/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) 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="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];32[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 7[label="",style="solid", color="burlywood", weight=3];
33[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="sequence (map vx3 [])",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="sequence (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="sequence []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="vx3 vx40 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="return []",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
13[label="vx5 >>= sequence1 (map vx3 vx41)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 17[label="",style="solid", color="burlywood", weight=3];
35[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];13 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="Just []",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="Nothing >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
18[label="Just vx50 >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="Nothing",fontsize=16,color="green",shape="box"];21[label="sequence1 (map vx3 vx41) vx50",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22 -> 23[label="",style="dashed", color="red", weight=0];
22[label="sequence (map vx3 vx41) >>= sequence0 vx50",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24 -> 6[label="",style="dashed", color="red", weight=0];
24[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
23[label="vx6 >>= sequence0 vx50",fontsize=16,color="burlywood",shape="triangle"];36[label="vx6/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 26[label="",style="solid", color="burlywood", weight=3];
37[label="vx6/Just vx60",fontsize=10,color="white",style="solid",shape="box"];23 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="vx41",fontsize=16,color="green",shape="box"];26[label="Nothing >>= sequence0 vx50",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="Just vx60 >>= sequence0 vx50",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="Nothing",fontsize=16,color="green",shape="box"];29[label="sequence0 vx50 vx60",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="return (vx50 : vx60)",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="Just (vx50 : vx60)",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
   new_sequence(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs(vx3, vx41, h, ba)

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

(9) 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(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs(vx3, vx41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*new_gtGtEs(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(10)
YES