/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) 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"];48[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 4[label="",style="solid", color="burlywood", weight=3];
49[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 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="burlywood",shape="triangle"];50[label="vx30/vx300 : vx301",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 8[label="",style="solid", color="burlywood", weight=3];
51[label="vx30/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="return []",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="vx300 : vx301 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="[] >>= sequence1 vx31",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="[] : []",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="sequence1 vx31 vx300 ++ (vx301 >>= sequence1 vx31)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="[]",fontsize=16,color="green",shape="box"];14 -> 6[label="",style="dashed", color="red", weight=0];
14[label="vx301 >>= sequence1 vx31",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="sequence1 vx31 vx300 ++ vx4",fontsize=16,color="black",shape="triangle"];13 -> 16[label="",style="solid", color="black", weight=3];
15[label="vx301",fontsize=16,color="green",shape="box"];16 -> 17[label="",style="dashed", color="red", weight=0];
16[label="(sequence vx31 >>= sequence0 vx300) ++ vx4",fontsize=16,color="magenta"];16 -> 18[label="",style="dashed", color="magenta", weight=3];
18 -> 3[label="",style="dashed", color="red", weight=0];
18[label="sequence vx31",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="(vx5 >>= sequence0 vx300) ++ vx4",fontsize=16,color="burlywood",shape="triangle"];52[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];17 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 20[label="",style="solid", color="burlywood", weight=3];
53[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 21[label="",style="solid", color="burlywood", weight=3];
19[label="vx31",fontsize=16,color="green",shape="box"];20[label="(vx50 : vx51 >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="([] >>= sequence0 vx300) ++ vx4",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="(sequence0 vx300 vx50 ++ (vx51 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="[] ++ vx4",fontsize=16,color="black",shape="triangle"];23 -> 25[label="",style="solid", color="black", weight=3];
24[label="(return (vx300 : vx50) ++ (vx51 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="vx4",fontsize=16,color="green",shape="box"];26[label="(((vx300 : vx50) : []) ++ (vx51 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27 -> 28[label="",style="dashed", color="red", weight=0];
27[label="((vx300 : vx50) : [] ++ (vx51 >>= sequence0 vx300)) ++ vx4",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3];
29 -> 23[label="",style="dashed", color="red", weight=0];
29[label="[] ++ (vx51 >>= sequence0 vx300)",fontsize=16,color="magenta"];29 -> 30[label="",style="dashed", color="magenta", weight=3];
28[label="((vx300 : vx50) : vx6) ++ vx4",fontsize=16,color="black",shape="triangle"];28 -> 31[label="",style="solid", color="black", weight=3];
30[label="vx51 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];54[label="vx51/vx510 : vx511",fontsize=10,color="white",style="solid",shape="box"];30 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 32[label="",style="solid", color="burlywood", weight=3];
55[label="vx51/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 33[label="",style="solid", color="burlywood", weight=3];
31[label="(vx300 : vx50) : vx6 ++ vx4",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3];
32[label="vx510 : vx511 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
33[label="[] >>= sequence0 vx300",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3];
34[label="vx6 ++ vx4",fontsize=16,color="burlywood",shape="triangle"];56[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];34 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 37[label="",style="solid", color="burlywood", weight=3];
57[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 38[label="",style="solid", color="burlywood", weight=3];
35 -> 34[label="",style="dashed", color="red", weight=0];
35[label="sequence0 vx300 vx510 ++ (vx511 >>= sequence0 vx300)",fontsize=16,color="magenta"];35 -> 39[label="",style="dashed", color="magenta", weight=3];
35 -> 40[label="",style="dashed", color="magenta", weight=3];
36[label="[]",fontsize=16,color="green",shape="box"];37[label="(vx60 : vx61) ++ vx4",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="[] ++ vx4",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="sequence0 vx300 vx510",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40 -> 30[label="",style="dashed", color="red", weight=0];
40[label="vx511 >>= sequence0 vx300",fontsize=16,color="magenta"];40 -> 44[label="",style="dashed", color="magenta", weight=3];
41[label="vx60 : vx61 ++ vx4",fontsize=16,color="green",shape="box"];41 -> 45[label="",style="dashed", color="green", weight=3];
42[label="vx4",fontsize=16,color="green",shape="box"];43[label="return (vx300 : vx510)",fontsize=16,color="black",shape="box"];43 -> 46[label="",style="solid", color="black", weight=3];
44[label="vx511",fontsize=16,color="green",shape="box"];45 -> 34[label="",style="dashed", color="red", weight=0];
45[label="vx61 ++ vx4",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
46[label="(vx300 : vx510) : []",fontsize=16,color="green",shape="box"];47[label="vx61",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(:(vx510, vx511), vx300, h) -> new_gtGtEs(vx511, 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(:(vx510, vx511), vx300, h) -> new_gtGtEs(vx511, 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), h) -> new_gtGtEs0(vx301, vx31, h)
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_gtGtEs0(vx301, vx31, h)
   new_sequence(:(:(vx300, vx301), vx31), h) -> new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
   new_gtGtEs0(:(vx300, vx301), vx31, h) -> new_psPs0(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
   new_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, h)

The TRS R consists of the following rules:

   new_psPs4(vx300, vx50, vx6, vx4, h) -> :(:(vx300, vx50), new_psPs5(vx6, vx4, h))
   new_gtGtEs1([], vx31, h) -> []
   new_gtGtEs2(:(vx510, vx511), vx300, h) -> new_psPs5(:(:(vx300, vx510), []), new_gtGtEs2(vx511, vx300, h), h)
   new_psPs5(:(vx60, vx61), vx4, h) -> :(vx60, new_psPs5(vx61, vx4, h))
   new_psPs3(:(vx50, vx51), vx300, vx4, h) -> new_psPs4(vx300, vx50, new_psPs1(new_gtGtEs2(vx51, vx300, h), h), vx4, h)
   new_sequence0(:(vx30, vx31), h) -> new_gtGtEs1(vx30, vx31, h)
   new_gtGtEs1(:(vx300, vx301), vx31, h) -> new_psPs2(vx31, vx300, new_gtGtEs1(vx301, vx31, h), h)
   new_psPs3([], vx300, vx4, h) -> new_psPs1(vx4, h)
   new_psPs1(vx4, h) -> vx4
   new_psPs2(vx31, vx300, vx4, h) -> new_psPs3(new_sequence0(vx31, h), vx300, vx4, h)
   new_gtGtEs2([], vx300, h) -> []
   new_psPs5([], vx4, h) -> vx4
   new_sequence0([], h) -> :([], [])

The set Q consists of the following terms:

   new_gtGtEs2(:(x0, x1), x2, x3)
   new_sequence0(:(x0, x1), x2)
   new_gtGtEs1(:(x0, x1), x2, x3)
   new_sequence0([], x0)
   new_psPs3([], x0, x1, x2)
   new_psPs3(:(x0, x1), x2, x3, x4)
   new_psPs1(x0, x1)
   new_psPs2(x0, x1, x2, x3)
   new_gtGtEs2([], x0, x1)
   new_psPs4(x0, x1, x2, x3, x4)
   new_gtGtEs1([], x0, x1)
   new_psPs5([], x0, x1)
   new_psPs5(:(x0, x1), x2, x3)

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_psPs0(vx31, vx300, vx4, h) -> new_sequence(vx31, h)
The graph contains the following edges 1 >= 1, 4 >= 2


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


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


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


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


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

(14)
YES

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

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

   new_psPs(:(vx60, vx61), vx4, h) -> new_psPs(vx61, vx4, 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_psPs(:(vx60, vx61), vx4, h) -> new_psPs(vx61, vx4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(17)
YES