/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 29 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


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

(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@(vy : vz)"
is replaced by the following term
"vy : vz"

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

(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 (vy : vz)|p vy(vy : ys,zs)|otherwise([],vy : vz) where {
vu43  = span p vz;
;
ys  = ys0 vu43;
;
ys0 (ys,wv) = ys;
;
zs  = zs0 vu43;
;
zs0 (wu,zs) = zs;
} 
;
"
is transformed to
"span p [] = span3 p [];
span p (vy : vz) = span2 p (vy : vz);
"
"span2 p (vy : vz) = span1 p vy vz (p vy) where {
span0 p vy vz True = ([],vy : vz);
;
span1 p vy vz True = (vy : ys,zs);
span1 p vy vz False = span0 p vy vz otherwise;
;
vu43  = span p vz;
;
ys  = ys0 vu43;
;
ys0 (ys,wv) = ys;
;
zs  = zs0 vu43;
;
zs0 (wu,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 vy vz (p vy) where {
span0 p vy vz True = ([],vy : vz);
;
span1 p vy vz True = (vy : ys,zs);
span1 p vy vz False = span0 p vy vz otherwise;
;
vu43  = span p vz;
;
ys  = ys0 vu43;
;
ys0 (ys,wv) = ys;
;
zs  = zs0 vu43;
;
zs0 (wu,zs) = zs;
} 
"
are unpacked to the following functions on top level
"span2Vu43 xx xy = span xx xy;
"
"span2Span1 xx xy p vy vz True = (vy : span2Ys xx xy,span2Zs xx xy);
span2Span1 xx xy p vy vz False = span2Span0 xx xy p vy vz otherwise;
"
"span2Zs0 xx xy (wu,zs) = zs;
"
"span2Span0 xx xy p vy vz True = ([],vy : vz);
"
"span2Ys0 xx xy (ys,wv) = ys;
"
"span2Zs xx xy = span2Zs0 xx xy (span2Vu43 xx xy);
"
"span2Ys xx xy = span2Ys0 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
"groupByZs1 xz yu yv (wy,zs) = zs;
"
"groupByVv10 xz yu yv = span (xz yu) yv;
"
"groupByYs1 xz yu yv (ys,wx) = ys;
"
"groupByYs xz yu yv = groupByYs1 xz yu yv (groupByVv10 xz yu yv);
"
"groupByZs xz yu yv = groupByZs1 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"];77[label="yw3/yw30 : yw31",fontsize=10,color="white",style="solid",shape="box"];4 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 5[label="",style="solid", color="burlywood", weight=3];
78[label="yw3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 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"];79[label="yw31/yw310 : yw311",fontsize=10,color="white",style="solid",shape="box"];13 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 15[label="",style="solid", color="burlywood", weight=3];
80[label="yw31/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 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"];81[label="yw31/yw310 : yw311",fontsize=10,color="white",style="solid",shape="box"];17 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 20[label="",style="solid", color="burlywood", weight=3];
82[label="yw31/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 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"];83[label="yw30/()",fontsize=10,color="white",style="solid",shape="box"];22 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 26[label="",style="solid", color="burlywood", weight=3];
23[label="List.groupByYs1 (==) yw30 [] ([],[])",fontsize=16,color="black",shape="box"];23 -> 27[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 -> 28[label="",style="solid", color="black", weight=3];
25[label="List.groupByZs1 (==) yw30 [] (span3 ((==) yw30) [])",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="List.groupByYs1 (==) () (yw310 : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) yw310 yw311 ((==) () yw310))",fontsize=16,color="burlywood",shape="box"];84[label="yw310/()",fontsize=10,color="white",style="solid",shape="box"];26 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 30[label="",style="solid", color="burlywood", weight=3];
27[label="[]",fontsize=16,color="green",shape="box"];28[label="List.groupByZs1 (==) yw30 (yw310 : yw311) (span2Span1 ((==) yw30) yw311 ((==) yw30) yw310 yw311 ((==) yw30 yw310))",fontsize=16,color="burlywood",shape="box"];85[label="yw30/()",fontsize=10,color="white",style="solid",shape="box"];28 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 31[label="",style="solid", color="burlywood", weight=3];
29[label="List.groupByZs1 (==) yw30 [] ([],[])",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="List.groupByYs1 (==) () (() : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) () yw311 ((==) () ()))",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3];
31[label="List.groupByZs1 (==) () (yw310 : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) yw310 yw311 ((==) () yw310))",fontsize=16,color="burlywood",shape="box"];86[label="yw310/()",fontsize=10,color="white",style="solid",shape="box"];31 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 34[label="",style="solid", color="burlywood", weight=3];
32[label="[]",fontsize=16,color="green",shape="box"];33[label="List.groupByYs1 (==) () (() : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) () yw311 True)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="List.groupByZs1 (==) () (() : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) () yw311 ((==) () ()))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="List.groupByYs1 (==) () (() : yw311) (() : span2Ys ((==) ()) yw311,span2Zs ((==) ()) yw311)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="List.groupByZs1 (==) () (() : yw311) (span2Span1 ((==) ()) yw311 ((==) ()) () yw311 True)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="() : span2Ys ((==) ()) yw311",fontsize=16,color="green",shape="box"];37 -> 39[label="",style="dashed", color="green", weight=3];
38[label="List.groupByZs1 (==) () (() : yw311) (() : span2Ys ((==) ()) yw311,span2Zs ((==) ()) yw311)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="span2Ys ((==) ()) yw311",fontsize=16,color="black",shape="triangle"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="span2Zs ((==) ()) yw311",fontsize=16,color="black",shape="triangle"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="span2Ys0 ((==) ()) yw311 (span2Vu43 ((==) ()) yw311)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="span2Zs0 ((==) ()) yw311 (span2Vu43 ((==) ()) yw311)",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="span2Ys0 ((==) ()) yw311 (span ((==) ()) yw311)",fontsize=16,color="burlywood",shape="box"];87[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];43 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 45[label="",style="solid", color="burlywood", weight=3];
88[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];43 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 46[label="",style="solid", color="burlywood", weight=3];
44[label="span2Zs0 ((==) ()) yw311 (span ((==) ()) yw311)",fontsize=16,color="burlywood",shape="box"];89[label="yw311/yw3110 : yw3111",fontsize=10,color="white",style="solid",shape="box"];44 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 47[label="",style="solid", color="burlywood", weight=3];
90[label="yw311/[]",fontsize=10,color="white",style="solid",shape="box"];44 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 48[label="",style="solid", color="burlywood", weight=3];
45[label="span2Ys0 ((==) ()) (yw3110 : yw3111) (span ((==) ()) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="span2Ys0 ((==) ()) [] (span ((==) ()) [])",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
47[label="span2Zs0 ((==) ()) (yw3110 : yw3111) (span ((==) ()) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
48[label="span2Zs0 ((==) ()) [] (span ((==) ()) [])",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
49[label="span2Ys0 ((==) ()) (yw3110 : yw3111) (span2 ((==) ()) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];49 -> 53[label="",style="solid", color="black", weight=3];
50[label="span2Ys0 ((==) ()) [] (span3 ((==) ()) [])",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
51[label="span2Zs0 ((==) ()) (yw3110 : yw3111) (span2 ((==) ()) (yw3110 : yw3111))",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
52[label="span2Zs0 ((==) ()) [] (span3 ((==) ()) [])",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="span2Ys0 ((==) ()) (yw3110 : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) yw3110 yw3111 ((==) () yw3110))",fontsize=16,color="burlywood",shape="box"];91[label="yw3110/()",fontsize=10,color="white",style="solid",shape="box"];53 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 57[label="",style="solid", color="burlywood", weight=3];
54[label="span2Ys0 ((==) ()) [] ([],[])",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55[label="span2Zs0 ((==) ()) (yw3110 : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) yw3110 yw3111 ((==) () yw3110))",fontsize=16,color="burlywood",shape="box"];92[label="yw3110/()",fontsize=10,color="white",style="solid",shape="box"];55 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 59[label="",style="solid", color="burlywood", weight=3];
56[label="span2Zs0 ((==) ()) [] ([],[])",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
57[label="span2Ys0 ((==) ()) (() : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) () yw3111 ((==) () ()))",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3];
58[label="[]",fontsize=16,color="green",shape="box"];59[label="span2Zs0 ((==) ()) (() : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) () yw3111 ((==) () ()))",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="[]",fontsize=16,color="green",shape="box"];61[label="span2Ys0 ((==) ()) (() : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) () yw3111 True)",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="span2Zs0 ((==) ()) (() : yw3111) (span2Span1 ((==) ()) yw3111 ((==) ()) () yw3111 True)",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63 -> 65[label="",style="dashed", color="red", weight=0];
63[label="span2Ys0 ((==) ()) (() : yw3111) (() : span2Ys ((==) ()) yw3111,span2Zs ((==) ()) yw3111)",fontsize=16,color="magenta"];63 -> 66[label="",style="dashed", color="magenta", weight=3];
63 -> 67[label="",style="dashed", color="magenta", weight=3];
64 -> 68[label="",style="dashed", color="red", weight=0];
64[label="span2Zs0 ((==) ()) (() : yw3111) (() : span2Ys ((==) ()) yw3111,span2Zs ((==) ()) yw3111)",fontsize=16,color="magenta"];64 -> 69[label="",style="dashed", color="magenta", weight=3];
64 -> 70[label="",style="dashed", color="magenta", weight=3];
66 -> 40[label="",style="dashed", color="red", weight=0];
66[label="span2Zs ((==) ()) yw3111",fontsize=16,color="magenta"];66 -> 71[label="",style="dashed", color="magenta", weight=3];
67 -> 39[label="",style="dashed", color="red", weight=0];
67[label="span2Ys ((==) ()) yw3111",fontsize=16,color="magenta"];67 -> 72[label="",style="dashed", color="magenta", weight=3];
65[label="span2Ys0 ((==) ()) (() : yw3111) (() : yw5,yw4)",fontsize=16,color="black",shape="triangle"];65 -> 73[label="",style="solid", color="black", weight=3];
69 -> 40[label="",style="dashed", color="red", weight=0];
69[label="span2Zs ((==) ()) yw3111",fontsize=16,color="magenta"];69 -> 74[label="",style="dashed", color="magenta", weight=3];
70 -> 39[label="",style="dashed", color="red", weight=0];
70[label="span2Ys ((==) ()) yw3111",fontsize=16,color="magenta"];70 -> 75[label="",style="dashed", color="magenta", weight=3];
68[label="span2Zs0 ((==) ()) (() : yw3111) (() : yw7,yw6)",fontsize=16,color="black",shape="triangle"];68 -> 76[label="",style="solid", color="black", weight=3];
71[label="yw3111",fontsize=16,color="green",shape="box"];72[label="yw3111",fontsize=16,color="green",shape="box"];73[label="() : yw5",fontsize=16,color="green",shape="box"];74[label="yw3111",fontsize=16,color="green",shape="box"];75[label="yw3111",fontsize=16,color="green",shape="box"];76[label="yw6",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_span2Ys(:(@0, yw3111)) -> new_span2Zs(yw3111)
   new_span2Zs(:(@0, yw3111)) -> new_span2Zs(yw3111)
   new_span2Zs(:(@0, yw3111)) -> new_span2Ys(yw3111)
   new_span2Ys(:(@0, yw3111)) -> new_span2Ys(yw3111)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_span2Zs(:(@0, yw3111)) -> new_span2Ys(yw3111)
The graph contains the following edges 1 > 1


*new_span2Zs(:(@0, yw3111)) -> new_span2Zs(yw3111)
The graph contains the following edges 1 > 1


*new_span2Ys(:(@0, yw3111)) -> new_span2Ys(yw3111)
The graph contains the following edges 1 > 1


*new_span2Ys(:(@0, yw3111)) -> new_span2Zs(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_groupBy(:(yw30, yw31)) -> new_groupBy(new_groupByZs1(yw30, yw31))

The TRS R consists of the following rules:

   new_span2Ys0(yw3111, yw5, yw4) -> :(@0, yw5)
   new_span2Ys1(:(@0, yw3111)) -> new_span2Ys0(yw3111, new_span2Ys1(yw3111), new_span2Zs0(yw3111))
   new_span2Zs0(:(@0, yw3111)) -> new_span2Zs00(yw3111, new_span2Ys1(yw3111), new_span2Zs0(yw3111))
   new_groupByZs1(@0, :(@0, yw311)) -> new_span2Zs0(yw311)
   new_span2Zs0([]) -> []
   new_groupByZs1(yw30, []) -> []
   new_span2Ys1([]) -> []
   new_span2Zs00(yw3111, yw7, yw6) -> yw6

The set Q consists of the following terms:

   new_span2Zs0([])
   new_groupByZs1(x0, [])
   new_span2Zs0(:(@0, x0))
   new_span2Ys1([])
   new_span2Ys1(:(@0, x0))
   new_span2Zs00(x0, x1, x2)
   new_span2Ys0(x0, x1, x2)
   new_groupByZs1(@0, :(@0, x0))

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

Order:Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_2
   POL(@0) = 0
   POL([]) = 1
   POL(new_groupByZs1(x_1, x_2)) = x_2
   POL(new_span2Ys0(x_1, x_2, x_3)) = 1 + x_2
   POL(new_span2Ys1(x_1)) = 1 + x_1
   POL(new_span2Zs0(x_1)) = x_1
   POL(new_span2Zs00(x_1, x_2, x_3)) = x_3




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_span2Zs00(yw3111, yw7, yw6) -> yw6
   new_span2Zs0([]) -> []
   new_span2Zs0(:(@0, yw3111)) -> new_span2Zs00(yw3111, new_span2Ys1(yw3111), new_span2Zs0(yw3111))
   new_span2Ys1([]) -> []
   new_span2Ys1(:(@0, yw3111)) -> new_span2Ys0(yw3111, new_span2Ys1(yw3111), new_span2Zs0(yw3111))
   new_span2Ys0(yw3111, yw5, yw4) -> :(@0, yw5)
   new_groupByZs1(yw30, []) -> []
   new_groupByZs1(@0, :(@0, yw311)) -> new_span2Zs0(yw311)

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(16)
YES