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


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


YES
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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
"\_->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="sequence_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence_ vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr (>>) (return ()) vy3",fontsize=16,color="burlywood",shape="triangle"];63[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 5[label="",style="solid", color="burlywood", weight=3];
64[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr (>>) (return ()) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="blue",shape="box"];65[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="blue", weight=9];
65 -> 21[label="",style="solid", color="blue", weight=3];
66[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="blue", weight=9];
66 -> 22[label="",style="solid", color="blue", weight=3];
67[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];7 -> 67[label="",style="solid", color="blue", weight=9];
67 -> 23[label="",style="solid", color="blue", weight=3];
8[label="return ()",fontsize=16,color="blue",shape="box"];68[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 68[label="",style="solid", color="blue", weight=9];
68 -> 11[label="",style="solid", color="blue", weight=3];
69[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 69[label="",style="solid", color="blue", weight=9];
69 -> 12[label="",style="solid", color="blue", weight=3];
70[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];8 -> 70[label="",style="solid", color="blue", weight=9];
70 -> 13[label="",style="solid", color="blue", weight=3];
21 -> 15[label="",style="dashed", color="red", weight=0];
21[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];21 -> 28[label="",style="dashed", color="magenta", weight=3];
22 -> 16[label="",style="dashed", color="red", weight=0];
22[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];22 -> 29[label="",style="dashed", color="magenta", weight=3];
23 -> 17[label="",style="dashed", color="red", weight=0];
23[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];23 -> 30[label="",style="dashed", color="magenta", weight=3];
11[label="return ()",fontsize=16,color="black",shape="box"];11 -> 18[label="",style="solid", color="black", weight=3];
12[label="return ()",fontsize=16,color="black",shape="box"];12 -> 19[label="",style="solid", color="black", weight=3];
13[label="return ()",fontsize=16,color="black",shape="box"];13 -> 20[label="",style="solid", color="black", weight=3];
28 -> 4[label="",style="dashed", color="red", weight=0];
28[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];28 -> 36[label="",style="dashed", color="magenta", weight=3];
15[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];15 -> 24[label="",style="solid", color="black", weight=3];
29 -> 4[label="",style="dashed", color="red", weight=0];
29[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];29 -> 37[label="",style="dashed", color="magenta", weight=3];
16[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];16 -> 25[label="",style="solid", color="black", weight=3];
30 -> 4[label="",style="dashed", color="red", weight=0];
30[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];30 -> 38[label="",style="dashed", color="magenta", weight=3];
17[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="primretIO ()",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="Just ()",fontsize=16,color="green",shape="box"];20[label="() : []",fontsize=16,color="green",shape="box"];36[label="vy31",fontsize=16,color="green",shape="box"];24[label="vy30 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];24 -> 31[label="",style="solid", color="black", weight=3];
37[label="vy31",fontsize=16,color="green",shape="box"];25[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="box"];71[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];25 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 32[label="",style="solid", color="burlywood", weight=3];
72[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];25 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 33[label="",style="solid", color="burlywood", weight=3];
38[label="vy31",fontsize=16,color="green",shape="box"];26[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];73[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];26 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 34[label="",style="solid", color="burlywood", weight=3];
74[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 35[label="",style="solid", color="burlywood", weight=3];
27[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];31[label="primbindIO vy30 (gtGt0 vy4)",fontsize=16,color="burlywood",shape="box"];75[label="vy30/IO vy300",fontsize=10,color="white",style="solid",shape="box"];31 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 39[label="",style="solid", color="burlywood", weight=3];
76[label="vy30/AProVE_IO vy300",fontsize=10,color="white",style="solid",shape="box"];31 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 40[label="",style="solid", color="burlywood", weight=3];
77[label="vy30/AProVE_Exception vy300",fontsize=10,color="white",style="solid",shape="box"];31 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 41[label="",style="solid", color="burlywood", weight=3];
78[label="vy30/AProVE_Error vy300",fontsize=10,color="white",style="solid",shape="box"];31 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 42[label="",style="solid", color="burlywood", weight=3];
32[label="Nothing >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];32 -> 43[label="",style="solid", color="black", weight=3];
33[label="Just vy300 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];33 -> 44[label="",style="solid", color="black", weight=3];
34[label="vy300 : vy301 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];34 -> 45[label="",style="solid", color="black", weight=3];
35[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];35 -> 46[label="",style="solid", color="black", weight=3];
39[label="primbindIO (IO vy300) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
40[label="primbindIO (AProVE_IO vy300) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="primbindIO (AProVE_Exception vy300) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
42[label="primbindIO (AProVE_Error vy300) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="Nothing",fontsize=16,color="green",shape="box"];44[label="gtGt0 vy4 vy300",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
45 -> 52[label="",style="dashed", color="red", weight=0];
45[label="gtGt0 vy4 vy300 ++ (vy301 >>= gtGt0 vy4)",fontsize=16,color="magenta"];45 -> 53[label="",style="dashed", color="magenta", weight=3];
46[label="[]",fontsize=16,color="green",shape="box"];47[label="error []",fontsize=16,color="red",shape="box"];48[label="gtGt0 vy4 vy300",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
49[label="AProVE_Exception vy300",fontsize=16,color="green",shape="box"];50[label="AProVE_Error vy300",fontsize=16,color="green",shape="box"];51[label="vy4",fontsize=16,color="green",shape="box"];53 -> 26[label="",style="dashed", color="red", weight=0];
53[label="vy301 >>= gtGt0 vy4",fontsize=16,color="magenta"];53 -> 55[label="",style="dashed", color="magenta", weight=3];
52[label="gtGt0 vy4 vy300 ++ vy5",fontsize=16,color="black",shape="triangle"];52 -> 56[label="",style="solid", color="black", weight=3];
54[label="vy4",fontsize=16,color="green",shape="box"];55[label="vy301",fontsize=16,color="green",shape="box"];56[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];79[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];56 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 57[label="",style="solid", color="burlywood", weight=3];
80[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];56 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 58[label="",style="solid", color="burlywood", weight=3];
57[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
58[label="[] ++ vy5",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
59[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];59 -> 61[label="",style="dashed", color="green", weight=3];
60[label="vy5",fontsize=16,color="green",shape="box"];61 -> 56[label="",style="dashed", color="red", weight=0];
61[label="vy41 ++ vy5",fontsize=16,color="magenta"];61 -> 62[label="",style="dashed", color="magenta", weight=3];
62[label="vy41",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(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, 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(:(vy300, vy301), vy4, h) -> new_gtGtEs(vy301, vy4, 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_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)
   new_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)
   new_foldr(:(vy30, vy31), ty_IO, h) -> new_foldr(vy31, ty_IO, h)

R is empty.
Q is empty.
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_foldr(:(vy30, vy31), ty_IO, h) -> new_foldr(vy31, ty_IO, h)

R is empty.
Q is empty.
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_foldr(:(vy30, vy31), ty_IO, h) -> new_foldr(vy31, ty_IO, 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_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)

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(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, 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_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], 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_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], 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(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5)

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(:(vy40, vy41), vy5) -> new_psPs(vy41, vy5)
The graph contains the following edges 1 > 1, 2 >= 2


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

(26)
YES