/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QDPOrderProof [EQUIVALENT, 38 ms]
        (14) QDP
        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) MNOCProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) NonTerminationLoopProof [COMPLETE, 5 ms]
        (20) NO
    (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) YES
(24) Narrow [COMPLETE, 0 ms]
(25) 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 b => (a  ->  d  ->  b c)  ->  [a]  ->  [d]  ->  b [c];
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 a => (c  ->  d  ->  a b)  ->  [c]  ->  [d]  ->  a [b];
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 c => (d  ->  b  ->  c a)  ->  [d]  ->  [b]  ->  c [a];
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 d => (b  ->  c  ->  d a)  ->  [b]  ->  [c]  ->  d [a];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];63[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 7[label="",style="solid", color="burlywood", weight=3];
64[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];65[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 9[label="",style="solid", color="burlywood", weight=3];
66[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="sequence (zipWith wv3 (wv40 : wv41) (wv50 : wv51))",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="sequence (zipWith wv3 (wv40 : wv41) [])",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="sequence []",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="sequence (wv3 wv40 wv50 : zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13 -> 11[label="",style="dashed", color="red", weight=0];
13[label="sequence []",fontsize=16,color="magenta"];14[label="return []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 17[label="",style="dashed", color="red", weight=0];
15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];15 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="[] : []",fontsize=16,color="green",shape="box"];18[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];18 -> 23[label="",style="dashed", color="green", weight=3];
18 -> 24[label="",style="dashed", color="green", weight=3];
17[label="wv6 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];67[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];17 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 21[label="",style="solid", color="burlywood", weight=3];
68[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 22[label="",style="solid", color="burlywood", weight=3];
23[label="wv40",fontsize=16,color="green",shape="box"];24[label="wv50",fontsize=16,color="green",shape="box"];21[label="wv60 : wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="[] >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
25 -> 27[label="",style="dashed", color="red", weight=0];
25[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ (wv61 >>= sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
26[label="[]",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
28[label="wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
27[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ wv7",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
29[label="wv61",fontsize=16,color="green",shape="box"];30 -> 31[label="",style="dashed", color="red", weight=0];
30[label="(sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60) ++ wv7",fontsize=16,color="magenta"];30 -> 32[label="",style="dashed", color="magenta", weight=3];
32 -> 6[label="",style="dashed", color="red", weight=0];
32[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];32 -> 33[label="",style="dashed", color="magenta", weight=3];
32 -> 34[label="",style="dashed", color="magenta", weight=3];
31[label="(wv8 >>= sequence0 wv60) ++ wv7",fontsize=16,color="burlywood",shape="triangle"];69[label="wv8/wv80 : wv81",fontsize=10,color="white",style="solid",shape="box"];31 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 35[label="",style="solid", color="burlywood", weight=3];
70[label="wv8/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="wv51",fontsize=16,color="green",shape="box"];34[label="wv41",fontsize=16,color="green",shape="box"];35[label="(wv80 : wv81 >>= sequence0 wv60) ++ wv7",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="([] >>= sequence0 wv60) ++ wv7",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="(sequence0 wv60 wv80 ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="[] ++ wv7",fontsize=16,color="black",shape="triangle"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="(return (wv60 : wv80) ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="wv7",fontsize=16,color="green",shape="box"];41[label="(((wv60 : wv80) : []) ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42 -> 43[label="",style="dashed", color="red", weight=0];
42[label="((wv60 : wv80) : [] ++ (wv81 >>= sequence0 wv60)) ++ wv7",fontsize=16,color="magenta"];42 -> 44[label="",style="dashed", color="magenta", weight=3];
44 -> 38[label="",style="dashed", color="red", weight=0];
44[label="[] ++ (wv81 >>= sequence0 wv60)",fontsize=16,color="magenta"];44 -> 45[label="",style="dashed", color="magenta", weight=3];
43[label="((wv60 : wv80) : wv9) ++ wv7",fontsize=16,color="black",shape="triangle"];43 -> 46[label="",style="solid", color="black", weight=3];
45[label="wv81 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];71[label="wv81/wv810 : wv811",fontsize=10,color="white",style="solid",shape="box"];45 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 47[label="",style="solid", color="burlywood", weight=3];
72[label="wv81/[]",fontsize=10,color="white",style="solid",shape="box"];45 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 48[label="",style="solid", color="burlywood", weight=3];
46[label="(wv60 : wv80) : wv9 ++ wv7",fontsize=16,color="green",shape="box"];46 -> 49[label="",style="dashed", color="green", weight=3];
47[label="wv810 : wv811 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
48[label="[] >>= sequence0 wv60",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
49[label="wv9 ++ wv7",fontsize=16,color="burlywood",shape="triangle"];73[label="wv9/wv90 : wv91",fontsize=10,color="white",style="solid",shape="box"];49 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 52[label="",style="solid", color="burlywood", weight=3];
74[label="wv9/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 53[label="",style="solid", color="burlywood", weight=3];
50 -> 49[label="",style="dashed", color="red", weight=0];
50[label="sequence0 wv60 wv810 ++ (wv811 >>= sequence0 wv60)",fontsize=16,color="magenta"];50 -> 54[label="",style="dashed", color="magenta", weight=3];
50 -> 55[label="",style="dashed", color="magenta", weight=3];
51[label="[]",fontsize=16,color="green",shape="box"];52[label="(wv90 : wv91) ++ wv7",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="[] ++ wv7",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
54[label="sequence0 wv60 wv810",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3];
55 -> 45[label="",style="dashed", color="red", weight=0];
55[label="wv811 >>= sequence0 wv60",fontsize=16,color="magenta"];55 -> 59[label="",style="dashed", color="magenta", weight=3];
56[label="wv90 : wv91 ++ wv7",fontsize=16,color="green",shape="box"];56 -> 60[label="",style="dashed", color="green", weight=3];
57[label="wv7",fontsize=16,color="green",shape="box"];58[label="return (wv60 : wv810)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="wv811",fontsize=16,color="green",shape="box"];60 -> 49[label="",style="dashed", color="red", weight=0];
60[label="wv91 ++ wv7",fontsize=16,color="magenta"];60 -> 62[label="",style="dashed", color="magenta", weight=3];
61[label="(wv60 : wv810) : []",fontsize=16,color="green",shape="box"];62[label="wv91",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(:(wv810, wv811), wv60, h) -> new_gtGtEs(wv811, 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(:(wv810, wv811), wv60, h) -> new_gtGtEs(wv811, 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_gtGtEs0(wv3, wv41, wv51, h, ba, bb) -> new_psPs0(wv3, wv41, wv51, h, ba, bb)
   new_psPs0(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, 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), h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, h, ba, bb)

The TRS R consists of the following rules:

   new_gtGtEs2(:(wv810, wv811), wv60, h) -> new_psPs1(:(:(wv60, wv810), []), new_gtGtEs2(wv811, wv60, h), h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs2(wv3, wv41, wv51, wv60, wv7, h, ba, bb) -> new_psPs4(new_sequence0(wv3, wv41, wv51, h, ba, bb), wv60, wv7, h)
   new_psPs1(:(wv90, wv91), wv7, h) -> :(wv90, new_psPs1(wv91, wv7, h))
   new_psPs3(wv60, wv80, wv9, wv7, h) -> :(:(wv60, wv80), new_psPs1(wv9, wv7, h))
   new_gtGtEs2([], wv60, h) -> []
   new_psPs1([], wv7, h) -> wv7
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs2(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_psPs4(:(wv80, wv81), wv60, wv7, h) -> new_psPs3(wv60, wv80, new_psPs5(new_gtGtEs2(wv81, wv60, h), h), wv7, h)
   new_psPs4([], wv60, wv7, h) -> new_psPs5(wv7, h)
   new_psPs5(wv7, h) -> wv7

The set Q consists of the following terms:

   new_psPs1(:(x0, x1), x2, x3)
   new_gtGtEs2(:(x0, x1), x2, x3)
   new_psPs2(x0, x1, x2, x3, x4, x5, x6, x7)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs4(:(x0, x1), x2, x3, x4)
   new_psPs4([], x0, x1, x2)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1)
   new_psPs1([], x0, x1)
   new_psPs5(x0, x1)

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

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


The following pairs can be oriented strictly and are deleted.

   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_gtGtEs0(wv3, wv41, wv51, 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)) = x_2 + x_3
   POL(new_psPs0(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_3
   POL(new_sequence(x_1, x_2, x_3, x_4, x_5, x_6)) = x_2 + x_3

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


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

(14)
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_psPs0(wv3, wv41, wv51, h, ba, bb) -> new_sequence(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_gtGtEs2(:(wv810, wv811), wv60, h) -> new_psPs1(:(:(wv60, wv810), []), new_gtGtEs2(wv811, wv60, h), h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs2(wv3, wv41, wv51, wv60, wv7, h, ba, bb) -> new_psPs4(new_sequence0(wv3, wv41, wv51, h, ba, bb), wv60, wv7, h)
   new_psPs1(:(wv90, wv91), wv7, h) -> :(wv90, new_psPs1(wv91, wv7, h))
   new_psPs3(wv60, wv80, wv9, wv7, h) -> :(:(wv60, wv80), new_psPs1(wv9, wv7, h))
   new_gtGtEs2([], wv60, h) -> []
   new_psPs1([], wv7, h) -> wv7
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs2(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_psPs4(:(wv80, wv81), wv60, wv7, h) -> new_psPs3(wv60, wv80, new_psPs5(new_gtGtEs2(wv81, wv60, h), h), wv7, h)
   new_psPs4([], wv60, wv7, h) -> new_psPs5(wv7, h)
   new_psPs5(wv7, h) -> wv7

The set Q consists of the following terms:

   new_psPs1(:(x0, x1), x2, x3)
   new_gtGtEs2(:(x0, x1), x2, x3)
   new_psPs2(x0, x1, x2, x3, x4, x5, x6, x7)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs4(:(x0, x1), x2, x3, x4)
   new_psPs4([], x0, x1, x2)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1)
   new_psPs1([], x0, x1)
   new_psPs5(x0, x1)

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

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

(16)
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_gtGtEs2(:(wv810, wv811), wv60, h) -> new_psPs1(:(:(wv60, wv810), []), new_gtGtEs2(wv811, wv60, h), h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs2(wv3, wv41, wv51, wv60, wv7, h, ba, bb) -> new_psPs4(new_sequence0(wv3, wv41, wv51, h, ba, bb), wv60, wv7, h)
   new_psPs1(:(wv90, wv91), wv7, h) -> :(wv90, new_psPs1(wv91, wv7, h))
   new_psPs3(wv60, wv80, wv9, wv7, h) -> :(:(wv60, wv80), new_psPs1(wv9, wv7, h))
   new_gtGtEs2([], wv60, h) -> []
   new_psPs1([], wv7, h) -> wv7
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs2(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_psPs4(:(wv80, wv81), wv60, wv7, h) -> new_psPs3(wv60, wv80, new_psPs5(new_gtGtEs2(wv81, wv60, h), h), wv7, h)
   new_psPs4([], wv60, wv7, h) -> new_psPs5(wv7, h)
   new_psPs5(wv7, h) -> wv7

The set Q consists of the following terms:

   new_psPs1(:(x0, x1), x2, x3)
   new_gtGtEs2(:(x0, x1), x2, x3)
   new_psPs2(x0, x1, x2, x3, x4, x5, x6, x7)
   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5, x6, x7)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs4(:(x0, x1), x2, x3, x4)
   new_psPs4([], x0, x1, x2)
   new_gtGtEs1([], x0, x1, x2, x3, x4, x5)
   new_gtGtEs2([], x0, x1)
   new_psPs1([], x0, x1)
   new_psPs5(x0, x1)

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

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

(18)
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_gtGtEs2(:(wv810, wv811), wv60, h) -> new_psPs1(:(:(wv60, wv810), []), new_gtGtEs2(wv811, wv60, h), h)
   new_gtGtEs1([], wv3, wv41, wv51, h, ba, bb) -> []
   new_psPs2(wv3, wv41, wv51, wv60, wv7, h, ba, bb) -> new_psPs4(new_sequence0(wv3, wv41, wv51, h, ba, bb), wv60, wv7, h)
   new_psPs1(:(wv90, wv91), wv7, h) -> :(wv90, new_psPs1(wv91, wv7, h))
   new_psPs3(wv60, wv80, wv9, wv7, h) -> :(:(wv60, wv80), new_psPs1(wv9, wv7, h))
   new_gtGtEs2([], wv60, h) -> []
   new_psPs1([], wv7, h) -> wv7
   new_gtGtEs1(:(wv60, wv61), wv3, wv41, wv51, h, ba, bb) -> new_psPs2(wv3, wv41, wv51, wv60, new_gtGtEs1(wv61, wv3, wv41, wv51, h, ba, bb), h, ba, bb)
   new_psPs4(:(wv80, wv81), wv60, wv7, h) -> new_psPs3(wv60, wv80, new_psPs5(new_gtGtEs2(wv81, wv60, h), h), wv7, h)
   new_psPs4([], wv60, wv7, h) -> new_psPs5(wv7, h)
   new_psPs5(wv7, h) -> wv7

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

(19) 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).




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

(20)
NO

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

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

   new_psPs(:(wv90, wv91), wv7, h) -> new_psPs(wv91, wv7, h)

R is empty.
Q is empty.
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_psPs(:(wv90, wv91), wv7, h) -> new_psPs(wv91, wv7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(23)
YES

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

(24) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];63[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 7[label="",style="solid", color="burlywood", weight=3];
64[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];65[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 9[label="",style="solid", color="burlywood", weight=3];
66[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
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13 -> 11[label="",style="dashed", color="red", weight=0];
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15 -> 17[label="",style="dashed", color="red", weight=0];
15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];15 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="[] : []",fontsize=16,color="green",shape="box"];18[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];18 -> 23[label="",style="dashed", color="green", weight=3];
18 -> 24[label="",style="dashed", color="green", weight=3];
17[label="wv6 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];67[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];17 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 21[label="",style="solid", color="burlywood", weight=3];
68[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 22[label="",style="solid", color="burlywood", weight=3];
23[label="wv40",fontsize=16,color="green",shape="box"];24[label="wv50",fontsize=16,color="green",shape="box"];21[label="wv60 : wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="[] >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
25 -> 27[label="",style="dashed", color="red", weight=0];
25[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ (wv61 >>= sequence1 (zipWith wv3 wv41 wv51))",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
26[label="[]",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
28[label="wv61 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
27[label="sequence1 (zipWith wv3 wv41 wv51) wv60 ++ wv7",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
29[label="wv61",fontsize=16,color="green",shape="box"];30 -> 31[label="",style="dashed", color="red", weight=0];
30[label="(sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60) ++ wv7",fontsize=16,color="magenta"];30 -> 32[label="",style="dashed", color="magenta", weight=3];
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69 -> 35[label="",style="solid", color="burlywood", weight=3];
70[label="wv8/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="wv51",fontsize=16,color="green",shape="box"];34[label="wv41",fontsize=16,color="green",shape="box"];35[label="(wv80 : wv81 >>= sequence0 wv60) ++ wv7",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
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57[label="wv7",fontsize=16,color="green",shape="box"];58[label="return (wv60 : wv810)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="wv811",fontsize=16,color="green",shape="box"];60 -> 49[label="",style="dashed", color="red", weight=0];
60[label="wv91 ++ wv7",fontsize=16,color="magenta"];60 -> 62[label="",style="dashed", color="magenta", weight=3];
61[label="(wv60 : wv810) : []",fontsize=16,color="green",shape="box"];62[label="wv91",fontsize=16,color="green",shape="box"];}

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

(25)
TRUE