/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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;
ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
ap = liftM2 id;

liftM2 :: Monad c => (d  ->  a  ->  b)  ->  c d  ->  c a  ->  c b;
liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\x2->return (f x1 x2)"
is transformed to
"liftM20 f x1 x2 = return (f x1 x2);
"
The following Lambda expression
"\x1->m2 >>= liftM20 f x1"
is transformed to
"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
"

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

(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;
ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
ap = liftM2 id;

liftM2 :: Monad b => (d  ->  a  ->  c)  ->  b d  ->  b a  ->  b c;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 f x1;

}

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

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

(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;
ap :: Monad b => b (c  ->  a)  ->  b c  ->  b a;
ap = liftM2 id;

liftM2 :: Monad c => (d  ->  b  ->  a)  ->  c d  ->  c b  ->  c a;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 f x1;

}

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

(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 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;
ap :: Monad b => b (c  ->  a)  ->  b c  ->  b a;
ap = liftM2 id;

liftM2 :: Monad b => (c  ->  a  ->  d)  ->  b c  ->  b a  ->  b d;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 f x1;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="triangle"];49[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 7[label="",style="solid", color="burlywood", weight=3];
50[label="vy3/[]",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];
7[label="vy30 : vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="[] >>= Monad.liftM21 vy4 id",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="Monad.liftM21 vy4 id vy30 ++ (vy31 >>= Monad.liftM21 vy4 id)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];12 -> 6[label="",style="dashed", color="red", weight=0];
12[label="vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="magenta"];12 -> 13[label="",style="dashed", color="magenta", weight=3];
11[label="Monad.liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
13[label="vy31",fontsize=16,color="green",shape="box"];14[label="(vy4 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];51[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];14 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 15[label="",style="solid", color="burlywood", weight=3];
52[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];14 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 16[label="",style="solid", color="burlywood", weight=3];
15[label="(vy40 : vy41 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="([] >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="(Monad.liftM20 id vy30 vy40 ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="(return (id vy30 vy40) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="vy5",fontsize=16,color="green",shape="box"];21[label="((id vy30 vy40 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",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="(id vy30 vy40 : [] ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24 -> 18[label="",style="dashed", color="red", weight=0];
24[label="[] ++ (vy41 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
23[label="(id vy30 vy40 : vy6) ++ vy5",fontsize=16,color="black",shape="triangle"];23 -> 26[label="",style="solid", color="black", weight=3];
25[label="vy41 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];53[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];25 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 27[label="",style="solid", color="burlywood", weight=3];
54[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];25 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 28[label="",style="solid", color="burlywood", weight=3];
26[label="id vy30 vy40 : vy6 ++ vy5",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
26 -> 30[label="",style="dashed", color="green", weight=3];
27[label="vy410 : vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="[] >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="vy6 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];55[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 34[label="",style="solid", color="burlywood", weight=3];
56[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 35[label="",style="solid", color="burlywood", weight=3];
31 -> 30[label="",style="dashed", color="red", weight=0];
31[label="Monad.liftM20 id vy30 vy410 ++ (vy411 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];31 -> 36[label="",style="dashed", color="magenta", weight=3];
31 -> 37[label="",style="dashed", color="magenta", weight=3];
32[label="[]",fontsize=16,color="green",shape="box"];33[label="vy30 vy40",fontsize=16,color="green",shape="box"];33 -> 38[label="",style="dashed", color="green", weight=3];
34[label="(vy60 : vy61) ++ vy5",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="[] ++ vy5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="Monad.liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
37 -> 25[label="",style="dashed", color="red", weight=0];
37[label="vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="magenta"];37 -> 42[label="",style="dashed", color="magenta", weight=3];
38[label="vy40",fontsize=16,color="green",shape="box"];39[label="vy60 : vy61 ++ vy5",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
40[label="vy5",fontsize=16,color="green",shape="box"];41 -> 44[label="",style="dashed", color="red", weight=0];
41[label="return (id vy30 vy410)",fontsize=16,color="magenta"];41 -> 45[label="",style="dashed", color="magenta", weight=3];
42[label="vy411",fontsize=16,color="green",shape="box"];43 -> 30[label="",style="dashed", color="red", weight=0];
43[label="vy61 ++ vy5",fontsize=16,color="magenta"];43 -> 46[label="",style="dashed", color="magenta", weight=3];
45 -> 29[label="",style="dashed", color="red", weight=0];
45[label="id vy30 vy410",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
44[label="return vy7",fontsize=16,color="black",shape="triangle"];44 -> 48[label="",style="solid", color="black", weight=3];
46[label="vy61",fontsize=16,color="green",shape="box"];47[label="vy410",fontsize=16,color="green",shape="box"];48[label="vy7 : []",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)

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_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)

R is empty.
Q is empty.
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_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(14)
YES

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

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

   new_psPs(:(vy60, vy61), vy5, h) -> new_psPs(vy61, vy5, 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(:(vy60, vy61), vy5, h) -> new_psPs(vy61, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(17)
YES