/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= (\flg ->filterM p xs >>= (\ys ->return ( if flg then x : ys else ys)));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\ys->return (if flg then x : ys else ys)"
is transformed to
"filterM0 flg x ys = return (if flg then x : ys else ys);
"
The following Lambda expression
"\flg->filterM p xs >>= filterM0 flg x"
is transformed to
"filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;
"

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

(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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return ( if flg then x : ys else ys);

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

(3) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if flg then x : ys else ys"
is transformed to
"filterM00 x ys True = x : ys;
filterM00 x ys False = ys;
"

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

(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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM _ [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

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

(6)
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;
filterM :: Monad b => (a  ->  b Bool)  ->  [a]  ->  b [a];
filterM vy [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

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

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

(8)
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;
filterM :: Monad a => (b  ->  a Bool)  ->  [b]  ->  a [b];
filterM vy [] = return [];
filterM p (x : xs) = p x >>= filterM1 p xs x;

filterM0 flg x ys = return (filterM00 x ys flg);

filterM00 x ys True = x : ys;
filterM00 x ys False = ys;

filterM1 p xs x flg = filterM p xs >>= filterM0 flg x;

}

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.filterM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.filterM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.filterM vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];45[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 5[label="",style="solid", color="burlywood", weight=3];
46[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="Monad.filterM vz3 (vz40 : vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Monad.filterM vz3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="vz3 vz40 >>= Monad.filterM1 vz3 vz41 vz40",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="return []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="primbindIO (vz3 vz40) (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="primretIO []",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
12[label="vz3 vz40",fontsize=16,color="green",shape="box"];12 -> 19[label="",style="dashed", color="green", weight=3];
11[label="primbindIO vz5 (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="burlywood",shape="triangle"];47[label="vz5/IO vz50",fontsize=10,color="white",style="solid",shape="box"];11 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 15[label="",style="solid", color="burlywood", weight=3];
48[label="vz5/AProVE_IO vz50",fontsize=10,color="white",style="solid",shape="box"];11 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 16[label="",style="solid", color="burlywood", weight=3];
49[label="vz5/AProVE_Exception vz50",fontsize=10,color="white",style="solid",shape="box"];11 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 17[label="",style="solid", color="burlywood", weight=3];
50[label="vz5/AProVE_Error vz50",fontsize=10,color="white",style="solid",shape="box"];11 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 18[label="",style="solid", color="burlywood", weight=3];
13[label="AProVE_IO []",fontsize=16,color="green",shape="box"];19[label="vz40",fontsize=16,color="green",shape="box"];15[label="primbindIO (IO vz50) (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];15 -> 20[label="",style="solid", color="black", weight=3];
16[label="primbindIO (AProVE_IO vz50) (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];16 -> 21[label="",style="solid", color="black", weight=3];
17[label="primbindIO (AProVE_Exception vz50) (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];17 -> 22[label="",style="solid", color="black", weight=3];
18[label="primbindIO (AProVE_Error vz50) (Monad.filterM1 vz3 vz41 vz40)",fontsize=16,color="black",shape="box"];18 -> 23[label="",style="solid", color="black", weight=3];
20[label="error []",fontsize=16,color="red",shape="box"];21[label="Monad.filterM1 vz3 vz41 vz40 vz50",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="AProVE_Exception vz50",fontsize=16,color="green",shape="box"];23[label="AProVE_Error vz50",fontsize=16,color="green",shape="box"];24 -> 25[label="",style="dashed", color="red", weight=0];
24[label="Monad.filterM vz3 vz41 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="magenta"];24 -> 26[label="",style="dashed", color="magenta", weight=3];
26 -> 4[label="",style="dashed", color="red", weight=0];
26[label="Monad.filterM vz3 vz41",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
25[label="vz6 >>= Monad.filterM0 vz50 vz40",fontsize=16,color="black",shape="triangle"];25 -> 28[label="",style="solid", color="black", weight=3];
27[label="vz41",fontsize=16,color="green",shape="box"];28[label="primbindIO vz6 (Monad.filterM0 vz50 vz40)",fontsize=16,color="burlywood",shape="box"];51[label="vz6/IO vz60",fontsize=10,color="white",style="solid",shape="box"];28 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 29[label="",style="solid", color="burlywood", weight=3];
52[label="vz6/AProVE_IO vz60",fontsize=10,color="white",style="solid",shape="box"];28 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 30[label="",style="solid", color="burlywood", weight=3];
53[label="vz6/AProVE_Exception vz60",fontsize=10,color="white",style="solid",shape="box"];28 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 31[label="",style="solid", color="burlywood", weight=3];
54[label="vz6/AProVE_Error vz60",fontsize=10,color="white",style="solid",shape="box"];28 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 32[label="",style="solid", color="burlywood", weight=3];
29[label="primbindIO (IO vz60) (Monad.filterM0 vz50 vz40)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="primbindIO (AProVE_IO vz60) (Monad.filterM0 vz50 vz40)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="primbindIO (AProVE_Exception vz60) (Monad.filterM0 vz50 vz40)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="primbindIO (AProVE_Error vz60) (Monad.filterM0 vz50 vz40)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="error []",fontsize=16,color="red",shape="box"];34[label="Monad.filterM0 vz50 vz40 vz60",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
35[label="AProVE_Exception vz60",fontsize=16,color="green",shape="box"];36[label="AProVE_Error vz60",fontsize=16,color="green",shape="box"];37[label="return (Monad.filterM00 vz40 vz60 vz50)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="primretIO (Monad.filterM00 vz40 vz60 vz50)",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39[label="AProVE_IO (Monad.filterM00 vz40 vz60 vz50)",fontsize=16,color="green",shape="box"];39 -> 40[label="",style="dashed", color="green", weight=3];
40[label="Monad.filterM00 vz40 vz60 vz50",fontsize=16,color="burlywood",shape="box"];55[label="vz50/False",fontsize=10,color="white",style="solid",shape="box"];40 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 41[label="",style="solid", color="burlywood", weight=3];
56[label="vz50/True",fontsize=10,color="white",style="solid",shape="box"];40 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 42[label="",style="solid", color="burlywood", weight=3];
41[label="Monad.filterM00 vz40 vz60 False",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="Monad.filterM00 vz40 vz60 True",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="vz60",fontsize=16,color="green",shape="box"];44[label="vz40 : vz60",fontsize=16,color="green",shape="box"];}

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

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

   new_filterM(vz3, :(vz40, vz41), h) -> new_primbindIO(vz3, vz41, vz40, h)
   new_primbindIO(vz3, vz41, vz40, h) -> new_filterM(vz3, vz41, 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_primbindIO(vz3, vz41, vz40, h) -> new_filterM(vz3, vz41, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3


*new_filterM(vz3, :(vz40, vz41), h) -> new_primbindIO(vz3, vz41, vz40, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4


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

(12)
YES