/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
join :: Monad b => b (b a)  ->  b a;
join x = x >>= id;

}

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

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
join :: Monad a => a (a b)  ->  a b;
join x = x >>= id;

}

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

(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 Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
join :: Monad b => b (b a)  ->  b a;
join x = x >>= id;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.join",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.join vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="vy3 >>= id",fontsize=16,color="blue",shape="box"];37[label=">>= :: ([] ([] a)) -> (([] a) -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];4 -> 37[label="",style="solid", color="blue", weight=9];
37 -> 5[label="",style="solid", color="blue", weight=3];
38[label=">>= :: (Maybe (Maybe a)) -> ((Maybe a) -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];4 -> 38[label="",style="solid", color="blue", weight=9];
38 -> 6[label="",style="solid", color="blue", weight=3];
39[label=">>= :: (IO (IO a)) -> ((IO a) -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="blue", weight=9];
39 -> 7[label="",style="solid", color="blue", weight=3];
5[label="vy3 >>= id",fontsize=16,color="burlywood",shape="triangle"];40[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];5 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 8[label="",style="solid", color="burlywood", weight=3];
41[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 9[label="",style="solid", color="burlywood", weight=3];
6[label="vy3 >>= id",fontsize=16,color="burlywood",shape="box"];42[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 10[label="",style="solid", color="burlywood", weight=3];
43[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];6 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 11[label="",style="solid", color="burlywood", weight=3];
7[label="vy3 >>= id",fontsize=16,color="black",shape="box"];7 -> 12[label="",style="solid", color="black", weight=3];
8[label="vy30 : vy31 >>= id",fontsize=16,color="black",shape="box"];8 -> 13[label="",style="solid", color="black", weight=3];
9[label="[] >>= id",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
10[label="Nothing >>= id",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="Just vy30 >>= id",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
12[label="primbindIO vy3 id",fontsize=16,color="burlywood",shape="box"];44[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 17[label="",style="solid", color="burlywood", weight=3];
45[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 18[label="",style="solid", color="burlywood", weight=3];
46[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 19[label="",style="solid", color="burlywood", weight=3];
47[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 20[label="",style="solid", color="burlywood", weight=3];
13 -> 21[label="",style="dashed", color="red", weight=0];
13[label="id vy30 ++ (vy31 >>= id)",fontsize=16,color="magenta"];13 -> 22[label="",style="dashed", color="magenta", weight=3];
14[label="[]",fontsize=16,color="green",shape="box"];15[label="Nothing",fontsize=16,color="green",shape="box"];16[label="id vy30",fontsize=16,color="black",shape="box"];16 -> 23[label="",style="solid", color="black", weight=3];
17[label="primbindIO (IO vy30) id",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3];
18[label="primbindIO (AProVE_IO vy30) id",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
19[label="primbindIO (AProVE_Exception vy30) id",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
20[label="primbindIO (AProVE_Error vy30) id",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
22 -> 5[label="",style="dashed", color="red", weight=0];
22[label="vy31 >>= id",fontsize=16,color="magenta"];22 -> 28[label="",style="dashed", color="magenta", weight=3];
21[label="id vy30 ++ vy4",fontsize=16,color="black",shape="triangle"];21 -> 29[label="",style="solid", color="black", weight=3];
23[label="vy30",fontsize=16,color="green",shape="box"];24[label="error []",fontsize=16,color="red",shape="box"];25[label="id vy30",fontsize=16,color="black",shape="box"];25 -> 30[label="",style="solid", color="black", weight=3];
26[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];27[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];28[label="vy31",fontsize=16,color="green",shape="box"];29[label="vy30 ++ vy4",fontsize=16,color="burlywood",shape="triangle"];48[label="vy30/vy300 : vy301",fontsize=10,color="white",style="solid",shape="box"];29 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 31[label="",style="solid", color="burlywood", weight=3];
49[label="vy30/[]",fontsize=10,color="white",style="solid",shape="box"];29 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 32[label="",style="solid", color="burlywood", weight=3];
30[label="vy30",fontsize=16,color="green",shape="box"];31[label="(vy300 : vy301) ++ vy4",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="[] ++ vy4",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="vy300 : vy301 ++ vy4",fontsize=16,color="green",shape="box"];33 -> 35[label="",style="dashed", color="green", weight=3];
34[label="vy4",fontsize=16,color="green",shape="box"];35 -> 29[label="",style="dashed", color="red", weight=0];
35[label="vy301 ++ vy4",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
36[label="vy301",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(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)

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(:(vy30, vy31), h) -> new_gtGtEs(vy31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

   new_psPs(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, 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(:(vy300, vy301), vy4, h) -> new_psPs(vy301, vy4, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(12)
YES