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


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


MAYBE
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) AND
    (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) MNOCProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) NonTerminationLoopProof [COMPLETE, 0 ms]
        (19) NO
    (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
(26) Narrow [COMPLETE, 0 ms]
(27) 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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= (\flg ->filterM p xs >>= (\ys ->return ( if flg then x : ys else ys)));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\ys->return (if flg then x : ys else ys)"
is transformed to
"filterM0 flg x ys = return (if flg then x : ys else ys);
"
The following Lambda expression
"\flg->filterM p xs >>= filterM0 flg x"
is transformed to
"filterM1 p xs x flg = filterM p xs >>= filterM0 flg 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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return ( if flg then x : ys else ys);

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

(3) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if flg then x : ys else ys"
is transformed to
"filterM00 x ys True = x : ys;
filterM00 x ys False = ys;
"

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

(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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

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

(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;
filterM :: Monad a => (b  ->  a Bool)  ->  [b]  ->  a [b];
filterM vy [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

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

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

(8)
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;
filterM :: Monad a => (b  ->  a Bool)  ->  [b]  ->  a [b];
filterM vy [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.filterM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.filterM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.filterM vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];62[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 5[label="",style="solid", color="burlywood", weight=3];
63[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="Monad.filterM vz3 (vz40 : vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Monad.filterM vz3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="vz3 vz40 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8 -> 56[label="",style="dashed", color="red", weight=0];
8[label="return []",fontsize=16,color="magenta"];8 -> 57[label="",style="dashed", color="magenta", weight=3];
10[label="vz3 vz40",fontsize=16,color="green",shape="box"];10 -> 15[label="",style="dashed", color="green", weight=3];
9[label="vz5 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="burlywood",shape="triangle"];64[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 13[label="",style="solid", color="burlywood", weight=3];
65[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 14[label="",style="solid", color="burlywood", weight=3];
57[label="[]",fontsize=16,color="green",shape="box"];56[label="return vz9",fontsize=16,color="black",shape="triangle"];56 -> 59[label="",style="solid", color="black", weight=3];
15[label="vz40",fontsize=16,color="green",shape="box"];13[label="vz50 : vz51 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="[] >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
59[label="vz9 : []",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="Monad.filterM1 vz3 vz41 vz40 vz50 ++ (vz51 >>= Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];19 -> 9[label="",style="dashed", color="red", weight=0];
19[label="vz51 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
18[label="Monad.filterM1 vz3 vz41 vz40 vz50 ++ vz6",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="vz51",fontsize=16,color="green",shape="box"];21 -> 22[label="",style="dashed", color="red", weight=0];
21[label="(Monad.filterM vz3 vz41 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="magenta"];21 -> 23[label="",style="dashed", color="magenta", weight=3];
23 -> 4[label="",style="dashed", color="red", weight=0];
23[label="Monad.filterM vz3 vz41",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
22[label="(vz7 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="burlywood",shape="triangle"];66[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];22 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 25[label="",style="solid", color="burlywood", weight=3];
67[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 26[label="",style="solid", color="burlywood", weight=3];
24[label="vz41",fontsize=16,color="green",shape="box"];25[label="(vz70 : vz71 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="([] >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(Monad.filterM0 vz50 vz40 vz70 ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="[] ++ vz6",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="(return (Monad.filterM00 vz40 vz70 vz50) ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="vz6",fontsize=16,color="green",shape="box"];31[label="((Monad.filterM00 vz40 vz70 vz50 : []) ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32 -> 33[label="",style="dashed", color="red", weight=0];
32[label="(Monad.filterM00 vz40 vz70 vz50 : [] ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="magenta"];32 -> 34[label="",style="dashed", color="magenta", weight=3];
34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="[] ++ (vz71 >>= Monad.filterM0 vz50 vz40)",fontsize=16,color="magenta"];34 -> 35[label="",style="dashed", color="magenta", weight=3];
33[label="(Monad.filterM00 vz40 vz70 vz50 : vz8) ++ vz6",fontsize=16,color="black",shape="triangle"];33 -> 36[label="",style="solid", color="black", weight=3];
35[label="vz71 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="burlywood",shape="triangle"];68[label="vz71/vz710 : vz711",fontsize=10,color="white",style="solid",shape="box"];35 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 37[label="",style="solid", color="burlywood", weight=3];
69[label="vz71/[]",fontsize=10,color="white",style="solid",shape="box"];35 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 38[label="",style="solid", color="burlywood", weight=3];
36[label="Monad.filterM00 vz40 vz70 vz50 : vz8 ++ vz6",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3];
36 -> 40[label="",style="dashed", color="green", weight=3];
37[label="vz710 : vz711 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="[] >>= Monad.filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="Monad.filterM00 vz40 vz70 vz50",fontsize=16,color="burlywood",shape="triangle"];70[label="vz50/False",fontsize=10,color="white",style="solid",shape="box"];39 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 43[label="",style="solid", color="burlywood", weight=3];
71[label="vz50/True",fontsize=10,color="white",style="solid",shape="box"];39 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 44[label="",style="solid", color="burlywood", weight=3];
40[label="vz8 ++ vz6",fontsize=16,color="burlywood",shape="triangle"];72[label="vz8/vz80 : vz81",fontsize=10,color="white",style="solid",shape="box"];40 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 45[label="",style="solid", color="burlywood", weight=3];
73[label="vz8/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 46[label="",style="solid", color="burlywood", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="Monad.filterM0 vz50 vz40 vz710 ++ (vz711 >>= Monad.filterM0 vz50 vz40)",fontsize=16,color="magenta"];41 -> 47[label="",style="dashed", color="magenta", weight=3];
41 -> 48[label="",style="dashed", color="magenta", weight=3];
42[label="[]",fontsize=16,color="green",shape="box"];43[label="Monad.filterM00 vz40 vz70 False",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="Monad.filterM00 vz40 vz70 True",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="(vz80 : vz81) ++ vz6",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="[] ++ vz6",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47 -> 35[label="",style="dashed", color="red", weight=0];
47[label="vz711 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
48[label="Monad.filterM0 vz50 vz40 vz710",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
49[label="vz70",fontsize=16,color="green",shape="box"];50[label="vz40 : vz70",fontsize=16,color="green",shape="box"];51[label="vz80 : vz81 ++ vz6",fontsize=16,color="green",shape="box"];51 -> 55[label="",style="dashed", color="green", weight=3];
52[label="vz6",fontsize=16,color="green",shape="box"];53[label="vz711",fontsize=16,color="green",shape="box"];54 -> 56[label="",style="dashed", color="red", weight=0];
54[label="return (Monad.filterM00 vz40 vz710 vz50)",fontsize=16,color="magenta"];54 -> 58[label="",style="dashed", color="magenta", weight=3];
55 -> 40[label="",style="dashed", color="red", weight=0];
55[label="vz81 ++ vz6",fontsize=16,color="magenta"];55 -> 60[label="",style="dashed", color="magenta", weight=3];
58 -> 39[label="",style="dashed", color="red", weight=0];
58[label="Monad.filterM00 vz40 vz710 vz50",fontsize=16,color="magenta"];58 -> 61[label="",style="dashed", color="magenta", weight=3];
60[label="vz81",fontsize=16,color="green",shape="box"];61[label="vz710",fontsize=16,color="green",shape="box"];}

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

(10)
Complex Obligation (AND)

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

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

   new_psPs0(vz3, vz41, vz40, h) -> new_filterM(vz3, vz41, h)
   new_gtGtEs0(vz3, vz41, vz40, h) -> new_psPs0(vz3, vz41, vz40, h)
   new_filterM(vz3, :(vz40, vz41), h) -> new_gtGtEs0(vz3, vz41, vz40, h)
   new_gtGtEs0(vz3, vz41, vz40, h) -> new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:

   new_psPs3(vz3, vz41, vz40, vz50, vz6, h) -> new_psPs4(new_filterM0(vz3, vz41, h), vz50, vz40, vz6, h)
   new_gtGtEs2([], vz50, vz40, h) -> []
   new_gtGtEs1(:(vz50, vz51), vz3, vz41, vz40, h) -> new_psPs3(vz3, vz41, vz40, vz50, new_gtGtEs1(vz51, vz3, vz41, vz40, h), h)
   new_filterM00(vz40, vz70, False, h) -> vz70
   new_psPs4([], vz50, vz40, vz6, h) -> new_psPs5(vz6, h)
   new_gtGtEs1([], vz3, vz41, vz40, h) -> []
   new_return(vz9, h) -> :(vz9, [])
   new_psPs2([], vz6, h) -> vz6
   new_gtGtEs2(:(vz710, vz711), vz50, vz40, h) -> new_psPs2(new_return(new_filterM00(vz40, vz710, vz50, h), h), new_gtGtEs2(vz711, vz50, vz40, h), h)
   new_psPs2(:(vz80, vz81), vz6, h) -> :(vz80, new_psPs2(vz81, vz6, h))
   new_psPs1(vz40, vz70, vz50, vz8, vz6, h) -> :(new_filterM00(vz40, vz70, vz50, h), new_psPs2(vz8, vz6, h))
   new_psPs5(vz6, h) -> vz6
   new_psPs4(:(vz70, vz71), vz50, vz40, vz6, h) -> new_psPs1(vz40, vz70, vz50, new_psPs5(new_gtGtEs2(vz71, vz50, vz40, h), h), vz6, h)
   new_filterM00(vz40, vz70, True, h) -> :(vz40, vz70)

The set Q consists of the following terms:

   new_filterM00(x0, x1, False, x2)
   new_filterM00(x0, x1, True, x2)
   new_return(x0, x1)
   new_gtGtEs1([], x0, x1, x2, x3)
   new_psPs2([], x0, x1)
   new_psPs4(:(x0, x1), x2, x3, x4, x5)
   new_psPs4([], x0, x1, x2, x3)
   new_psPs1(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2(:(x0, x1), x2, x3, x4)
   new_psPs3(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1, x2)
   new_psPs2(:(x0, x1), x2, x3)
   new_psPs5(x0, x1)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)

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

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


The following pairs can be oriented strictly and are deleted.

   new_gtGtEs0(vz3, vz41, vz40, h) -> new_psPs0(vz3, vz41, vz40, h)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_1 + x_2
   POL(new_filterM(x_1, x_2, x_3)) = x_2
   POL(new_gtGtEs0(x_1, x_2, x_3, x_4)) = 1 + x_2
   POL(new_psPs0(x_1, x_2, x_3, x_4)) = x_2

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


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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs0(vz3, vz41, vz40, h) -> new_filterM(vz3, vz41, h)
   new_filterM(vz3, :(vz40, vz41), h) -> new_gtGtEs0(vz3, vz41, vz40, h)
   new_gtGtEs0(vz3, vz41, vz40, h) -> new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:

   new_psPs3(vz3, vz41, vz40, vz50, vz6, h) -> new_psPs4(new_filterM0(vz3, vz41, h), vz50, vz40, vz6, h)
   new_gtGtEs2([], vz50, vz40, h) -> []
   new_gtGtEs1(:(vz50, vz51), vz3, vz41, vz40, h) -> new_psPs3(vz3, vz41, vz40, vz50, new_gtGtEs1(vz51, vz3, vz41, vz40, h), h)
   new_filterM00(vz40, vz70, False, h) -> vz70
   new_psPs4([], vz50, vz40, vz6, h) -> new_psPs5(vz6, h)
   new_gtGtEs1([], vz3, vz41, vz40, h) -> []
   new_return(vz9, h) -> :(vz9, [])
   new_psPs2([], vz6, h) -> vz6
   new_gtGtEs2(:(vz710, vz711), vz50, vz40, h) -> new_psPs2(new_return(new_filterM00(vz40, vz710, vz50, h), h), new_gtGtEs2(vz711, vz50, vz40, h), h)
   new_psPs2(:(vz80, vz81), vz6, h) -> :(vz80, new_psPs2(vz81, vz6, h))
   new_psPs1(vz40, vz70, vz50, vz8, vz6, h) -> :(new_filterM00(vz40, vz70, vz50, h), new_psPs2(vz8, vz6, h))
   new_psPs5(vz6, h) -> vz6
   new_psPs4(:(vz70, vz71), vz50, vz40, vz6, h) -> new_psPs1(vz40, vz70, vz50, new_psPs5(new_gtGtEs2(vz71, vz50, vz40, h), h), vz6, h)
   new_filterM00(vz40, vz70, True, h) -> :(vz40, vz70)

The set Q consists of the following terms:

   new_filterM00(x0, x1, False, x2)
   new_filterM00(x0, x1, True, x2)
   new_return(x0, x1)
   new_gtGtEs1([], x0, x1, x2, x3)
   new_psPs2([], x0, x1)
   new_psPs4(:(x0, x1), x2, x3, x4, x5)
   new_psPs4([], x0, x1, x2, x3)
   new_psPs1(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2(:(x0, x1), x2, x3, x4)
   new_psPs3(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1, x2)
   new_psPs2(:(x0, x1), x2, x3)
   new_psPs5(x0, x1)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)

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

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

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

   new_gtGtEs0(vz3, vz41, vz40, h) -> new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:

   new_psPs3(vz3, vz41, vz40, vz50, vz6, h) -> new_psPs4(new_filterM0(vz3, vz41, h), vz50, vz40, vz6, h)
   new_gtGtEs2([], vz50, vz40, h) -> []
   new_gtGtEs1(:(vz50, vz51), vz3, vz41, vz40, h) -> new_psPs3(vz3, vz41, vz40, vz50, new_gtGtEs1(vz51, vz3, vz41, vz40, h), h)
   new_filterM00(vz40, vz70, False, h) -> vz70
   new_psPs4([], vz50, vz40, vz6, h) -> new_psPs5(vz6, h)
   new_gtGtEs1([], vz3, vz41, vz40, h) -> []
   new_return(vz9, h) -> :(vz9, [])
   new_psPs2([], vz6, h) -> vz6
   new_gtGtEs2(:(vz710, vz711), vz50, vz40, h) -> new_psPs2(new_return(new_filterM00(vz40, vz710, vz50, h), h), new_gtGtEs2(vz711, vz50, vz40, h), h)
   new_psPs2(:(vz80, vz81), vz6, h) -> :(vz80, new_psPs2(vz81, vz6, h))
   new_psPs1(vz40, vz70, vz50, vz8, vz6, h) -> :(new_filterM00(vz40, vz70, vz50, h), new_psPs2(vz8, vz6, h))
   new_psPs5(vz6, h) -> vz6
   new_psPs4(:(vz70, vz71), vz50, vz40, vz6, h) -> new_psPs1(vz40, vz70, vz50, new_psPs5(new_gtGtEs2(vz71, vz50, vz40, h), h), vz6, h)
   new_filterM00(vz40, vz70, True, h) -> :(vz40, vz70)

The set Q consists of the following terms:

   new_filterM00(x0, x1, False, x2)
   new_filterM00(x0, x1, True, x2)
   new_return(x0, x1)
   new_gtGtEs1([], x0, x1, x2, x3)
   new_psPs2([], x0, x1)
   new_psPs4(:(x0, x1), x2, x3, x4, x5)
   new_psPs4([], x0, x1, x2, x3)
   new_psPs1(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2(:(x0, x1), x2, x3, x4)
   new_psPs3(x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1, x2)
   new_psPs2(:(x0, x1), x2, x3)
   new_psPs5(x0, x1)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)

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

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

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

   new_gtGtEs0(vz3, vz41, vz40, h) -> new_gtGtEs0(vz3, vz41, vz40, h)

The TRS R consists of the following rules:

   new_psPs3(vz3, vz41, vz40, vz50, vz6, h) -> new_psPs4(new_filterM0(vz3, vz41, h), vz50, vz40, vz6, h)
   new_gtGtEs2([], vz50, vz40, h) -> []
   new_gtGtEs1(:(vz50, vz51), vz3, vz41, vz40, h) -> new_psPs3(vz3, vz41, vz40, vz50, new_gtGtEs1(vz51, vz3, vz41, vz40, h), h)
   new_filterM00(vz40, vz70, False, h) -> vz70
   new_psPs4([], vz50, vz40, vz6, h) -> new_psPs5(vz6, h)
   new_gtGtEs1([], vz3, vz41, vz40, h) -> []
   new_return(vz9, h) -> :(vz9, [])
   new_psPs2([], vz6, h) -> vz6
   new_gtGtEs2(:(vz710, vz711), vz50, vz40, h) -> new_psPs2(new_return(new_filterM00(vz40, vz710, vz50, h), h), new_gtGtEs2(vz711, vz50, vz40, h), h)
   new_psPs2(:(vz80, vz81), vz6, h) -> :(vz80, new_psPs2(vz81, vz6, h))
   new_psPs1(vz40, vz70, vz50, vz8, vz6, h) -> :(new_filterM00(vz40, vz70, vz50, h), new_psPs2(vz8, vz6, h))
   new_psPs5(vz6, h) -> vz6
   new_psPs4(:(vz70, vz71), vz50, vz40, vz6, h) -> new_psPs1(vz40, vz70, vz50, new_psPs5(new_gtGtEs2(vz71, vz50, vz40, h), h), vz6, h)
   new_filterM00(vz40, vz70, True, h) -> :(vz40, vz70)

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

(18) 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(vz3, vz41, vz40, h) evaluates to  t =new_gtGtEs0(vz3, vz41, vz40, h)

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(vz3, vz41, vz40, h) to new_gtGtEs0(vz3, vz41, vz40, h).




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

(19)
NO

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(:(vz710, vz711), vz50, vz40, h) -> new_gtGtEs(vz711, vz50, vz40, h)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_gtGtEs(:(vz710, vz711), vz50, vz40, h) -> new_gtGtEs(vz711, vz50, vz40, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(22)
YES

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(vz80, vz81), vz6, h) -> new_psPs(vz81, vz6, h)

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

(24) 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(:(vz80, vz81), vz6, h) -> new_psPs(vz81, vz6, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(25)
YES

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

(26) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.filterM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.filterM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.filterM vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];62[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 5[label="",style="solid", color="burlywood", weight=3];
63[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="Monad.filterM vz3 (vz40 : vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Monad.filterM vz3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="vz3 vz40 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8 -> 56[label="",style="dashed", color="red", weight=0];
8[label="return []",fontsize=16,color="magenta"];8 -> 57[label="",style="dashed", color="magenta", weight=3];
10[label="vz3 vz40",fontsize=16,color="green",shape="box"];10 -> 15[label="",style="dashed", color="green", weight=3];
9[label="vz5 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="burlywood",shape="triangle"];64[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];9 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 13[label="",style="solid", color="burlywood", weight=3];
65[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 14[label="",style="solid", color="burlywood", weight=3];
57[label="[]",fontsize=16,color="green",shape="box"];56[label="return vz9",fontsize=16,color="black",shape="triangle"];56 -> 59[label="",style="solid", color="black", weight=3];
15[label="vz40",fontsize=16,color="green",shape="box"];13[label="vz50 : vz51 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="[] >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
59[label="vz9 : []",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="Monad.filterM1 vz3 vz41 vz40 vz50 ++ (vz51 >>= Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];19 -> 9[label="",style="dashed", color="red", weight=0];
19[label="vz51 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
18[label="Monad.filterM1 vz3 vz41 vz40 vz50 ++ vz6",fontsize=16,color="black",shape="triangle"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="vz51",fontsize=16,color="green",shape="box"];21 -> 22[label="",style="dashed", color="red", weight=0];
21[label="(Monad.filterM vz3 vz41 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="magenta"];21 -> 23[label="",style="dashed", color="magenta", weight=3];
23 -> 4[label="",style="dashed", color="red", weight=0];
23[label="Monad.filterM vz3 vz41",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
22[label="(vz7 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="burlywood",shape="triangle"];66[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];22 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 25[label="",style="solid", color="burlywood", weight=3];
67[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];22 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 26[label="",style="solid", color="burlywood", weight=3];
24[label="vz41",fontsize=16,color="green",shape="box"];25[label="(vz70 : vz71 >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="([] >>= Monad.filterM0 vz50 vz40) ++ vz6",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="(Monad.filterM0 vz50 vz40 vz70 ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="[] ++ vz6",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="(return (Monad.filterM00 vz40 vz70 vz50) ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="vz6",fontsize=16,color="green",shape="box"];31[label="((Monad.filterM00 vz40 vz70 vz50 : []) ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32 -> 33[label="",style="dashed", color="red", weight=0];
32[label="(Monad.filterM00 vz40 vz70 vz50 : [] ++ (vz71 >>= Monad.filterM0 vz50 vz40)) ++ vz6",fontsize=16,color="magenta"];32 -> 34[label="",style="dashed", color="magenta", weight=3];
34 -> 28[label="",style="dashed", color="red", weight=0];
34[label="[] ++ (vz71 >>= Monad.filterM0 vz50 vz40)",fontsize=16,color="magenta"];34 -> 35[label="",style="dashed", color="magenta", weight=3];
33[label="(Monad.filterM00 vz40 vz70 vz50 : vz8) ++ vz6",fontsize=16,color="black",shape="triangle"];33 -> 36[label="",style="solid", color="black", weight=3];
35[label="vz71 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="burlywood",shape="triangle"];68[label="vz71/vz710 : vz711",fontsize=10,color="white",style="solid",shape="box"];35 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 37[label="",style="solid", color="burlywood", weight=3];
69[label="vz71/[]",fontsize=10,color="white",style="solid",shape="box"];35 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 38[label="",style="solid", color="burlywood", weight=3];
36[label="Monad.filterM00 vz40 vz70 vz50 : vz8 ++ vz6",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3];
36 -> 40[label="",style="dashed", color="green", weight=3];
37[label="vz710 : vz711 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="[] >>= Monad.filterM0 vz50 vz40",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="Monad.filterM00 vz40 vz70 vz50",fontsize=16,color="burlywood",shape="triangle"];70[label="vz50/False",fontsize=10,color="white",style="solid",shape="box"];39 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 43[label="",style="solid", color="burlywood", weight=3];
71[label="vz50/True",fontsize=10,color="white",style="solid",shape="box"];39 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 44[label="",style="solid", color="burlywood", weight=3];
40[label="vz8 ++ vz6",fontsize=16,color="burlywood",shape="triangle"];72[label="vz8/vz80 : vz81",fontsize=10,color="white",style="solid",shape="box"];40 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 45[label="",style="solid", color="burlywood", weight=3];
73[label="vz8/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 46[label="",style="solid", color="burlywood", weight=3];
41 -> 40[label="",style="dashed", color="red", weight=0];
41[label="Monad.filterM0 vz50 vz40 vz710 ++ (vz711 >>= Monad.filterM0 vz50 vz40)",fontsize=16,color="magenta"];41 -> 47[label="",style="dashed", color="magenta", weight=3];
41 -> 48[label="",style="dashed", color="magenta", weight=3];
42[label="[]",fontsize=16,color="green",shape="box"];43[label="Monad.filterM00 vz40 vz70 False",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="Monad.filterM00 vz40 vz70 True",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="(vz80 : vz81) ++ vz6",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="[] ++ vz6",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47 -> 35[label="",style="dashed", color="red", weight=0];
47[label="vz711 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="magenta"];47 -> 53[label="",style="dashed", color="magenta", weight=3];
48[label="Monad.filterM0 vz50 vz40 vz710",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
49[label="vz70",fontsize=16,color="green",shape="box"];50[label="vz40 : vz70",fontsize=16,color="green",shape="box"];51[label="vz80 : vz81 ++ vz6",fontsize=16,color="green",shape="box"];51 -> 55[label="",style="dashed", color="green", weight=3];
52[label="vz6",fontsize=16,color="green",shape="box"];53[label="vz711",fontsize=16,color="green",shape="box"];54 -> 56[label="",style="dashed", color="red", weight=0];
54[label="return (Monad.filterM00 vz40 vz710 vz50)",fontsize=16,color="magenta"];54 -> 58[label="",style="dashed", color="magenta", weight=3];
55 -> 40[label="",style="dashed", color="red", weight=0];
55[label="vz81 ++ vz6",fontsize=16,color="magenta"];55 -> 60[label="",style="dashed", color="magenta", weight=3];
58 -> 39[label="",style="dashed", color="red", weight=0];
58[label="Monad.filterM00 vz40 vz710 vz50",fontsize=16,color="magenta"];58 -> 61[label="",style="dashed", color="magenta", weight=3];
60[label="vz81",fontsize=16,color="green",shape="box"];61[label="vz710",fontsize=16,color="green",shape="box"];}

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

(27)
TRUE