/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
        (17) AND
            (18) QDP
                (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (20) YES
            (21) QDP
                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (23) YES
            (24) QDP
                (25) 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
"\_->q"
is transformed to
"gtGt0 q _ = q;
"

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

(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="mapM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapM_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence_ . map vy3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_ (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];79[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 8[label="",style="solid", color="burlywood", weight=3];
80[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (>>) (return ()) (map vy3 (vy40 : vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldr (>>) (return ()) (map vy3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (vy3 vy40 : map vy3 vy41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="blue",shape="box"];81[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 81[label="",style="solid", color="blue", weight=9];
81 -> 26[label="",style="solid", color="blue", weight=3];
82[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="blue", weight=9];
82 -> 27[label="",style="solid", color="blue", weight=3];
83[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="blue", weight=9];
83 -> 28[label="",style="solid", color="blue", weight=3];
13[label="return ()",fontsize=16,color="blue",shape="box"];84[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 84[label="",style="solid", color="blue", weight=9];
84 -> 16[label="",style="solid", color="blue", weight=3];
85[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 85[label="",style="solid", color="blue", weight=9];
85 -> 17[label="",style="solid", color="blue", weight=3];
86[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="blue", weight=9];
86 -> 18[label="",style="solid", color="blue", weight=3];
26 -> 20[label="",style="dashed", color="red", weight=0];
26[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];26 -> 33[label="",style="dashed", color="magenta", weight=3];
27 -> 21[label="",style="dashed", color="red", weight=0];
27[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];27 -> 34[label="",style="dashed", color="magenta", weight=3];
28 -> 22[label="",style="dashed", color="red", weight=0];
28[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];28 -> 35[label="",style="dashed", color="magenta", weight=3];
16[label="return ()",fontsize=16,color="black",shape="box"];16 -> 23[label="",style="solid", color="black", weight=3];
17[label="return ()",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3];
18[label="return ()",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
33 -> 7[label="",style="dashed", color="red", weight=0];
33[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];33 -> 38[label="",style="dashed", color="magenta", weight=3];
20[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];20 -> 29[label="",style="solid", color="black", weight=3];
34 -> 7[label="",style="dashed", color="red", weight=0];
34[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];34 -> 39[label="",style="dashed", color="magenta", weight=3];
21[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];21 -> 30[label="",style="solid", color="black", weight=3];
35 -> 7[label="",style="dashed", color="red", weight=0];
35[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];35 -> 40[label="",style="dashed", color="magenta", weight=3];
22[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="Just ()",fontsize=16,color="green",shape="box"];24[label="() : []",fontsize=16,color="green",shape="box"];25[label="primretIO ()",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3];
38[label="vy41",fontsize=16,color="green",shape="box"];29 -> 36[label="",style="dashed", color="red", weight=0];
29[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];29 -> 37[label="",style="dashed", color="magenta", weight=3];
39[label="vy41",fontsize=16,color="green",shape="box"];30 -> 41[label="",style="dashed", color="red", weight=0];
30[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];30 -> 42[label="",style="dashed", color="magenta", weight=3];
40[label="vy41",fontsize=16,color="green",shape="box"];31[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3];
32[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];37[label="vy3 vy40",fontsize=16,color="green",shape="box"];37 -> 44[label="",style="dashed", color="green", weight=3];
36[label="vy6 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];87[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];36 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 45[label="",style="solid", color="burlywood", weight=3];
88[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];36 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 46[label="",style="solid", color="burlywood", weight=3];
42[label="vy3 vy40",fontsize=16,color="green",shape="box"];42 -> 50[label="",style="dashed", color="green", weight=3];
41[label="vy7 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];89[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];41 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 48[label="",style="solid", color="burlywood", weight=3];
90[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 49[label="",style="solid", color="burlywood", weight=3];
43 -> 51[label="",style="dashed", color="red", weight=0];
43[label="primbindIO (vy3 vy40) (gtGt0 vy5)",fontsize=16,color="magenta"];43 -> 52[label="",style="dashed", color="magenta", weight=3];
44[label="vy40",fontsize=16,color="green",shape="box"];45[label="Nothing >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46[label="Just vy60 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];46 -> 54[label="",style="solid", color="black", weight=3];
50[label="vy40",fontsize=16,color="green",shape="box"];48[label="vy70 : vy71 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
49[label="[] >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3];
52[label="vy3 vy40",fontsize=16,color="green",shape="box"];52 -> 62[label="",style="dashed", color="green", weight=3];
51[label="primbindIO vy8 (gtGt0 vy5)",fontsize=16,color="burlywood",shape="triangle"];91[label="vy8/IO vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 58[label="",style="solid", color="burlywood", weight=3];
92[label="vy8/AProVE_IO vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 59[label="",style="solid", color="burlywood", weight=3];
93[label="vy8/AProVE_Exception vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 60[label="",style="solid", color="burlywood", weight=3];
94[label="vy8/AProVE_Error vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 61[label="",style="solid", color="burlywood", weight=3];
53[label="Nothing",fontsize=16,color="green",shape="box"];54[label="gtGt0 vy5 vy60",fontsize=16,color="black",shape="box"];54 -> 63[label="",style="solid", color="black", weight=3];
55 -> 64[label="",style="dashed", color="red", weight=0];
55[label="gtGt0 vy5 vy70 ++ (vy71 >>= gtGt0 vy5)",fontsize=16,color="magenta"];55 -> 65[label="",style="dashed", color="magenta", weight=3];
56[label="[]",fontsize=16,color="green",shape="box"];62[label="vy40",fontsize=16,color="green",shape="box"];58[label="primbindIO (IO vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3];
59[label="primbindIO (AProVE_IO vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3];
60[label="primbindIO (AProVE_Exception vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];60 -> 68[label="",style="solid", color="black", weight=3];
61[label="primbindIO (AProVE_Error vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];61 -> 69[label="",style="solid", color="black", weight=3];
63[label="vy5",fontsize=16,color="green",shape="box"];65 -> 41[label="",style="dashed", color="red", weight=0];
65[label="vy71 >>= gtGt0 vy5",fontsize=16,color="magenta"];65 -> 70[label="",style="dashed", color="magenta", weight=3];
64[label="gtGt0 vy5 vy70 ++ vy9",fontsize=16,color="black",shape="triangle"];64 -> 71[label="",style="solid", color="black", weight=3];
66[label="error []",fontsize=16,color="red",shape="box"];67[label="gtGt0 vy5 vy80",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3];
68[label="AProVE_Exception vy80",fontsize=16,color="green",shape="box"];69[label="AProVE_Error vy80",fontsize=16,color="green",shape="box"];70[label="vy71",fontsize=16,color="green",shape="box"];71[label="vy5 ++ vy9",fontsize=16,color="burlywood",shape="triangle"];95[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];71 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 73[label="",style="solid", color="burlywood", weight=3];
96[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];71 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 74[label="",style="solid", color="burlywood", weight=3];
72[label="vy5",fontsize=16,color="green",shape="box"];73[label="(vy50 : vy51) ++ vy9",fontsize=16,color="black",shape="box"];73 -> 75[label="",style="solid", color="black", weight=3];
74[label="[] ++ vy9",fontsize=16,color="black",shape="box"];74 -> 76[label="",style="solid", color="black", weight=3];
75[label="vy50 : vy51 ++ vy9",fontsize=16,color="green",shape="box"];75 -> 77[label="",style="dashed", color="green", weight=3];
76[label="vy9",fontsize=16,color="green",shape="box"];77 -> 71[label="",style="dashed", color="red", weight=0];
77[label="vy51 ++ vy9",fontsize=16,color="magenta"];77 -> 78[label="",style="dashed", color="magenta", weight=3];
78[label="vy51",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(:(vy70, vy71), vy5, h) -> new_gtGtEs(vy71, vy5, 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(:(vy70, vy71), vy5, h) -> new_gtGtEs(vy71, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(11)
YES

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

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

   new_psPs(:(vy50, vy51), vy9) -> new_psPs(vy51, vy9)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_psPs(:(vy50, vy51), vy9) -> new_psPs(vy51, vy9)
The graph contains the following edges 1 > 1, 2 >= 2


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

(14)
YES

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

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

   new_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba)
   new_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba)
   new_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba)

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

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

(17)
Complex Obligation (AND)

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

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

   new_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba)

R is empty.
Q is empty.
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_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


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

(20)
YES

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

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

   new_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba)

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_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


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

(23)
YES

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

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

   new_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba)

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_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5


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

(26)
YES