/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
        (17) AND
            (18) QDP
                (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (20) YES
            (21) QDP
                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (23) YES
            (24) QDP
                (25) QDPOrderProof [EQUIVALENT, 3 ms]
                (26) QDP
                (27) DependencyGraphProof [EQUIVALENT, 0 ms]
                (28) QDP
                (29) MNOCProof [EQUIVALENT, 0 ms]
                (30) QDP
                (31) NonTerminationLoopProof [COMPLETE, 3 ms]
                (32) NO
(33) Narrow [COMPLETE, 0 ms]
(34) TRUE


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad c => (d  ->  b  ->  c a)  ->  [d]  ->  [b]  ->  c [a];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\xs->return (x : xs)"
is transformed to
"sequence0 x xs = return (x : xs);
"
The following Lambda expression
"\x->sequence cs >>= sequence0 x"
is transformed to
"sequence1 cs x = sequence cs >>= sequence0 x;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad b => (a  ->  c  ->  b d)  ->  [a]  ->  [c]  ->  b [d];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad d => (a  ->  b  ->  d c)  ->  [a]  ->  [b]  ->  d [c];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad c => (a  ->  d  ->  c b)  ->  [a]  ->  [d]  ->  c [b];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
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124 -> 8[label="",style="solid", color="burlywood", weight=3];
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125 -> 9[label="",style="solid", color="burlywood", weight=3];
126[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 10[label="",style="solid", color="burlywood", weight=3];
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129 -> 18[label="",style="solid", color="blue", weight=3];
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133 -> 33[label="",style="solid", color="burlywood", weight=3];
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135 -> 37[label="",style="solid", color="burlywood", weight=3];
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29 -> 41[label="",style="dashed", color="red", weight=0];
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30[label="AProVE_IO []",fontsize=16,color="green",shape="box"];31[label="wv40",fontsize=16,color="green",shape="box"];32[label="wv50",fontsize=16,color="green",shape="box"];33[label="wv60 : wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];33 -> 43[label="",style="solid", color="black", weight=3];
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39[label="wv40",fontsize=16,color="green",shape="box"];40[label="wv50",fontsize=16,color="green",shape="box"];37[label="Nothing >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
38[label="Just wv70 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
42[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];42 -> 53[label="",style="dashed", color="green", weight=3];
42 -> 54[label="",style="dashed", color="green", weight=3];
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137 -> 49[label="",style="solid", color="burlywood", weight=3];
138[label="wv8/AProVE_IO wv80",fontsize=10,color="white",style="solid",shape="box"];41 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 50[label="",style="solid", color="burlywood", weight=3];
139[label="wv8/AProVE_Exception wv80",fontsize=10,color="white",style="solid",shape="box"];41 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 51[label="",style="solid", color="burlywood", weight=3];
140[label="wv8/AProVE_Error wv80",fontsize=10,color="white",style="solid",shape="box"];41 -> 140[label="",style="solid", color="burlywood", weight=9];
140 -> 52[label="",style="solid", color="burlywood", weight=3];
43 -> 55[label="",style="dashed", color="red", weight=0];
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44[label="[]",fontsize=16,color="green",shape="box"];45[label="Nothing",fontsize=16,color="green",shape="box"];46[label="sequence1 (zipWith wv3 wv41 wv51) wv70",fontsize=16,color="black",shape="box"];46 -> 57[label="",style="solid", color="black", weight=3];
53[label="wv40",fontsize=16,color="green",shape="box"];54[label="wv50",fontsize=16,color="green",shape="box"];49[label="primbindIO (IO wv80) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];49 -> 58[label="",style="solid", color="black", weight=3];
50[label="primbindIO (AProVE_IO wv80) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
51[label="primbindIO (AProVE_Exception wv80) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];51 -> 60[label="",style="solid", color="black", weight=3];
52[label="primbindIO (AProVE_Error wv80) (sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="black",shape="box"];52 -> 61[label="",style="solid", color="black", weight=3];
56 -> 25[label="",style="dashed", color="red", weight=0];
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58[label="error []",fontsize=16,color="red",shape="box"];59[label="sequence1 (zipWith wv3 wv41 wv51) wv80",fontsize=16,color="black",shape="box"];59 -> 66[label="",style="solid", color="black", weight=3];
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65 -> 6[label="",style="dashed", color="red", weight=0];
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65 -> 70[label="",style="dashed", color="magenta", weight=3];
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141 -> 71[label="",style="solid", color="burlywood", weight=3];
142[label="wv10/Just wv100",fontsize=10,color="white",style="solid",shape="box"];64 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 72[label="",style="solid", color="burlywood", weight=3];
66 -> 73[label="",style="dashed", color="red", weight=0];
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143 -> 77[label="",style="solid", color="burlywood", weight=3];
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75[label="wv51",fontsize=16,color="green",shape="box"];76[label="wv41",fontsize=16,color="green",shape="box"];77[label="(wv110 : wv111 >>= sequence0 wv60) ++ wv9",fontsize=16,color="black",shape="box"];77 -> 84[label="",style="solid", color="black", weight=3];
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79[label="Nothing",fontsize=16,color="green",shape="box"];80[label="sequence0 wv70 wv100",fontsize=16,color="black",shape="box"];80 -> 86[label="",style="solid", color="black", weight=3];
81[label="wv51",fontsize=16,color="green",shape="box"];82[label="wv41",fontsize=16,color="green",shape="box"];83[label="primbindIO wv12 (sequence0 wv80)",fontsize=16,color="burlywood",shape="box"];145[label="wv12/IO wv120",fontsize=10,color="white",style="solid",shape="box"];83 -> 145[label="",style="solid", color="burlywood", weight=9];
145 -> 87[label="",style="solid", color="burlywood", weight=3];
146[label="wv12/AProVE_IO wv120",fontsize=10,color="white",style="solid",shape="box"];83 -> 146[label="",style="solid", color="burlywood", weight=9];
146 -> 88[label="",style="solid", color="burlywood", weight=3];
147[label="wv12/AProVE_Exception wv120",fontsize=10,color="white",style="solid",shape="box"];83 -> 147[label="",style="solid", color="burlywood", weight=9];
147 -> 89[label="",style="solid", color="burlywood", weight=3];
148[label="wv12/AProVE_Error wv120",fontsize=10,color="white",style="solid",shape="box"];83 -> 148[label="",style="solid", color="burlywood", weight=9];
148 -> 90[label="",style="solid", color="burlywood", weight=3];
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85[label="[] ++ wv9",fontsize=16,color="black",shape="triangle"];85 -> 92[label="",style="solid", color="black", weight=3];
86[label="return (wv70 : wv100)",fontsize=16,color="black",shape="box"];86 -> 93[label="",style="solid", color="black", weight=3];
87[label="primbindIO (IO wv120) (sequence0 wv80)",fontsize=16,color="black",shape="box"];87 -> 94[label="",style="solid", color="black", weight=3];
88[label="primbindIO (AProVE_IO wv120) (sequence0 wv80)",fontsize=16,color="black",shape="box"];88 -> 95[label="",style="solid", color="black", weight=3];
89[label="primbindIO (AProVE_Exception wv120) (sequence0 wv80)",fontsize=16,color="black",shape="box"];89 -> 96[label="",style="solid", color="black", weight=3];
90[label="primbindIO (AProVE_Error wv120) (sequence0 wv80)",fontsize=16,color="black",shape="box"];90 -> 97[label="",style="solid", color="black", weight=3];
91[label="(return (wv60 : wv110) ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3];
92[label="wv9",fontsize=16,color="green",shape="box"];93[label="Just (wv70 : wv100)",fontsize=16,color="green",shape="box"];94[label="error []",fontsize=16,color="red",shape="box"];95[label="sequence0 wv80 wv120",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3];
96[label="AProVE_Exception wv120",fontsize=16,color="green",shape="box"];97[label="AProVE_Error wv120",fontsize=16,color="green",shape="box"];98[label="(((wv60 : wv110) : []) ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="black",shape="box"];98 -> 100[label="",style="solid", color="black", weight=3];
99[label="return (wv80 : wv120)",fontsize=16,color="black",shape="box"];99 -> 101[label="",style="solid", color="black", weight=3];
100 -> 102[label="",style="dashed", color="red", weight=0];
100[label="((wv60 : wv110) : [] ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="magenta"];100 -> 103[label="",style="dashed", color="magenta", weight=3];
101[label="primretIO (wv80 : wv120)",fontsize=16,color="black",shape="box"];101 -> 104[label="",style="solid", color="black", weight=3];
103 -> 85[label="",style="dashed", color="red", weight=0];
103[label="[] ++ (wv111 >>= sequence0 wv60)",fontsize=16,color="magenta"];103 -> 105[label="",style="dashed", color="magenta", weight=3];
102[label="((wv60 : wv110) : wv13) ++ wv9",fontsize=16,color="black",shape="triangle"];102 -> 106[label="",style="solid", color="black", weight=3];
104[label="AProVE_IO (wv80 : wv120)",fontsize=16,color="green",shape="box"];105[label="wv111 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];149[label="wv111/wv1110 : wv1111",fontsize=10,color="white",style="solid",shape="box"];105 -> 149[label="",style="solid", color="burlywood", weight=9];
149 -> 107[label="",style="solid", color="burlywood", weight=3];
150[label="wv111/[]",fontsize=10,color="white",style="solid",shape="box"];105 -> 150[label="",style="solid", color="burlywood", weight=9];
150 -> 108[label="",style="solid", color="burlywood", weight=3];
106[label="(wv60 : wv110) : wv13 ++ wv9",fontsize=16,color="green",shape="box"];106 -> 109[label="",style="dashed", color="green", weight=3];
107[label="wv1110 : wv1111 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];107 -> 110[label="",style="solid", color="black", weight=3];
108[label="[] >>= sequence0 wv60",fontsize=16,color="black",shape="box"];108 -> 111[label="",style="solid", color="black", weight=3];
109[label="wv13 ++ wv9",fontsize=16,color="burlywood",shape="triangle"];151[label="wv13/wv130 : wv131",fontsize=10,color="white",style="solid",shape="box"];109 -> 151[label="",style="solid", color="burlywood", weight=9];
151 -> 112[label="",style="solid", color="burlywood", weight=3];
152[label="wv13/[]",fontsize=10,color="white",style="solid",shape="box"];109 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 113[label="",style="solid", color="burlywood", weight=3];
110 -> 109[label="",style="dashed", color="red", weight=0];
110[label="sequence0 wv60 wv1110 ++ (wv1111 >>= sequence0 wv60)",fontsize=16,color="magenta"];110 -> 114[label="",style="dashed", color="magenta", weight=3];
110 -> 115[label="",style="dashed", color="magenta", weight=3];
111[label="[]",fontsize=16,color="green",shape="box"];112[label="(wv130 : wv131) ++ wv9",fontsize=16,color="black",shape="box"];112 -> 116[label="",style="solid", color="black", weight=3];
113[label="[] ++ wv9",fontsize=16,color="black",shape="box"];113 -> 117[label="",style="solid", color="black", weight=3];
114 -> 105[label="",style="dashed", color="red", weight=0];
114[label="wv1111 >>= sequence0 wv60",fontsize=16,color="magenta"];114 -> 118[label="",style="dashed", color="magenta", weight=3];
115[label="sequence0 wv60 wv1110",fontsize=16,color="black",shape="box"];115 -> 119[label="",style="solid", color="black", weight=3];
116[label="wv130 : wv131 ++ wv9",fontsize=16,color="green",shape="box"];116 -> 120[label="",style="dashed", color="green", weight=3];
117[label="wv9",fontsize=16,color="green",shape="box"];118[label="wv1111",fontsize=16,color="green",shape="box"];119[label="return (wv60 : wv1110)",fontsize=16,color="black",shape="box"];119 -> 121[label="",style="solid", color="black", weight=3];
120 -> 109[label="",style="dashed", color="red", weight=0];
120[label="wv131 ++ wv9",fontsize=16,color="magenta"];120 -> 122[label="",style="dashed", color="magenta", weight=3];
121[label="(wv60 : wv1110) : []",fontsize=16,color="green",shape="box"];122[label="wv131",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(:(wv1110, wv1111), wv60, h) -> new_gtGtEs(wv1111, wv60, h)

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

(10) 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_gtGtEs(:(wv1110, wv1111), wv60, h) -> new_gtGtEs(wv1111, wv60, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(wv130, wv131), wv9, h) -> new_psPs(wv131, wv9, h)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_psPs(:(wv130, wv131), wv9, h) -> new_psPs(wv131, wv9, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_psPs0(wv3, wv41, wv51, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_IO, h, ba, bb) -> new_primbindIO(wv3, wv41, wv51, h, ba, bb)
   new_psPs0(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_Maybe, h, ba, bb) -> new_gtGtEs2(wv3, wv41, wv51, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
   new_primbindIO(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)
   new_gtGtEs2(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_Maybe, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

(16) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
----------------------------------------

(17)
Complex Obligation (AND)

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs2(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_Maybe, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_Maybe, h, ba, bb) -> new_gtGtEs2(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

(19) 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_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_Maybe, h, ba, bb) -> new_gtGtEs2(wv3, wv41, wv51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 5 >= 4, 6 >= 5, 7 >= 6


*new_gtGtEs2(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_Maybe, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 5, 5 >= 6, 6 >= 7


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

(20)
YES

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

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

   new_primbindIO(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_IO, h, ba, bb) -> new_primbindIO(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_IO, h, ba, bb) -> new_primbindIO(wv3, wv41, wv51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 5 >= 4, 6 >= 5, 7 >= 6


*new_primbindIO(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_IO, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 5, 5 >= 6, 6 >= 7


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

(23)
YES

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

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

   new_psPs0(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_psPs0(wv3, wv41, wv51, h, ba, bb)
   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

(25) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   new_psPs0(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, ty_[], h, ba, bb)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_2
   POL(new_gtGtEs0(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + x_3
   POL(new_psPs0(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + x_3
   POL(new_sequence(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_3 + x_4
   POL(ty_[]) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


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

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

   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), ty_[], h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)
   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_psPs0(wv3, wv41, wv51, h, ba, bb)
   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

(27) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

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

   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

The set Q consists of the following terms:

   new_psPs2(x0, x1, x2, x3, x4)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs1([], x0, x1, x2)
   new_gtGtEs3(:(x0, x1), x2, x3)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_psPs3(x0, x1)
   new_gtGtEs3([], x0, x1)
   new_psPs4([], x0, x1)
   new_psPs1(:(x0, x1), x2, x3, x4)
   new_psPs5(x0, x1, x2, x3, x4, x5, x6, x7)
   new_psPs4(:(x0, x1), x2, x3)

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

(29) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
----------------------------------------

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

   new_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_psPs1([], wv60, wv9, h) -> new_psPs3(wv9, h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs5(wv3, wv41, wv51, wv60, wv9, h, ba, bb) -> new_psPs1(new_sequence0(wv3, wv41, wv51, ty_[], h, ba, bb), wv60, wv9, h)
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs5(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_gtGtEs3([], wv60, h) -> []
   new_psPs4([], wv9, h) -> wv9
   new_psPs4(:(wv130, wv131), wv9, h) -> :(wv130, new_psPs4(wv131, wv9, h))
   new_psPs1(:(wv110, wv111), wv60, wv9, h) -> new_psPs2(wv60, wv110, new_psPs3(new_gtGtEs3(wv111, wv60, h), h), wv9, h)
   new_gtGtEs3(:(wv1110, wv1111), wv60, h) -> new_psPs4(:(:(wv60, wv1110), []), new_gtGtEs3(wv1111, wv60, h), h)
   new_psPs3(wv9, h) -> wv9
   new_psPs2(wv60, wv110, wv13, wv9, h) -> :(:(wv60, wv110), new_psPs4(wv13, wv9, h))

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

(31) 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_gtGtEs0(wv3, wv41, wv51, h, ba, bb) evaluates to  t =new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

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_gtGtEs0(wv3, wv41, wv51, h, ba, bb) to new_gtGtEs0(wv3, wv41, wv51, h, ba, bb).




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

(32)
NO

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

(33) Narrow (COMPLETE)
Haskell To QDPs

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90[label="primbindIO (AProVE_Error wv120) (sequence0 wv80)",fontsize=16,color="black",shape="box"];90 -> 97[label="",style="solid", color="black", weight=3];
91[label="(return (wv60 : wv110) ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3];
92[label="wv9",fontsize=16,color="green",shape="box"];93[label="Just (wv70 : wv100)",fontsize=16,color="green",shape="box"];94[label="error []",fontsize=16,color="red",shape="box"];95[label="sequence0 wv80 wv120",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3];
96[label="AProVE_Exception wv120",fontsize=16,color="green",shape="box"];97[label="AProVE_Error wv120",fontsize=16,color="green",shape="box"];98[label="(((wv60 : wv110) : []) ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="black",shape="box"];98 -> 100[label="",style="solid", color="black", weight=3];
99[label="return (wv80 : wv120)",fontsize=16,color="black",shape="box"];99 -> 101[label="",style="solid", color="black", weight=3];
100 -> 102[label="",style="dashed", color="red", weight=0];
100[label="((wv60 : wv110) : [] ++ (wv111 >>= sequence0 wv60)) ++ wv9",fontsize=16,color="magenta"];100 -> 103[label="",style="dashed", color="magenta", weight=3];
101[label="primretIO (wv80 : wv120)",fontsize=16,color="black",shape="box"];101 -> 104[label="",style="solid", color="black", weight=3];
103 -> 85[label="",style="dashed", color="red", weight=0];
103[label="[] ++ (wv111 >>= sequence0 wv60)",fontsize=16,color="magenta"];103 -> 105[label="",style="dashed", color="magenta", weight=3];
102[label="((wv60 : wv110) : wv13) ++ wv9",fontsize=16,color="black",shape="triangle"];102 -> 106[label="",style="solid", color="black", weight=3];
104[label="AProVE_IO (wv80 : wv120)",fontsize=16,color="green",shape="box"];105[label="wv111 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];149[label="wv111/wv1110 : wv1111",fontsize=10,color="white",style="solid",shape="box"];105 -> 149[label="",style="solid", color="burlywood", weight=9];
149 -> 107[label="",style="solid", color="burlywood", weight=3];
150[label="wv111/[]",fontsize=10,color="white",style="solid",shape="box"];105 -> 150[label="",style="solid", color="burlywood", weight=9];
150 -> 108[label="",style="solid", color="burlywood", weight=3];
106[label="(wv60 : wv110) : wv13 ++ wv9",fontsize=16,color="green",shape="box"];106 -> 109[label="",style="dashed", color="green", weight=3];
107[label="wv1110 : wv1111 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];107 -> 110[label="",style="solid", color="black", weight=3];
108[label="[] >>= sequence0 wv60",fontsize=16,color="black",shape="box"];108 -> 111[label="",style="solid", color="black", weight=3];
109[label="wv13 ++ wv9",fontsize=16,color="burlywood",shape="triangle"];151[label="wv13/wv130 : wv131",fontsize=10,color="white",style="solid",shape="box"];109 -> 151[label="",style="solid", color="burlywood", weight=9];
151 -> 112[label="",style="solid", color="burlywood", weight=3];
152[label="wv13/[]",fontsize=10,color="white",style="solid",shape="box"];109 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 113[label="",style="solid", color="burlywood", weight=3];
110 -> 109[label="",style="dashed", color="red", weight=0];
110[label="sequence0 wv60 wv1110 ++ (wv1111 >>= sequence0 wv60)",fontsize=16,color="magenta"];110 -> 114[label="",style="dashed", color="magenta", weight=3];
110 -> 115[label="",style="dashed", color="magenta", weight=3];
111[label="[]",fontsize=16,color="green",shape="box"];112[label="(wv130 : wv131) ++ wv9",fontsize=16,color="black",shape="box"];112 -> 116[label="",style="solid", color="black", weight=3];
113[label="[] ++ wv9",fontsize=16,color="black",shape="box"];113 -> 117[label="",style="solid", color="black", weight=3];
114 -> 105[label="",style="dashed", color="red", weight=0];
114[label="wv1111 >>= sequence0 wv60",fontsize=16,color="magenta"];114 -> 118[label="",style="dashed", color="magenta", weight=3];
115[label="sequence0 wv60 wv1110",fontsize=16,color="black",shape="box"];115 -> 119[label="",style="solid", color="black", weight=3];
116[label="wv130 : wv131 ++ wv9",fontsize=16,color="green",shape="box"];116 -> 120[label="",style="dashed", color="green", weight=3];
117[label="wv9",fontsize=16,color="green",shape="box"];118[label="wv1111",fontsize=16,color="green",shape="box"];119[label="return (wv60 : wv1110)",fontsize=16,color="black",shape="box"];119 -> 121[label="",style="solid", color="black", weight=3];
120 -> 109[label="",style="dashed", color="red", weight=0];
120[label="wv131 ++ wv9",fontsize=16,color="magenta"];120 -> 122[label="",style="dashed", color="magenta", weight=3];
121[label="(wv60 : wv1110) : []",fontsize=16,color="green",shape="box"];122[label="wv131",fontsize=16,color="green",shape="box"];}

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

(34)
TRUE