/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES


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

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

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

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

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

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(>>=)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="(>>=) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="(>>=) vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];25[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 5[label="",style="solid", color="burlywood", weight=3];
26[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="(>>=) vx30 : vx31 vx4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="(>>=) [] vx4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 12[label="",style="dashed", color="red", weight=0];
7[label="vx4 vx30 ++ (vx31 >>= vx4)",fontsize=16,color="magenta"];7 -> 13[label="",style="dashed", color="magenta", weight=3];
7 -> 14[label="",style="dashed", color="magenta", weight=3];
8[label="[]",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
13[label="vx31 >>= vx4",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="vx4 vx30",fontsize=16,color="green",shape="box"];14 -> 17[label="",style="dashed", color="green", weight=3];
12[label="vx6 ++ vx5",fontsize=16,color="burlywood",shape="triangle"];27[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];12 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 18[label="",style="solid", color="burlywood", weight=3];
28[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 19[label="",style="solid", color="burlywood", weight=3];
16[label="vx31",fontsize=16,color="green",shape="box"];17[label="vx30",fontsize=16,color="green",shape="box"];18[label="(vx60 : vx61) ++ vx5",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="[] ++ vx5",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
21[label="vx60 : vx61 ++ vx5",fontsize=16,color="green",shape="box"];21 -> 23[label="",style="dashed", color="green", weight=3];
22[label="vx5",fontsize=16,color="green",shape="box"];23 -> 12[label="",style="dashed", color="red", weight=0];
23[label="vx61 ++ vx5",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
24[label="vx61",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)

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

(8) 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(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(9)
YES

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)

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

(11) 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), vx5, h) -> new_psPs(vx61, vx5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(12)
YES