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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) DependencyGraphProof [EQUIVALENT, 0 ms]
        (14) AND
            (15) QDP
                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (17) YES
            (18) QDP
                (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (20) YES
            (21) QDP
                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (23) YES
    (24) QDP
        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (26) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(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 Main where {
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

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

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence vx3",fontsize=16,color="burlywood",shape="triangle"];96[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 4[label="",style="solid", color="burlywood", weight=3];
97[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="sequence (vx30 : vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="sequence []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="vx30 >>= sequence1 vx31",fontsize=16,color="blue",shape="box"];98[label=">>= :: (Maybe a) -> (a -> Maybe ([] a)) -> Maybe ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 98[label="",style="solid", color="blue", weight=9];
98 -> 8[label="",style="solid", color="blue", weight=3];
99[label=">>= :: ([] a) -> (a -> [] ([] a)) -> [] ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 99[label="",style="solid", color="blue", weight=9];
99 -> 9[label="",style="solid", color="blue", weight=3];
100[label=">>= :: (IO a) -> (a -> IO ([] a)) -> IO ([] a)",fontsize=10,color="white",style="solid",shape="box"];6 -> 100[label="",style="solid", color="blue", weight=9];
100 -> 10[label="",style="solid", color="blue", weight=3];
7[label="return []",fontsize=16,color="blue",shape="box"];101[label="return :: ([] a) -> Maybe ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 101[label="",style="solid", color="blue", weight=9];
101 -> 11[label="",style="solid", color="blue", weight=3];
102[label="return :: ([] a) -> [] ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 102[label="",style="solid", color="blue", weight=9];
102 -> 12[label="",style="solid", color="blue", weight=3];
103[label="return :: ([] a) -> IO ([] a)",fontsize=10,color="white",style="solid",shape="box"];7 -> 103[label="",style="solid", color="blue", weight=9];
103 -> 13[label="",style="solid", color="blue", weight=3];
8[label="vx30 >>= sequence1 vx31",fontsize=16,color="burlywood",shape="box"];104[label="vx30/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 14[label="",style="solid", color="burlywood", weight=3];
105[label="vx30/Just vx300",fontsize=10,color="white",style="solid",shape="box"];8 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 15[label="",style="solid", color="burlywood", weight=3];
9[label="vx30 >>= sequence1 vx31",fontsize=16,color="burlywood",shape="triangle"];106[label="vx30/vx300 : vx301",fontsize=10,color="white",style="solid",shape="box"];9 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 16[label="",style="solid", color="burlywood", weight=3];
107[label="vx30/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 17[label="",style="solid", color="burlywood", weight=3];
10[label="vx30 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];10 -> 18[label="",style="solid", color="black", weight=3];
11[label="return []",fontsize=16,color="black",shape="box"];11 -> 19[label="",style="solid", color="black", weight=3];
12[label="return []",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="return []",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="Nothing >>= sequence1 vx31",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="Just vx300 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="vx300 : vx301 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="[] >>= sequence1 vx31",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primbindIO vx30 (sequence1 vx31)",fontsize=16,color="burlywood",shape="box"];108[label="vx30/IO vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 26[label="",style="solid", color="burlywood", weight=3];
109[label="vx30/AProVE_IO vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 27[label="",style="solid", color="burlywood", weight=3];
110[label="vx30/AProVE_Exception vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 28[label="",style="solid", color="burlywood", weight=3];
111[label="vx30/AProVE_Error vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 29[label="",style="solid", color="burlywood", weight=3];
19[label="Just []",fontsize=16,color="green",shape="box"];20[label="[] : []",fontsize=16,color="green",shape="box"];21[label="primretIO []",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="Nothing",fontsize=16,color="green",shape="box"];23[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
24 -> 32[label="",style="dashed", color="red", weight=0];
24[label="sequence1 vx31 vx300 ++ (vx301 >>= sequence1 vx31)",fontsize=16,color="magenta"];24 -> 33[label="",style="dashed", color="magenta", weight=3];
25[label="[]",fontsize=16,color="green",shape="box"];26[label="primbindIO (IO vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];26 -> 34[label="",style="solid", color="black", weight=3];
27[label="primbindIO (AProVE_IO vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];27 -> 35[label="",style="solid", color="black", weight=3];
28[label="primbindIO (AProVE_Exception vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];28 -> 36[label="",style="solid", color="black", weight=3];
29[label="primbindIO (AProVE_Error vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];29 -> 37[label="",style="solid", color="black", weight=3];
30[label="AProVE_IO []",fontsize=16,color="green",shape="box"];31 -> 38[label="",style="dashed", color="red", weight=0];
31[label="sequence vx31 >>= sequence0 vx300",fontsize=16,color="magenta"];31 -> 39[label="",style="dashed", color="magenta", weight=3];
33 -> 9[label="",style="dashed", color="red", weight=0];
33[label="vx301 >>= sequence1 vx31",fontsize=16,color="magenta"];33 -> 40[label="",style="dashed", color="magenta", weight=3];
32[label="sequence1 vx31 vx300 ++ vx4",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
34[label="error []",fontsize=16,color="red",shape="box"];35[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];35 -> 42[label="",style="solid", color="black", weight=3];
36[label="AProVE_Exception vx300",fontsize=16,color="green",shape="box"];37[label="AProVE_Error vx300",fontsize=16,color="green",shape="box"];39 -> 3[label="",style="dashed", color="red", weight=0];
39[label="sequence vx31",fontsize=16,color="magenta"];39 -> 43[label="",style="dashed", color="magenta", weight=3];
38[label="vx5 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];112[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];38 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 44[label="",style="solid", color="burlywood", weight=3];
113[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];38 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 45[label="",style="solid", color="burlywood", weight=3];
40[label="vx301",fontsize=16,color="green",shape="box"];41 -> 46[label="",style="dashed", color="red", weight=0];
41[label="(sequence vx31 >>= sequence0 vx300) ++ vx4",fontsize=16,color="magenta"];41 -> 47[label="",style="dashed", color="magenta", weight=3];
42 -> 48[label="",style="dashed", color="red", weight=0];
42[label="sequence vx31 >>= sequence0 vx300",fontsize=16,color="magenta"];42 -> 49[label="",style="dashed", color="magenta", weight=3];
43[label="vx31",fontsize=16,color="green",shape="box"];44[label="Nothing >>= sequence0 vx300",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="Just vx50 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
47 -> 3[label="",style="dashed", color="red", weight=0];
47[label="sequence vx31",fontsize=16,color="magenta"];47 -> 52[label="",style="dashed", color="magenta", weight=3];
46[label="(vx6 >>= sequence0 vx300) ++ vx4",fontsize=16,color="burlywood",shape="triangle"];114[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];46 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 53[label="",style="solid", color="burlywood", weight=3];
115[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];46 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 54[label="",style="solid", color="burlywood", weight=3];
49 -> 3[label="",style="dashed", color="red", weight=0];
49[label="sequence vx31",fontsize=16,color="magenta"];49 -> 55[label="",style="dashed", color="magenta", weight=3];
48[label="vx7 >>= sequence0 vx300",fontsize=16,color="black",shape="triangle"];48 -> 56[label="",style="solid", color="black", weight=3];
50[label="Nothing",fontsize=16,color="green",shape="box"];51[label="sequence0 vx300 vx50",fontsize=16,color="black",shape="box"];51 -> 57[label="",style="solid", color="black", weight=3];
52[label="vx31",fontsize=16,color="green",shape="box"];53[label="(vx60 : vx61 >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3];
54[label="([] >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];54 -> 59[label="",style="solid", color="black", weight=3];
55[label="vx31",fontsize=16,color="green",shape="box"];56[label="primbindIO vx7 (sequence0 vx300)",fontsize=16,color="burlywood",shape="box"];116[label="vx7/IO vx70",fontsize=10,color="white",style="solid",shape="box"];56 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 60[label="",style="solid", color="burlywood", weight=3];
117[label="vx7/AProVE_IO vx70",fontsize=10,color="white",style="solid",shape="box"];56 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 61[label="",style="solid", color="burlywood", weight=3];
118[label="vx7/AProVE_Exception vx70",fontsize=10,color="white",style="solid",shape="box"];56 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 62[label="",style="solid", color="burlywood", weight=3];
119[label="vx7/AProVE_Error vx70",fontsize=10,color="white",style="solid",shape="box"];56 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 63[label="",style="solid", color="burlywood", weight=3];
57[label="return (vx300 : vx50)",fontsize=16,color="black",shape="box"];57 -> 64[label="",style="solid", color="black", weight=3];
58[label="(sequence0 vx300 vx60 ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];58 -> 65[label="",style="solid", color="black", weight=3];
59[label="[] ++ vx4",fontsize=16,color="black",shape="triangle"];59 -> 66[label="",style="solid", color="black", weight=3];
60[label="primbindIO (IO vx70) (sequence0 vx300)",fontsize=16,color="black",shape="box"];60 -> 67[label="",style="solid", color="black", weight=3];
61[label="primbindIO (AProVE_IO vx70) (sequence0 vx300)",fontsize=16,color="black",shape="box"];61 -> 68[label="",style="solid", color="black", weight=3];
62[label="primbindIO (AProVE_Exception vx70) (sequence0 vx300)",fontsize=16,color="black",shape="box"];62 -> 69[label="",style="solid", color="black", weight=3];
63[label="primbindIO (AProVE_Error vx70) (sequence0 vx300)",fontsize=16,color="black",shape="box"];63 -> 70[label="",style="solid", color="black", weight=3];
64[label="Just (vx300 : vx50)",fontsize=16,color="green",shape="box"];65[label="(return (vx300 : vx60) ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];65 -> 71[label="",style="solid", color="black", weight=3];
66[label="vx4",fontsize=16,color="green",shape="box"];67[label="error []",fontsize=16,color="red",shape="box"];68[label="sequence0 vx300 vx70",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="AProVE_Exception vx70",fontsize=16,color="green",shape="box"];70[label="AProVE_Error vx70",fontsize=16,color="green",shape="box"];71[label="(((vx300 : vx60) : []) ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];71 -> 73[label="",style="solid", color="black", weight=3];
72[label="return (vx300 : vx70)",fontsize=16,color="black",shape="box"];72 -> 74[label="",style="solid", color="black", weight=3];
73 -> 75[label="",style="dashed", color="red", weight=0];
73[label="((vx300 : vx60) : [] ++ (vx61 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="magenta"];73 -> 76[label="",style="dashed", color="magenta", weight=3];
74[label="primretIO (vx300 : vx70)",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3];
76 -> 59[label="",style="dashed", color="red", weight=0];
76[label="[] ++ (vx61 >>= sequence0 vx300)",fontsize=16,color="magenta"];76 -> 78[label="",style="dashed", color="magenta", weight=3];
75[label="((vx300 : vx60) : vx8) ++ vx4",fontsize=16,color="black",shape="triangle"];75 -> 79[label="",style="solid", color="black", weight=3];
77[label="AProVE_IO (vx300 : vx70)",fontsize=16,color="green",shape="box"];78[label="vx61 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];120[label="vx61/vx610 : vx611",fontsize=10,color="white",style="solid",shape="box"];78 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 80[label="",style="solid", color="burlywood", weight=3];
121[label="vx61/[]",fontsize=10,color="white",style="solid",shape="box"];78 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 81[label="",style="solid", color="burlywood", weight=3];
79[label="(vx300 : vx60) : vx8 ++ vx4",fontsize=16,color="green",shape="box"];79 -> 82[label="",style="dashed", color="green", weight=3];
80[label="vx610 : vx611 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];80 -> 83[label="",style="solid", color="black", weight=3];
81[label="[] >>= sequence0 vx300",fontsize=16,color="black",shape="box"];81 -> 84[label="",style="solid", color="black", weight=3];
82[label="vx8 ++ vx4",fontsize=16,color="burlywood",shape="triangle"];122[label="vx8/vx80 : vx81",fontsize=10,color="white",style="solid",shape="box"];82 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 85[label="",style="solid", color="burlywood", weight=3];
123[label="vx8/[]",fontsize=10,color="white",style="solid",shape="box"];82 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 86[label="",style="solid", color="burlywood", weight=3];
83 -> 82[label="",style="dashed", color="red", weight=0];
83[label="sequence0 vx300 vx610 ++ (vx611 >>= sequence0 vx300)",fontsize=16,color="magenta"];83 -> 87[label="",style="dashed", color="magenta", weight=3];
83 -> 88[label="",style="dashed", color="magenta", weight=3];
84[label="[]",fontsize=16,color="green",shape="box"];85[label="(vx80 : vx81) ++ vx4",fontsize=16,color="black",shape="box"];85 -> 89[label="",style="solid", color="black", weight=3];
86[label="[] ++ vx4",fontsize=16,color="black",shape="box"];86 -> 90[label="",style="solid", color="black", weight=3];
87 -> 78[label="",style="dashed", color="red", weight=0];
87[label="vx611 >>= sequence0 vx300",fontsize=16,color="magenta"];87 -> 91[label="",style="dashed", color="magenta", weight=3];
88[label="sequence0 vx300 vx610",fontsize=16,color="black",shape="box"];88 -> 92[label="",style="solid", color="black", weight=3];
89[label="vx80 : vx81 ++ vx4",fontsize=16,color="green",shape="box"];89 -> 93[label="",style="dashed", color="green", weight=3];
90[label="vx4",fontsize=16,color="green",shape="box"];91[label="vx611",fontsize=16,color="green",shape="box"];92[label="return (vx300 : vx610)",fontsize=16,color="black",shape="box"];92 -> 94[label="",style="solid", color="black", weight=3];
93 -> 82[label="",style="dashed", color="red", weight=0];
93[label="vx81 ++ vx4",fontsize=16,color="magenta"];93 -> 95[label="",style="dashed", color="magenta", weight=3];
94[label="(vx300 : vx610) : []",fontsize=16,color="green",shape="box"];95[label="vx81",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(:(vx610, vx611), vx300, h) -> new_gtGtEs(vx611, vx300, 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(:(vx610, vx611), vx300, h) -> new_gtGtEs(vx611, vx300, 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_sequence(:(:(vx300, vx301), vx31), ty_[], h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_gtGtEs1(vx301, vx31, h)
   new_sequence(:(:(vx300, vx301), vx31), ty_[], h) -> new_gtGtEs1(vx301, vx31, h)
   new_sequence(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_sequence(vx31, ty_IO, h)
   new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, ty_[], h)
   new_sequence(:(Just(vx300), vx31), ty_Maybe, h) -> new_sequence(vx31, ty_Maybe, h)

The TRS R consists of the following rules:

   new_psPs4(vx4, h) -> vx4
   new_gtGtEs0([], vx31, h) -> []
   new_sequence0(:(AProVE_Exception(vx300), vx31), ty_IO, h) -> AProVE_Exception(vx300)
   new_sequence0(:(Just(vx300), vx31), ty_Maybe, h) -> new_gtGtEs3(new_sequence0(vx31, ty_Maybe, h), vx300, h)
   new_sequence0([], ty_[], h) -> :([], [])
   new_gtGtEs2(AProVE_IO(vx70), vx300, h) -> AProVE_IO(:(vx300, vx70))
   new_sequence0(:(IO(vx300), vx31), ty_IO, h) -> error([])
   new_sequence0(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_gtGtEs2(new_sequence0(vx31, ty_IO, h), vx300, h)
   new_sequence0([], ty_Maybe, h) -> Just([])
   new_gtGtEs4([], vx300, h) -> []
   new_psPs1(:(vx80, vx81), vx4, h) -> :(vx80, new_psPs1(vx81, vx4, h))
   new_sequence0(:(Nothing, vx31), ty_Maybe, h) -> Nothing
   new_psPs3(vx300, vx60, vx8, vx4, h) -> :(:(vx300, vx60), new_psPs1(vx8, vx4, h))
   new_gtGtEs2(AProVE_Error(vx70), vx300, h) -> AProVE_Error(vx70)
   new_gtGtEs2(IO(vx70), vx300, h) -> error([])
   new_gtGtEs2(AProVE_Exception(vx70), vx300, h) -> AProVE_Exception(vx70)
   new_sequence0(:(AProVE_Error(vx300), vx31), ty_IO, h) -> AProVE_Error(vx300)
   new_psPs2([], vx300, vx4, h) -> new_psPs4(vx4, h)
   new_psPs5(vx31, vx300, vx4, h) -> new_psPs2(new_sequence0(vx31, ty_[], h), vx300, vx4, h)
   new_gtGtEs4(:(vx610, vx611), vx300, h) -> new_psPs1(:(:(vx300, vx610), []), new_gtGtEs4(vx611, vx300, h), h)
   new_psPs2(:(vx60, vx61), vx300, vx4, h) -> new_psPs3(vx300, vx60, new_psPs4(new_gtGtEs4(vx61, vx300, h), h), vx4, h)
   new_psPs1([], vx4, h) -> vx4
   new_gtGtEs3(Nothing, vx300, h) -> Nothing
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_psPs5(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_sequence0(:(vx30, vx31), ty_[], h) -> new_gtGtEs0(vx30, vx31, h)
   new_sequence0([], ty_IO, h) -> AProVE_IO([])
   new_gtGtEs3(Just(vx50), vx300, h) -> Just(:(vx300, vx50))

The set Q consists of the following terms:

   new_gtGtEs3(Just(x0), x1, x2)
   new_gtGtEs2(AProVE_IO(x0), x1, x2)
   new_psPs4(x0, x1)
   new_psPs2(:(x0, x1), x2, x3, x4)
   new_gtGtEs2(AProVE_Exception(x0), x1, x2)
   new_sequence0([], ty_IO, x0)
   new_gtGtEs2(AProVE_Error(x0), x1, x2)
   new_psPs1(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_IO(x0), x1), ty_IO, x2)
   new_gtGtEs0(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Error(x0), x1), ty_IO, x2)
   new_gtGtEs3(Nothing, x0, x1)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs5(x0, x1, x2, x3)
   new_sequence0(:(Nothing, x0), ty_Maybe, x1)
   new_sequence0(:(Just(x0), x1), ty_Maybe, x2)
   new_gtGtEs0([], x0, x1)
   new_gtGtEs4([], x0, x1)
   new_sequence0([], ty_[], x0)
   new_sequence0([], ty_Maybe, x0)
   new_sequence0(:(x0, x1), ty_[], x2)
   new_psPs2([], x0, x1, x2)
   new_sequence0(:(IO(x0), x1), ty_IO, x2)
   new_gtGtEs2(IO(x0), x1, x2)
   new_psPs1([], x0, x1)
   new_gtGtEs4(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Exception(x0), x1), ty_IO, x2)

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

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

(14)
Complex Obligation (AND)

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

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

   new_sequence(:(Just(vx300), vx31), ty_Maybe, h) -> new_sequence(vx31, ty_Maybe, h)

The TRS R consists of the following rules:

   new_psPs4(vx4, h) -> vx4
   new_gtGtEs0([], vx31, h) -> []
   new_sequence0(:(AProVE_Exception(vx300), vx31), ty_IO, h) -> AProVE_Exception(vx300)
   new_sequence0(:(Just(vx300), vx31), ty_Maybe, h) -> new_gtGtEs3(new_sequence0(vx31, ty_Maybe, h), vx300, h)
   new_sequence0([], ty_[], h) -> :([], [])
   new_gtGtEs2(AProVE_IO(vx70), vx300, h) -> AProVE_IO(:(vx300, vx70))
   new_sequence0(:(IO(vx300), vx31), ty_IO, h) -> error([])
   new_sequence0(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_gtGtEs2(new_sequence0(vx31, ty_IO, h), vx300, h)
   new_sequence0([], ty_Maybe, h) -> Just([])
   new_gtGtEs4([], vx300, h) -> []
   new_psPs1(:(vx80, vx81), vx4, h) -> :(vx80, new_psPs1(vx81, vx4, h))
   new_sequence0(:(Nothing, vx31), ty_Maybe, h) -> Nothing
   new_psPs3(vx300, vx60, vx8, vx4, h) -> :(:(vx300, vx60), new_psPs1(vx8, vx4, h))
   new_gtGtEs2(AProVE_Error(vx70), vx300, h) -> AProVE_Error(vx70)
   new_gtGtEs2(IO(vx70), vx300, h) -> error([])
   new_gtGtEs2(AProVE_Exception(vx70), vx300, h) -> AProVE_Exception(vx70)
   new_sequence0(:(AProVE_Error(vx300), vx31), ty_IO, h) -> AProVE_Error(vx300)
   new_psPs2([], vx300, vx4, h) -> new_psPs4(vx4, h)
   new_psPs5(vx31, vx300, vx4, h) -> new_psPs2(new_sequence0(vx31, ty_[], h), vx300, vx4, h)
   new_gtGtEs4(:(vx610, vx611), vx300, h) -> new_psPs1(:(:(vx300, vx610), []), new_gtGtEs4(vx611, vx300, h), h)
   new_psPs2(:(vx60, vx61), vx300, vx4, h) -> new_psPs3(vx300, vx60, new_psPs4(new_gtGtEs4(vx61, vx300, h), h), vx4, h)
   new_psPs1([], vx4, h) -> vx4
   new_gtGtEs3(Nothing, vx300, h) -> Nothing
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_psPs5(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_sequence0(:(vx30, vx31), ty_[], h) -> new_gtGtEs0(vx30, vx31, h)
   new_sequence0([], ty_IO, h) -> AProVE_IO([])
   new_gtGtEs3(Just(vx50), vx300, h) -> Just(:(vx300, vx50))

The set Q consists of the following terms:

   new_gtGtEs3(Just(x0), x1, x2)
   new_gtGtEs2(AProVE_IO(x0), x1, x2)
   new_psPs4(x0, x1)
   new_psPs2(:(x0, x1), x2, x3, x4)
   new_gtGtEs2(AProVE_Exception(x0), x1, x2)
   new_sequence0([], ty_IO, x0)
   new_gtGtEs2(AProVE_Error(x0), x1, x2)
   new_psPs1(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_IO(x0), x1), ty_IO, x2)
   new_gtGtEs0(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Error(x0), x1), ty_IO, x2)
   new_gtGtEs3(Nothing, x0, x1)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs5(x0, x1, x2, x3)
   new_sequence0(:(Nothing, x0), ty_Maybe, x1)
   new_sequence0(:(Just(x0), x1), ty_Maybe, x2)
   new_gtGtEs0([], x0, x1)
   new_gtGtEs4([], x0, x1)
   new_sequence0([], ty_[], x0)
   new_sequence0([], ty_Maybe, x0)
   new_sequence0(:(x0, x1), ty_[], x2)
   new_psPs2([], x0, x1, x2)
   new_sequence0(:(IO(x0), x1), ty_IO, x2)
   new_gtGtEs2(IO(x0), x1, x2)
   new_psPs1([], x0, x1)
   new_gtGtEs4(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Exception(x0), x1), ty_IO, x2)

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

(16) 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(:(Just(vx300), vx31), ty_Maybe, h) -> new_sequence(vx31, ty_Maybe, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(17)
YES

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

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

   new_sequence(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_sequence(vx31, ty_IO, h)

The TRS R consists of the following rules:

   new_psPs4(vx4, h) -> vx4
   new_gtGtEs0([], vx31, h) -> []
   new_sequence0(:(AProVE_Exception(vx300), vx31), ty_IO, h) -> AProVE_Exception(vx300)
   new_sequence0(:(Just(vx300), vx31), ty_Maybe, h) -> new_gtGtEs3(new_sequence0(vx31, ty_Maybe, h), vx300, h)
   new_sequence0([], ty_[], h) -> :([], [])
   new_gtGtEs2(AProVE_IO(vx70), vx300, h) -> AProVE_IO(:(vx300, vx70))
   new_sequence0(:(IO(vx300), vx31), ty_IO, h) -> error([])
   new_sequence0(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_gtGtEs2(new_sequence0(vx31, ty_IO, h), vx300, h)
   new_sequence0([], ty_Maybe, h) -> Just([])
   new_gtGtEs4([], vx300, h) -> []
   new_psPs1(:(vx80, vx81), vx4, h) -> :(vx80, new_psPs1(vx81, vx4, h))
   new_sequence0(:(Nothing, vx31), ty_Maybe, h) -> Nothing
   new_psPs3(vx300, vx60, vx8, vx4, h) -> :(:(vx300, vx60), new_psPs1(vx8, vx4, h))
   new_gtGtEs2(AProVE_Error(vx70), vx300, h) -> AProVE_Error(vx70)
   new_gtGtEs2(IO(vx70), vx300, h) -> error([])
   new_gtGtEs2(AProVE_Exception(vx70), vx300, h) -> AProVE_Exception(vx70)
   new_sequence0(:(AProVE_Error(vx300), vx31), ty_IO, h) -> AProVE_Error(vx300)
   new_psPs2([], vx300, vx4, h) -> new_psPs4(vx4, h)
   new_psPs5(vx31, vx300, vx4, h) -> new_psPs2(new_sequence0(vx31, ty_[], h), vx300, vx4, h)
   new_gtGtEs4(:(vx610, vx611), vx300, h) -> new_psPs1(:(:(vx300, vx610), []), new_gtGtEs4(vx611, vx300, h), h)
   new_psPs2(:(vx60, vx61), vx300, vx4, h) -> new_psPs3(vx300, vx60, new_psPs4(new_gtGtEs4(vx61, vx300, h), h), vx4, h)
   new_psPs1([], vx4, h) -> vx4
   new_gtGtEs3(Nothing, vx300, h) -> Nothing
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_psPs5(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_sequence0(:(vx30, vx31), ty_[], h) -> new_gtGtEs0(vx30, vx31, h)
   new_sequence0([], ty_IO, h) -> AProVE_IO([])
   new_gtGtEs3(Just(vx50), vx300, h) -> Just(:(vx300, vx50))

The set Q consists of the following terms:

   new_gtGtEs3(Just(x0), x1, x2)
   new_gtGtEs2(AProVE_IO(x0), x1, x2)
   new_psPs4(x0, x1)
   new_psPs2(:(x0, x1), x2, x3, x4)
   new_gtGtEs2(AProVE_Exception(x0), x1, x2)
   new_sequence0([], ty_IO, x0)
   new_gtGtEs2(AProVE_Error(x0), x1, x2)
   new_psPs1(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_IO(x0), x1), ty_IO, x2)
   new_gtGtEs0(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Error(x0), x1), ty_IO, x2)
   new_gtGtEs3(Nothing, x0, x1)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs5(x0, x1, x2, x3)
   new_sequence0(:(Nothing, x0), ty_Maybe, x1)
   new_sequence0(:(Just(x0), x1), ty_Maybe, x2)
   new_gtGtEs0([], x0, x1)
   new_gtGtEs4([], x0, x1)
   new_sequence0([], ty_[], x0)
   new_sequence0([], ty_Maybe, x0)
   new_sequence0(:(x0, x1), ty_[], x2)
   new_psPs2([], x0, x1, x2)
   new_sequence0(:(IO(x0), x1), ty_IO, x2)
   new_gtGtEs2(IO(x0), x1, x2)
   new_psPs1([], x0, x1)
   new_gtGtEs4(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Exception(x0), x1), ty_IO, x2)

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(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_sequence(vx31, ty_IO, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(20)
YES

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

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

   new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, ty_[], h)
   new_sequence(:(:(vx300, vx301), vx31), ty_[], h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_sequence(:(:(vx300, vx301), vx31), ty_[], h) -> new_gtGtEs1(vx301, vx31, h)
   new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_gtGtEs1(vx301, vx31, h)
   new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)

The TRS R consists of the following rules:

   new_psPs4(vx4, h) -> vx4
   new_gtGtEs0([], vx31, h) -> []
   new_sequence0(:(AProVE_Exception(vx300), vx31), ty_IO, h) -> AProVE_Exception(vx300)
   new_sequence0(:(Just(vx300), vx31), ty_Maybe, h) -> new_gtGtEs3(new_sequence0(vx31, ty_Maybe, h), vx300, h)
   new_sequence0([], ty_[], h) -> :([], [])
   new_gtGtEs2(AProVE_IO(vx70), vx300, h) -> AProVE_IO(:(vx300, vx70))
   new_sequence0(:(IO(vx300), vx31), ty_IO, h) -> error([])
   new_sequence0(:(AProVE_IO(vx300), vx31), ty_IO, h) -> new_gtGtEs2(new_sequence0(vx31, ty_IO, h), vx300, h)
   new_sequence0([], ty_Maybe, h) -> Just([])
   new_gtGtEs4([], vx300, h) -> []
   new_psPs1(:(vx80, vx81), vx4, h) -> :(vx80, new_psPs1(vx81, vx4, h))
   new_sequence0(:(Nothing, vx31), ty_Maybe, h) -> Nothing
   new_psPs3(vx300, vx60, vx8, vx4, h) -> :(:(vx300, vx60), new_psPs1(vx8, vx4, h))
   new_gtGtEs2(AProVE_Error(vx70), vx300, h) -> AProVE_Error(vx70)
   new_gtGtEs2(IO(vx70), vx300, h) -> error([])
   new_gtGtEs2(AProVE_Exception(vx70), vx300, h) -> AProVE_Exception(vx70)
   new_sequence0(:(AProVE_Error(vx300), vx31), ty_IO, h) -> AProVE_Error(vx300)
   new_psPs2([], vx300, vx4, h) -> new_psPs4(vx4, h)
   new_psPs5(vx31, vx300, vx4, h) -> new_psPs2(new_sequence0(vx31, ty_[], h), vx300, vx4, h)
   new_gtGtEs4(:(vx610, vx611), vx300, h) -> new_psPs1(:(:(vx300, vx610), []), new_gtGtEs4(vx611, vx300, h), h)
   new_psPs2(:(vx60, vx61), vx300, vx4, h) -> new_psPs3(vx300, vx60, new_psPs4(new_gtGtEs4(vx61, vx300, h), h), vx4, h)
   new_psPs1([], vx4, h) -> vx4
   new_gtGtEs3(Nothing, vx300, h) -> Nothing
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_psPs5(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
   new_sequence0(:(vx30, vx31), ty_[], h) -> new_gtGtEs0(vx30, vx31, h)
   new_sequence0([], ty_IO, h) -> AProVE_IO([])
   new_gtGtEs3(Just(vx50), vx300, h) -> Just(:(vx300, vx50))

The set Q consists of the following terms:

   new_gtGtEs3(Just(x0), x1, x2)
   new_gtGtEs2(AProVE_IO(x0), x1, x2)
   new_psPs4(x0, x1)
   new_psPs2(:(x0, x1), x2, x3, x4)
   new_gtGtEs2(AProVE_Exception(x0), x1, x2)
   new_sequence0([], ty_IO, x0)
   new_gtGtEs2(AProVE_Error(x0), x1, x2)
   new_psPs1(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_IO(x0), x1), ty_IO, x2)
   new_gtGtEs0(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Error(x0), x1), ty_IO, x2)
   new_gtGtEs3(Nothing, x0, x1)
   new_psPs3(x0, x1, x2, x3, x4)
   new_psPs5(x0, x1, x2, x3)
   new_sequence0(:(Nothing, x0), ty_Maybe, x1)
   new_sequence0(:(Just(x0), x1), ty_Maybe, x2)
   new_gtGtEs0([], x0, x1)
   new_gtGtEs4([], x0, x1)
   new_sequence0([], ty_[], x0)
   new_sequence0([], ty_Maybe, x0)
   new_sequence0(:(x0, x1), ty_[], x2)
   new_psPs2([], x0, x1, x2)
   new_sequence0(:(IO(x0), x1), ty_IO, x2)
   new_gtGtEs2(IO(x0), x1, x2)
   new_psPs1([], x0, x1)
   new_gtGtEs4(:(x0, x1), x2, x3)
   new_sequence0(:(AProVE_Exception(x0), x1), ty_IO, x2)

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(:(:(vx300, vx301), vx31), ty_[], h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 4


*new_sequence(:(:(vx300, vx301), vx31), ty_[], h) -> new_gtGtEs1(vx301, vx31, h)
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEs0(vx301, vx31, h), h)
The graph contains the following edges 2 >= 1, 1 > 2, 3 >= 4


*new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, ty_[], h)
The graph contains the following edges 1 >= 1, 4 >= 3


*new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_gtGtEs1(vx301, vx31, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(23)
YES

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

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

   new_psPs(:(vx80, vx81), vx4, h) -> new_psPs(vx81, vx4, h)

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

(25) 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(:(vx80, vx81), vx4, h) -> new_psPs(vx81, vx4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(26)
YES