((==) False) yw310 yw311 ((==) False yw310))",fontsize=16,color="burlywood",shape="box"];180[label="yw310/False",fontsize=10,color="white",style="solid",shape="box"];26 -> 180[label="",style="solid", color="burlywood", weight=9];
180 -> 31[label="",style="solid", color="burlywood", weight=3];
181[label="yw310/True",fontsize=10,color="white",style="solid",shape="box"];26 -> 181[label="",style="solid", color="burlywood", weight=9];
181 -> 32[label="",style="solid", color="burlywood", weight=3];
27[label="List.groupByYs1 (==) True (yw310 : yw311) (span2Span1 ((==) True) yw311 ((==) True) yw310 yw311 ((==) True yw310))",fontsize=16,color="burlywood",shape="box"];182[label="yw310/False",fontsize=10,color="white",style="solid",shape="box"];27 -> 182[label="",style="solid", color="burlywood", weight=9];
182 -> 33[label="",style="solid", color="burlywood", weight=3];
183[label="yw310/True",fontsize=10,color="white",style="solid",shape="box"];27 -> 183[label="",style="solid", color="burlywood", weight=9];
183 -> 34[label="",style="solid", color="burlywood", weight=3];
28[label="[]",fontsize=16,color="green",shape="box"];29[label="List.groupByZs1 (==) yw30 (yw310 : yw311) (span2Span1 ((==) yw30) yw311 ((==) yw30) yw310 yw311 ((==) yw30 yw310))",fontsize=16,color="burlywood",shape="box"];184[label="yw30/False",fontsize=10,color="white",style="solid",shape="box"];29 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 35[label="",style="solid", color="burlywood", weight=3];
185[label="yw30/True",fontsize=10,color="white",style="solid",shape="box"];29 -> 185[label="",style="solid", color="burlywood", weight=9];
185 -> 36[label="",style="solid", color="burlywood", weight=3];
30[label="List.groupByZs1 (==) yw30 [] ([],[])",fontsize=16,color="black",shape="box"];30 -> 37[label="",style="solid", color="black", weight=3];
31[label="List.groupByYs1 (==) False (False : yw311) (span2Span1 ((==) False) yw311 ((==) False) False yw311 ((==) False False))",fontsize=16,color="black",shape="box"];31 -> 38[label="",style="solid", color="black", weight=3];
32[label="List.groupByYs1 (==) False (True : yw311) (span2Span1 ((==) False) yw311 ((==) False) True yw311 ((==) False True))",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3];
33[label="List.groupByYs1 (==) True (False : yw311) (span2Span1 ((==) True) yw311 ((==) True) False yw311 ((==) True False))",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
34[label="List.groupByYs1 (==) True (True : yw311) (span2Span1 ((==) True) yw311 ((==) True) True yw311 ((==) True True))",fontsize=16,color="black",shape="box"];34 -> 41[label="",style="solid", color="black", weight=3];
35[label="List.groupByZs1 (==) False (yw310 : yw311) (span2Span1 ((==) False) yw311 ((==) False) yw310 yw311 ((==) False yw310))",fontsize=16,color="burlywood",shape="box"];186[label="yw310/False",fontsize=10,color="white",style="solid",shape="box"];35 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 42[label="",style="solid", color="burlywood", weight=3];
187[label="yw310/True",fontsize=10,color="white",style="solid",shape="box"];35 -> 187[label="",style="solid", color="burlywood", weight=9];
187 -> 43[label="",style="solid", color="burlywood", weight=3];
36[label="List.groupByZs1 (==) True (yw310 : yw311) (span2Span1 ((==) True) yw311 ((==) True) yw310 yw311 ((==) True yw310))",fontsize=16,color="burlywood",shape="box"];188[label="yw310/False",fontsize=10,color="white",style="solid",shape="box"];36 -> 188[label="",style="solid", color="burlywood", weight=9];
188 -> 44[label="",style="solid", color="burlywood", weight=3];
189[label="yw310/True",fontsize=10,color="white",style="solid",shape="box"];36 -> 189[label="",style="solid", color="burlywood", weight=9];
189 -> 45[label="",style="solid", color="burlywood", weight=3];
37[label="[]",fontsize=16,color="green",shape="box"];38[label="List.groupByYs1 (==) False (False : yw311) (span2Span1 ((==) False) yw311 ((==) False) False yw311 True)",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
39[label="List.groupByYs1 (==) False (True : yw311) (span2Span1 ((==) False) yw311 ((==) False) True yw311 False)",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
40[label="List.groupByYs1 (==) True (False : yw311) (span2Span1 ((==) True) yw311 ((==) True) False yw311 False)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="List.groupByYs1 (==) True (True : yw311) (span2Span1 ((==) True) yw311 ((==) True) True yw311 True)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
42[label="List.groupByZs1 (==) False (False : yw311) (span2Span1 ((==) False) yw311 ((==) False) False yw311 ((==) False False))",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="List.groupByZs1 (==) False (True : yw311) (span2Span1 ((==) False) yw311 ((==) False) True yw311 ((==) False True))",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="List.groupByZs1 (==) True (False : yw311) (span2Span1 ((==) True) yw311 ((==) True) False yw311 ((==) True False))",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="List.groupByZs1 (==) True (True : yw311) (span2Span1 ((==) True) yw311 ((==) True) True yw311 ((==) True True))",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46[label="List.groupByYs1 (==) False (False : yw311) (False : span2Ys ((==) False) yw311,span2Zs ((==) False) yw311)",fontsize=16,color="black",shape="box"];46 -> 54[label="",style="solid", color="black", weight=3];
47[label="List.groupByYs1 (==) False (True : yw311) (span2Span0 ((==) False) yw311 ((==) False) True yw311 otherwise)",fontsize=16,color="black",shape="box"];47 -> 55[label="",style="solid", color="black", weight=3];
48[label="List.groupByYs1 (==) True (False : yw311) (span2Span0 ((==) True) yw311 ((==) True) False yw311 otherwise)",fontsize=16,color="black",shape="box"];48 -> 56[label="",style="solid", color="black", weight=3];
49[label="List.groupByYs1 (==) True (True : yw311) (True : span2Ys ((==) True) yw311,span2Zs ((==) True) yw311)",fontsize=16,color="black",shape="box"];49 -> 57[label="",style="solid", color="black", weight=3];
50[label="List.groupByZs1 (==) False (False : yw311) (span2Span1 ((==) False) yw311 ((==) False) False yw311 True)",fontsize=16,color="black",shape="box"];50 -> 58[label="",style="solid", color="black", weight=3];
51[label="List.groupByZs1 (==) False (True : yw311) (span2Span1 ((==) False) yw311 ((==) False) True yw311 False)",fontsize=16,color="black",shape="box"];51 -> 59[label="",style="solid", color="black", weight=3];
52[label="List.groupByZs1 (==) True (False : yw311) (span2Span1 ((==) True) yw311 ((==) True) False yw311 False)",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
53[label="List.groupByZs1 (==) True (True : yw311) (span2Span1 ((==) True) yw311 ((==) True) True yw311 True)",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
54[label="False : span2Ys ((==) False) yw311",fontsize=16,color="green",shape="box"];54 -> 62[label="",style="dashed", color="green", weight=3];
55[label="List.groupByYs1 (==) False (True : yw311) (span2Span0 ((==) False) yw311 ((==) False) True yw311 True)",fontsize=16,color="black",shape="box"];55 -> 63[label="",style="solid", color="black", weight=3];
56[label="List.groupByYs1 (==) True (False : yw311) (span2Span0 ((==) True) yw311 ((==) True) False yw311 True)",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3];
57[label="True : span2Ys ((==) True) yw311",fontsize=16,color="green",shape="box"];57 -> 65[label="",style="dashed", color="green", weight=3];
58[label="List.groupByZs1 (==) False (False : yw311) (False : span2Ys ((==) False) yw311,span2Zs ((==) False) yw311)",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3];
59[label="List.groupByZs1 (==) False (True : yw311) (span2Span0 ((==) False) yw311 ((==) False) True yw311 otherwise)",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3];
60[label="List.groupByZs1 (==) True (False : yw311) (span2Span0 ((==) True) yw311 ((==) True) False yw311 otherwise)",fontsize=16,color="black",shape="box"];60 -> 68[label="",style="solid", color="black", weight=3];
61[label="List.groupByZs1 (==) True (True : yw311) (True : span2Ys ((==) True) yw311,span2Zs ((==) True) yw311)",fontsize=16,color="black",shape="box"];61 -> 69[label="",style="solid", color="black", weight=3];
62[label="span2Ys ((==) False) yw311",fontsize=16,color="black",shape="triangle"];62 -> 70[label="",style="solid", color="black", weight=3];
63[label="List.groupByYs1 (==) False (True : yw311) ([],True : yw311)",fontsize=16,color="black",shape="box"];63 -> 71[label="",style="solid", color="black", weight=3];
64[label="List.groupByYs1 (==) True (False : yw311) ([],False : yw311)",fontsize=16,color="black",shape="box"];64 -> 72[label="",style="solid", color="black", weight=3];
65[label="span2Ys ((==) True) yw311",fontsize=16,color="black",shape="triangle"];65 -> 73[label="",style="solid", color="black", weight=3];
66[label="span2Zs ((==) False) yw311",fontsize=16,color="black",shape="triangle"];66 -> 74[label="",style="solid", color="black", weight=3];
67[label="List.groupByZs1 (==) False (True : yw311) (span2Span0 ((==) False) yw311 ((==) False) True yw311 True)",fontsize=16,color="black",shape="box"];67 -> 75[label="",style="solid", color="black", weight=3];
68[label="List.groupByZs1 (==) True (False : yw311) (span2Span0 ((==) True) yw311 ((==) True) False yw311 True)",fontsize=16,color="black",shape="box"];68 -> 76[label="",style="solid", color="black", weight=3];
69[label="span2Zs ((==) True) yw311",fontsize=16,color="black",shape="triangle"];69 -> 77[label="",style="solid", color="black", weight=3];
70[label="span2Ys0 ((==) False) yw311 (span2Vu43 ((==) False) yw311)",fontsize=16,color="black",shape="box"];70 -> 78[label="",style="solid", color="black", weight=3];
71[label="[]",fontsize=16,color="green",shape="box"];72[label="[]",fontsize=16,color="green",shape="box"];73[label="span2Ys0 ((==) True) yw311 (span2Vu43 ((==) True) yw311)",fontsize=16,color="black",shape="box"];73 -> 79[label="",style="solid", color="black", weight=3];
74[label="span2Zs0 ((==) False) yw311 (span2Vu43 ((==) False) yw311)",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
75[label="List.groupByZs1 (==) False (True : yw311) ([],True : yw311)",fontsize=16,color="black",shape="box"];75 -> 81[label="",style="solid", color="black", weight=3];
76[label="List.groupByZs1 (==) True (False : yw311) ([],False : yw311)",fontsize=16,color="black",shape="box"];76 -> 82[label="",style="solid", color="black", weight=3];
77[label="span2Zs0 ((==) True) yw311 (span2Vu43 ((==) True) yw311)",fontsize=16,color="black",shape="box"];77 -> 83[label="",style="solid", color="black", weight=3];
78[label="span2Ys0 ((==) False) yw311 (span ((==) False) yw311)",fontsize=16,color="burlywood",shape="box"];190[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];78 -> 190[label="",style="solid", color="burlywood", weight=9];
190 -> 84[label="",style="solid", color="burlywood", weight=3];
191[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];78 -> 191[label="",style="solid", color="burlywood", weight=9];
191 -> 85[label="",style="solid", color="burlywood", weight=3];
79[label="span2Ys0 ((==) True) yw311 (span ((==) True) yw311)",fontsize=16,color="burlywood",shape="box"];192[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];79 -> 192[label="",style="solid", color="burlywood", weight=9];
192 -> 86[label="",style="solid", color="burlywood", weight=3];
193[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];79 -> 193[label="",style="solid", color="burlywood", weight=9];
193 -> 87[label="",style="solid", color="burlywood", weight=3];
80[label="span2Zs0 ((==) False) yw311 (span ((==) False) yw311)",fontsize=16,color="burlywood",shape="box"];194[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];80 -> 194[label="",style="solid", color="burlywood", weight=9];
194 -> 88[label="",style="solid", color="burlywood", weight=3];
195[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];80 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 89[label="",style="solid", color="burlywood", weight=3];
81[label="True : yw311",fontsize=16,color="green",shape="box"];82[label="False : yw311",fontsize=16,color="green",shape="box"];83[label="span2Zs0 ((==) True) yw311 (span ((==) True) yw311)",fontsize=16,color="burlywood",shape="box"];196[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];83 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 90[label="",style="solid", color="burlywood", weight=3];
197[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];83 -> 197[label="",style="solid", color="burlywood", weight=9];
197 -> 91[label="",style="solid", color="burlywood", weight=3];
84[label="span2Ys0 ((==) False) (yw3110 : yw3111) (span ((==) False) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3];
85[label="span2Ys0 ((==) False) [] (span ((==) False) [])",fontsize=16,color="black",shape="box"];85 -> 93[label="",style="solid", color="black", weight=3];
86[label="span2Ys0 ((==) True) (yw3110 : yw3111) (span ((==) True) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];86 -> 94[label="",style="solid", color="black", weight=3];
87[label="span2Ys0 ((==) True) [] (span ((==) True) [])",fontsize=16,color="black",shape="box"];87 -> 95[label="",style="solid", color="black", weight=3];
88[label="span2Zs0 ((==) False) (yw3110 : yw3111) (span ((==) False) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];88 -> 96[label="",style="solid", color="black", weight=3];
89[label="span2Zs0 ((==) False) [] (span ((==) False) [])",fontsize=16,color="black",shape="box"];89 -> 97[label="",style="solid", color="black", weight=3];
90[label="span2Zs0 ((==) True) (yw3110 : yw3111) (span ((==) True) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];90 -> 98[label="",style="solid", color="black", weight=3];
91[label="span2Zs0 ((==) True) [] (span ((==) True) [])",fontsize=16,color="black",shape="box"];91 -> 99[label="",style="solid", color="black", weight=3];
92[label="span2Ys0 ((==) False) (yw3110 : yw3111) (span2 ((==) False) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];92 -> 100[label="",style="solid", color="black", weight=3];
93[label="span2Ys0 ((==) False) [] (span3 ((==) False) [])",fontsize=16,color="black",shape="box"];93 -> 101[label="",style="solid", color="black", weight=3];
94[label="span2Ys0 ((==) True) (yw3110 : yw3111) (span2 ((==) True) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];94 -> 102[label="",style="solid", color="black", weight=3];
95[label="span2Ys0 ((==) True) [] (span3 ((==) True) [])",fontsize=16,color="black",shape="box"];95 -> 103[label="",style="solid", color="black", weight=3];
96[label="span2Zs0 ((==) False) (yw3110 : yw3111) (span2 ((==) False) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];96 -> 104[label="",style="solid", color="black", weight=3];
97[label="span2Zs0 ((==) False) [] (span3 ((==) False) [])",fontsize=16,color="black",shape="box"];97 -> 105[label="",style="solid", color="black", weight=3];
98[label="span2Zs0 ((==) True) (yw3110 : yw3111) (span2 ((==) True) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];98 -> 106[label="",style="solid", color="black", weight=3];
99[label="span2Zs0 ((==) True) [] (span3 ((==) True) [])",fontsize=16,color="black",shape="box"];99 -> 107[label="",style="solid", color="black", weight=3];
100[label="span2Ys0 ((==) False) (yw3110 : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) yw3110 yw3111 ((==) False yw3110))",fontsize=16,color="burlywood",shape="box"];198[label="yw3110/False",fontsize=10,color="white",style="solid",shape="box"];100 -> 198[label="",style="solid", color="burlywood", weight=9];
198 -> 108[label="",style="solid", color="burlywood", weight=3];
199[label="yw3110/True",fontsize=10,color="white",style="solid",shape="box"];100 -> 199[label="",style="solid", color="burlywood", weight=9];
199 -> 109[label="",style="solid", color="burlywood", weight=3];
101[label="span2Ys0 ((==) False) [] ([],[])",fontsize=16,color="black",shape="box"];101 -> 110[label="",style="solid", color="black", weight=3];
102[label="span2Ys0 ((==) True) (yw3110 : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) yw3110 yw3111 ((==) True yw3110))",fontsize=16,color="burlywood",shape="box"];200[label="yw3110/False",fontsize=10,color="white",style="solid",shape="box"];102 -> 200[label="",style="solid", color="burlywood", weight=9];
200 -> 111[label="",style="solid", color="burlywood", weight=3];
201[label="yw3110/True",fontsize=10,color="white",style="solid",shape="box"];102 -> 201[label="",style="solid", color="burlywood", weight=9];
201 -> 112[label="",style="solid", color="burlywood", weight=3];
103[label="span2Ys0 ((==) True) [] ([],[])",fontsize=16,color="black",shape="box"];103 -> 113[label="",style="solid", color="black", weight=3];
104[label="span2Zs0 ((==) False) (yw3110 : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) yw3110 yw3111 ((==) False yw3110))",fontsize=16,color="burlywood",shape="box"];202[label="yw3110/False",fontsize=10,color="white",style="solid",shape="box"];104 -> 202[label="",style="solid", color="burlywood", weight=9];
202 -> 114[label="",style="solid", color="burlywood", weight=3];
203[label="yw3110/True",fontsize=10,color="white",style="solid",shape="box"];104 -> 203[label="",style="solid", color="burlywood", weight=9];
203 -> 115[label="",style="solid", color="burlywood", weight=3];
105[label="span2Zs0 ((==) False) [] ([],[])",fontsize=16,color="black",shape="box"];105 -> 116[label="",style="solid", color="black", weight=3];
106[label="span2Zs0 ((==) True) (yw3110 : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) yw3110 yw3111 ((==) True yw3110))",fontsize=16,color="burlywood",shape="box"];204[label="yw3110/False",fontsize=10,color="white",style="solid",shape="box"];106 -> 204[label="",style="solid", color="burlywood", weight=9];
204 -> 117[label="",style="solid", color="burlywood", weight=3];
205[label="yw3110/True",fontsize=10,color="white",style="solid",shape="box"];106 -> 205[label="",style="solid", color="burlywood", weight=9];
205 -> 118[label="",style="solid", color="burlywood", weight=3];
107[label="span2Zs0 ((==) True) [] ([],[])",fontsize=16,color="black",shape="box"];107 -> 119[label="",style="solid", color="black", weight=3];
108[label="span2Ys0 ((==) False) (False : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) False yw3111 ((==) False False))",fontsize=16,color="black",shape="box"];108 -> 120[label="",style="solid", color="black", weight=3];
109[label="span2Ys0 ((==) False) (True : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) True yw3111 ((==) False True))",fontsize=16,color="black",shape="box"];109 -> 121[label="",style="solid", color="black", weight=3];
110[label="[]",fontsize=16,color="green",shape="box"];111[label="span2Ys0 ((==) True) (False : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) False yw3111 ((==) True False))",fontsize=16,color="black",shape="box"];111 -> 122[label="",style="solid", color="black", weight=3];
112[label="span2Ys0 ((==) True) (True : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) True yw3111 ((==) True True))",fontsize=16,color="black",shape="box"];112 -> 123[label="",style="solid", color="black", weight=3];
113[label="[]",fontsize=16,color="green",shape="box"];114[label="span2Zs0 ((==) False) (False : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) False yw3111 ((==) False False))",fontsize=16,color="black",shape="box"];114 -> 124[label="",style="solid", color="black", weight=3];
115[label="span2Zs0 ((==) False) (True : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) True yw3111 ((==) False True))",fontsize=16,color="black",shape="box"];115 -> 125[label="",style="solid", color="black", weight=3];
116[label="[]",fontsize=16,color="green",shape="box"];117[label="span2Zs0 ((==) True) (False : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) False yw3111 ((==) True False))",fontsize=16,color="black",shape="box"];117 -> 126[label="",style="solid", color="black", weight=3];
118[label="span2Zs0 ((==) True) (True : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) True yw3111 ((==) True True))",fontsize=16,color="black",shape="box"];118 -> 127[label="",style="solid", color="black", weight=3];
119[label="[]",fontsize=16,color="green",shape="box"];120[label="span2Ys0 ((==) False) (False : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) False yw3111 True)",fontsize=16,color="black",shape="box"];120 -> 128[label="",style="solid", color="black", weight=3];
121[label="span2Ys0 ((==) False) (True : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) True yw3111 False)",fontsize=16,color="black",shape="box"];121 -> 129[label="",style="solid", color="black", weight=3];
122[label="span2Ys0 ((==) True) (False : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) False yw3111 False)",fontsize=16,color="black",shape="box"];122 -> 130[label="",style="solid", color="black", weight=3];
123[label="span2Ys0 ((==) True) (True : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) True yw3111 True)",fontsize=16,color="black",shape="box"];123 -> 131[label="",style="solid", color="black", weight=3];
124[label="span2Zs0 ((==) False) (False : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) False yw3111 True)",fontsize=16,color="black",shape="box"];124 -> 132[label="",style="solid", color="black", weight=3];
125[label="span2Zs0 ((==) False) (True : yw3111) (span2Span1 ((==) False) yw3111 ((==) False) True yw3111 False)",fontsize=16,color="black",shape="box"];125 -> 133[label="",style="solid", color="black", weight=3];
126[label="span2Zs0 ((==) True) (False : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) False yw3111 False)",fontsize=16,color="black",shape="box"];126 -> 134[label="",style="solid", color="black", weight=3];
127[label="span2Zs0 ((==) True) (True : yw3111) (span2Span1 ((==) True) yw3111 ((==) True) True yw3111 True)",fontsize=16,color="black",shape="box"];127 -> 135[label="",style="solid", color="black", weight=3];
128 -> 136[label="",style="dashed", color="red", weight=0];
128[label="span2Ys0 ((==) False) (False : yw3111) (False : span2Ys ((==) False) yw3111,span2Zs ((==) False) yw3111)",fontsize=16,color="magenta"];128 -> 137[label="",style="dashed", color="magenta", weight=3];
128 -> 138[label="",style="dashed", color="magenta", weight=3];
129[label="span2Ys0 ((==) False) (True : yw3111) (span2Span0 ((==) False) yw3111 ((==) False) True yw3111 otherwise)",fontsize=16,color="black",shape="box"];129 -> 139[label="",style="solid", color="black", weight=3];
130[label="span2Ys0 ((==) True) (False : yw3111) (span2Span0 ((==) True) yw3111 ((==) True) False yw3111 otherwise)",fontsize=16,color="black",shape="box"];130 -> 140[label="",style="solid", color="black", weight=3];
131 -> 141[label="",style="dashed", color="red", weight=0];
131[label="span2Ys0 ((==) True) (True : yw3111) (True : span2Ys ((==) True) yw3111,span2Zs ((==) True) yw3111)",fontsize=16,color="magenta"];131 -> 142[label="",style="dashed", color="magenta", weight=3];
131 -> 143[label="",style="dashed", color="magenta", weight=3];
132 -> 144[label="",style="dashed", color="red", weight=0];
132[label="span2Zs0 ((==) False) (False : yw3111) (False : span2Ys ((==) False) yw3111,span2Zs ((==) False) yw3111)",fontsize=16,color="magenta"];132 -> 145[label="",style="dashed", color="magenta", weight=3];
132 -> 146[label="",style="dashed", color="magenta", weight=3];
133[label="span2Zs0 ((==) False) (True : yw3111) (span2Span0 ((==) False) yw3111 ((==) False) True yw3111 otherwise)",fontsize=16,color="black",shape="box"];133 -> 147[label="",style="solid", color="black", weight=3];
134[label="span2Zs0 ((==) True) (False : yw3111) (span2Span0 ((==) True) yw3111 ((==) True) False yw3111 otherwise)",fontsize=16,color="black",shape="box"];134 -> 148[label="",style="solid", color="black", weight=3];
135 -> 149[label="",style="dashed", color="red", weight=0];
135[label="span2Zs0 ((==) True) (True : yw3111) (True : span2Ys ((==) True) yw3111,span2Zs ((==) True) yw3111)",fontsize=16,color="magenta"];135 -> 150[label="",style="dashed", color="magenta", weight=3];
135 -> 151[label="",style="dashed", color="magenta", weight=3];
137 -> 66[label="",style="dashed", color="red", weight=0];
137[label="span2Zs ((==) False) yw3111",fontsize=16,color="magenta"];137 -> 152[label="",style="dashed", color="magenta", weight=3];
138 -> 62[label="",style="dashed", color="red", weight=0];
138[label="span2Ys ((==) False) yw3111",fontsize=16,color="magenta"];138 -> 153[label="",style="dashed", color="magenta", weight=3];
136[label="span2Ys0 ((==) False) (False : yw3111) (False : yw5,yw4)",fontsize=16,color="black",shape="triangle"];136 -> 154[label="",style="solid", color="black", weight=3];
139[label="span2Ys0 ((==) False) (True : yw3111) (span2Span0 ((==) False) yw3111 ((==) False) True yw3111 True)",fontsize=16,color="black",shape="box"];139 -> 155[label="",style="solid", color="black", weight=3];
140[label="span2Ys0 ((==) True) (False : yw3111) (span2Span0 ((==) True) yw3111 ((==) True) False yw3111 True)",fontsize=16,color="black",shape="box"];140 -> 156[label="",style="solid", color="black", weight=3];
142 -> 65[label="",style="dashed", color="red", weight=0];
142[label="span2Ys ((==) True) yw3111",fontsize=16,color="magenta"];142 -> 157[label="",style="dashed", color="magenta", weight=3];
143 -> 69[label="",style="dashed", color="red", weight=0];
143[label="span2Zs ((==) True) yw3111",fontsize=16,color="magenta"];143 -> 158[label="",style="dashed", color="magenta", weight=3];
141[label="span2Ys0 ((==) True) (True : yw3111) (True : yw7,yw6)",fontsize=16,color="black",shape="triangle"];141 -> 159[label="",style="solid", color="black", weight=3];
145 -> 66[label="",style="dashed", color="red", weight=0];
145[label="span2Zs ((==) False) yw3111",fontsize=16,color="magenta"];145 -> 160[label="",style="dashed", color="magenta", weight=3];
146 -> 62[label="",style="dashed", color="red", weight=0];
146[label="span2Ys ((==) False) yw3111",fontsize=16,color="magenta"];146 -> 161[label="",style="dashed", color="magenta", weight=3];
144[label="span2Zs0 ((==) False) (False : yw3111) (False : yw9,yw8)",fontsize=16,color="black",shape="triangle"];144 -> 162[label="",style="solid", color="black", weight=3];
147[label="span2Zs0 ((==) False) (True : yw3111) (span2Span0 ((==) False) yw3111 ((==) False) True yw3111 True)",fontsize=16,color="black",shape="box"];147 -> 163[label="",style="solid", color="black", weight=3];
148[label="span2Zs0 ((==) True) (False : yw3111) (span2Span0 ((==) True) yw3111 ((==) True) False yw3111 True)",fontsize=16,color="black",shape="box"];148 -> 164[label="",style="solid", color="black", weight=3];
150 -> 69[label="",style="dashed", color="red", weight=0];
150[label="span2Zs ((==) True) yw3111",fontsize=16,color="magenta"];150 -> 165[label="",style="dashed", color="magenta", weight=3];
151 -> 65[label="",style="dashed", color="red", weight=0];
151[label="span2Ys ((==) True) yw3111",fontsize=16,color="magenta"];151 -> 166[label="",style="dashed", color="magenta", weight=3];
149[label="span2Zs0 ((==) True) (True : yw3111) (True : yw11,yw10)",fontsize=16,color="black",shape="triangle"];149 -> 167[label="",style="solid", color="black", weight=3];
152[label="yw3111",fontsize=16,color="green",shape="box"];153[label="yw3111",fontsize=16,color="green",shape="box"];154[label="False : yw5",fontsize=16,color="green",shape="box"];155[label="span2Ys0 ((==) False) (True : yw3111) ([],True : yw3111)",fontsize=16,color="black",shape="box"];155 -> 168[label="",style="solid", color="black", weight=3];
156[label="span2Ys0 ((==) True) (False : yw3111) ([],False : yw3111)",fontsize=16,color="black",shape="box"];156 -> 169[label="",style="solid", color="black", weight=3];
157[label="yw3111",fontsize=16,color="green",shape="box"];158[label="yw3111",fontsize=16,color="green",shape="box"];159[label="True : yw7",fontsize=16,color="green",shape="box"];160[label="yw3111",fontsize=16,color="green",shape="box"];161[label="yw3111",fontsize=16,color="green",shape="box"];162[label="yw8",fontsize=16,color="green",shape="box"];163[label="span2Zs0 ((==) False) (True : yw3111) ([],True : yw3111)",fontsize=16,color="black",shape="box"];163 -> 170[label="",style="solid", color="black", weight=3];
164[label="span2Zs0 ((==) True) (False : yw3111) ([],False : yw3111)",fontsize=16,color="black",shape="box"];164 -> 171[label="",style="solid", color="black", weight=3];
165[label="yw3111",fontsize=16,color="green",shape="box"];166[label="yw3111",fontsize=16,color="green",shape="box"];167[label="yw10",fontsize=16,color="green",shape="box"];168[label="[]",fontsize=16,color="green",shape="box"];169[label="[]",fontsize=16,color="green",shape="box"];170[label="True : yw3111",fontsize=16,color="green",shape="box"];171[label="False : yw3111",fontsize=16,color="green",shape="box"];}

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(10)
Complex Obligation (AND)

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

   new_span2Zs(:(True, yw3111)) -> new_span2Ys(yw3111)
   new_span2Ys(:(True, yw3111)) -> new_span2Ys(yw3111)
   new_span2Ys(:(True, yw3111)) -> new_span2Zs(yw3111)
   new_span2Zs(:(True, yw3111)) -> new_span2Zs(yw3111)

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(:(True, yw3111)) -> new_span2Zs(yw3111)
The graph contains the following edges 1 > 1


*new_span2Ys(:(True, yw3111)) -> new_span2Ys(yw3111)
The graph contains the following edges 1 > 1


*new_span2Zs(:(True, yw3111)) -> new_span2Zs(yw3111)
The graph contains the following edges 1 > 1


*new_span2Zs(:(True, yw3111)) -> new_span2Ys(yw3111)
The graph contains the following edges 1 > 1


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(13)
YES

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

   new_span2Ys0(:(False, yw3111)) -> new_span2Zs0(yw3111)
   new_span2Zs0(:(False, yw3111)) -> new_span2Zs0(yw3111)
   new_span2Zs0(:(False, yw3111)) -> new_span2Ys0(yw3111)
   new_span2Ys0(:(False, yw3111)) -> new_span2Ys0(yw3111)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_span2Zs0(:(False, yw3111)) -> new_span2Ys0(yw3111)
The graph contains the following edges 1 > 1


*new_span2Zs0(:(False, yw3111)) -> new_span2Zs0(yw3111)
The graph contains the following edges 1 > 1


*new_span2Ys0(:(False, yw3111)) -> new_span2Ys0(yw3111)
The graph contains the following edges 1 > 1


*new_span2Ys0(:(False, yw3111)) -> new_span2Zs0(yw3111)
The graph contains the following edges 1 > 1


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

(16)
YES

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

   new_groupBy(:(yw30, yw31)) -> new_groupBy(new_groupByZs1(yw30, yw31))

The TRS R consists of the following rules:

   new_span2Ys00(yw3111, yw5, yw4) -> :(False, yw5)
   new_groupByZs1(False, :(False, yw311)) -> new_span2Zs1(yw311)
   new_span2Ys1(:(False, yw3111)) -> new_span2Ys00(yw3111, new_span2Ys1(yw3111), new_span2Zs1(yw3111))
   new_span2Ys2(:(False, yw3111)) -> []
   new_groupByZs1(False, :(True, yw311)) -> :(True, yw311)
   new_span2Zs2(:(False, yw3111)) -> :(False, yw3111)
   new_span2Zs1(:(True, yw3111)) -> :(True, yw3111)
   new_groupByZs1(True, :(False, yw311)) -> :(False, yw311)
   new_span2Zs1(:(False, yw3111)) -> new_span2Zs01(yw3111, new_span2Ys1(yw3111), new_span2Zs1(yw3111))
   new_span2Ys2(:(True, yw3111)) -> new_span2Ys01(yw3111, new_span2Ys2(yw3111), new_span2Zs2(yw3111))
   new_span2Ys2([]) -> []
   new_span2Zs01(yw3111, yw9, yw8) -> yw8
   new_span2Zs1([]) -> []
   new_span2Ys1(:(True, yw3111)) -> []
   new_span2Zs2([]) -> []
   new_groupByZs1(yw30, []) -> []
   new_span2Zs2(:(True, yw3111)) -> new_span2Zs00(yw3111, new_span2Ys2(yw3111), new_span2Zs2(yw3111))
   new_span2Ys1([]) -> []
   new_span2Zs00(yw3111, yw11, yw10) -> yw10
   new_span2Ys01(yw3111, yw7, yw6) -> :(True, yw7)
   new_groupByZs1(True, :(True, yw311)) -> new_span2Zs2(yw311)

The set Q consists of the following terms:

   new_groupByZs1(False, :(True, x0))
   new_span2Zs1(:(False, x0))
   new_groupByZs1(x0, [])
   new_groupByZs1(True, :(False, x0))
   new_span2Zs1([])
   new_span2Ys1(:(True, x0))
   new_groupByZs1(True, :(True, x0))
   new_span2Ys1(:(False, x0))
   new_span2Zs01(x0, x1, x2)
   new_groupByZs1(False, :(False, x0))
   new_span2Zs2(:(False, x0))
   new_span2Ys2(:(False, x0))
   new_span2Zs2(:(True, x0))
   new_span2Zs00(x0, x1, x2)
   new_span2Ys1([])
   new_span2Ys2(:(True, x0))
   new_span2Ys00(x0, x1, x2)
   new_span2Zs1(:(True, x0))
   new_span2Ys2([])
   new_span2Ys01(x0, x1, x2)
   new_span2Zs2([])

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

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

Order:Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_2
   POL(False) = 0
   POL(True) = 0
   POL([]) = 1
   POL(new_groupByZs1(x_1, x_2)) = x_2
   POL(new_span2Ys00(x_1, x_2, x_3)) = 1 + x_2
   POL(new_span2Ys01(x_1, x_2, x_3)) = 1 + x_2
   POL(new_span2Ys1(x_1)) = 1 + x_1
   POL(new_span2Ys2(x_1)) = 1 + x_1
   POL(new_span2Zs00(x_1, x_2, x_3)) = 1 + x_3
   POL(new_span2Zs01(x_1, x_2, x_3)) = 1 + x_3
   POL(new_span2Zs1(x_1)) = 1 + x_1
   POL(new_span2Zs2(x_1)) = x_1




From the DPs we obtained the following set of size-change graphs:
*new_groupBy(:(yw30, yw31)) -> new_groupBy(new_groupByZs1(yw30, yw31)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1



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

   new_span2Zs2([]) -> []
   new_span2Zs2(:(True, yw3111)) -> new_span2Zs00(yw3111, new_span2Ys2(yw3111), new_span2Zs2(yw3111))
   new_span2Zs2(:(False, yw3111)) -> :(False, yw3111)
   new_span2Zs1([]) -> []
   new_span2Zs1(:(True, yw3111)) -> :(True, yw3111)
   new_span2Zs1(:(False, yw3111)) -> new_span2Zs01(yw3111, new_span2Ys1(yw3111), new_span2Zs1(yw3111))
   new_span2Zs01(yw3111, yw9, yw8) -> yw8
   new_span2Zs00(yw3111, yw11, yw10) -> yw10
   new_span2Ys2([]) -> []
   new_span2Ys2(:(True, yw3111)) -> new_span2Ys01(yw3111, new_span2Ys2(yw3111), new_span2Zs2(yw3111))
   new_span2Ys2(:(False, yw3111)) -> []
   new_span2Ys1([]) -> []
   new_span2Ys1(:(True, yw3111)) -> []
   new_span2Ys1(:(False, yw3111)) -> new_span2Ys00(yw3111, new_span2Ys1(yw3111), new_span2Zs1(yw3111))
   new_span2Ys01(yw3111, yw7, yw6) -> :(True, yw7)
   new_span2Ys00(yw3111, yw5, yw4) -> :(False, yw5)
   new_groupByZs1(yw30, []) -> []
   new_groupByZs1(True, :(True, yw311)) -> new_span2Zs2(yw311)
   new_groupByZs1(True, :(False, yw311)) -> :(False, yw311)
   new_groupByZs1(False, :(True, yw311)) -> :(True, yw311)
   new_groupByZs1(False, :(False, yw311)) -> new_span2Zs1(yw311)

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(19)
YES