/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) 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;
liftM4 :: Monad c => (b  ->  e  ->  f  ->  a  ->  d)  ->  c b  ->  c e  ->  c f  ->  c a  ->  c d;
liftM4 f m1 m2 m3 m4 = m1 >>= (\x1 ->m2 >>= (\x2 ->m3 >>= (\x3 ->m4 >>= (\x4 ->return (f x1 x2 x3 x4)))));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\x4->return (f x1 x2 x3 x4)"
is transformed to
"liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4);
"
The following Lambda expression
"\x3->m4 >>= liftM40 f x1 x2 x3"
is transformed to
"liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3;
"
The following Lambda expression
"\x2->m3 >>= liftM41 m4 f x1 x2"
is transformed to
"liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2;
"
The following Lambda expression
"\x1->m2 >>= liftM42 m3 m4 f x1"
is transformed to
"liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 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;
liftM4 :: Monad b => (e  ->  f  ->  a  ->  d  ->  c)  ->  b e  ->  b f  ->  b a  ->  b d  ->  b c;
liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f;

liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4);

liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3;

liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2;

liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 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;
liftM4 :: Monad c => (e  ->  b  ->  a  ->  d  ->  f)  ->  c e  ->  c b  ->  c a  ->  c d  ->  c f;
liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f;

liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4);

liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3;

liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2;

liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 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;
liftM4 :: Monad f => (b  ->  c  ->  a  ->  d  ->  e)  ->  f b  ->  f c  ->  f a  ->  f d  ->  f e;
liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f;

liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4);

liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3;

liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2;

liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 f x1;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.liftM4",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.liftM4 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.liftM4 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.liftM4 vy3 vy4 vy5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="Monad.liftM4 vy3 vy4 vy5 vy6",fontsize=16,color="grey",shape="box"];6 -> 7[label="",style="dashed", color="grey", weight=3];
7[label="Monad.liftM4 vy3 vy4 vy5 vy6 vy7",fontsize=16,color="black",shape="triangle"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="vy4 >>= Monad.liftM43 vy5 vy6 vy7 vy3",fontsize=16,color="burlywood",shape="triangle"];77[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];8 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 9[label="",style="solid", color="burlywood", weight=3];
78[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 10[label="",style="solid", color="burlywood", weight=3];
9[label="vy40 : vy41 >>= Monad.liftM43 vy5 vy6 vy7 vy3",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="[] >>= Monad.liftM43 vy5 vy6 vy7 vy3",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 28[label="",style="dashed", color="red", weight=0];
11[label="Monad.liftM43 vy5 vy6 vy7 vy3 vy40 ++ (vy41 >>= Monad.liftM43 vy5 vy6 vy7 vy3)",fontsize=16,color="magenta"];11 -> 29[label="",style="dashed", color="magenta", weight=3];
11 -> 30[label="",style="dashed", color="magenta", weight=3];
12[label="[]",fontsize=16,color="green",shape="box"];29[label="Monad.liftM43 vy5 vy6 vy7 vy3 vy40",fontsize=16,color="black",shape="box"];29 -> 39[label="",style="solid", color="black", weight=3];
30 -> 8[label="",style="dashed", color="red", weight=0];
30[label="vy41 >>= Monad.liftM43 vy5 vy6 vy7 vy3",fontsize=16,color="magenta"];30 -> 40[label="",style="dashed", color="magenta", weight=3];
28[label="vy9 ++ vy8",fontsize=16,color="burlywood",shape="triangle"];79[label="vy9/vy90 : vy91",fontsize=10,color="white",style="solid",shape="box"];28 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 41[label="",style="solid", color="burlywood", weight=3];
80[label="vy9/[]",fontsize=10,color="white",style="solid",shape="box"];28 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 42[label="",style="solid", color="burlywood", weight=3];
39[label="vy5 >>= Monad.liftM42 vy6 vy7 vy3 vy40",fontsize=16,color="burlywood",shape="triangle"];81[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];39 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 43[label="",style="solid", color="burlywood", weight=3];
82[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];39 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 44[label="",style="solid", color="burlywood", weight=3];
40[label="vy41",fontsize=16,color="green",shape="box"];41[label="(vy90 : vy91) ++ vy8",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="[] ++ vy8",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="vy50 : vy51 >>= Monad.liftM42 vy6 vy7 vy3 vy40",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44[label="[] >>= Monad.liftM42 vy6 vy7 vy3 vy40",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="vy90 : vy91 ++ vy8",fontsize=16,color="green",shape="box"];45 -> 49[label="",style="dashed", color="green", weight=3];
46[label="vy8",fontsize=16,color="green",shape="box"];47 -> 28[label="",style="dashed", color="red", weight=0];
47[label="Monad.liftM42 vy6 vy7 vy3 vy40 vy50 ++ (vy51 >>= Monad.liftM42 vy6 vy7 vy3 vy40)",fontsize=16,color="magenta"];47 -> 50[label="",style="dashed", color="magenta", weight=3];
47 -> 51[label="",style="dashed", color="magenta", weight=3];
48[label="[]",fontsize=16,color="green",shape="box"];49 -> 28[label="",style="dashed", color="red", weight=0];
49[label="vy91 ++ vy8",fontsize=16,color="magenta"];49 -> 52[label="",style="dashed", color="magenta", weight=3];
50[label="Monad.liftM42 vy6 vy7 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
51 -> 39[label="",style="dashed", color="red", weight=0];
51[label="vy51 >>= Monad.liftM42 vy6 vy7 vy3 vy40",fontsize=16,color="magenta"];51 -> 54[label="",style="dashed", color="magenta", weight=3];
52[label="vy91",fontsize=16,color="green",shape="box"];53[label="vy6 >>= Monad.liftM41 vy7 vy3 vy40 vy50",fontsize=16,color="burlywood",shape="triangle"];83[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];53 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 55[label="",style="solid", color="burlywood", weight=3];
84[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 56[label="",style="solid", color="burlywood", weight=3];
54[label="vy51",fontsize=16,color="green",shape="box"];55[label="vy60 : vy61 >>= Monad.liftM41 vy7 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
56[label="[] >>= Monad.liftM41 vy7 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57 -> 28[label="",style="dashed", color="red", weight=0];
57[label="Monad.liftM41 vy7 vy3 vy40 vy50 vy60 ++ (vy61 >>= Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="magenta"];57 -> 59[label="",style="dashed", color="magenta", weight=3];
57 -> 60[label="",style="dashed", color="magenta", weight=3];
58[label="[]",fontsize=16,color="green",shape="box"];59[label="Monad.liftM41 vy7 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3];
60 -> 53[label="",style="dashed", color="red", weight=0];
60[label="vy61 >>= Monad.liftM41 vy7 vy3 vy40 vy50",fontsize=16,color="magenta"];60 -> 62[label="",style="dashed", color="magenta", weight=3];
61[label="vy7 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="burlywood",shape="triangle"];85[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];61 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 63[label="",style="solid", color="burlywood", weight=3];
86[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 64[label="",style="solid", color="burlywood", weight=3];
62[label="vy61",fontsize=16,color="green",shape="box"];63[label="vy70 : vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];63 -> 65[label="",style="solid", color="black", weight=3];
64[label="[] >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
65 -> 28[label="",style="dashed", color="red", weight=0];
65[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70 ++ (vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="magenta"];65 -> 67[label="",style="dashed", color="magenta", weight=3];
65 -> 68[label="",style="dashed", color="magenta", weight=3];
66[label="[]",fontsize=16,color="green",shape="box"];67[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70",fontsize=16,color="black",shape="box"];67 -> 69[label="",style="solid", color="black", weight=3];
68 -> 61[label="",style="dashed", color="red", weight=0];
68[label="vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="magenta"];68 -> 70[label="",style="dashed", color="magenta", weight=3];
69[label="return (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="black",shape="box"];69 -> 71[label="",style="solid", color="black", weight=3];
70[label="vy71",fontsize=16,color="green",shape="box"];71[label="vy3 vy40 vy50 vy60 vy70 : []",fontsize=16,color="green",shape="box"];71 -> 72[label="",style="dashed", color="green", weight=3];
72[label="vy3 vy40 vy50 vy60 vy70",fontsize=16,color="green",shape="box"];72 -> 73[label="",style="dashed", color="green", weight=3];
72 -> 74[label="",style="dashed", color="green", weight=3];
72 -> 75[label="",style="dashed", color="green", weight=3];
72 -> 76[label="",style="dashed", color="green", weight=3];
73[label="vy40",fontsize=16,color="green",shape="box"];74[label="vy50",fontsize=16,color="green",shape="box"];75[label="vy60",fontsize=16,color="green",shape="box"];76[label="vy70",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(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) -> new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd)

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(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) -> new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10


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

(11)
YES

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

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

   new_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) -> new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd)

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_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) -> new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10


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

(14)
YES

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

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

   new_gtGtEs(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) -> new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd)

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_gtGtEs(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) -> new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10


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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(vy90, vy91), vy8, h) -> new_psPs(vy91, vy8, h)

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

(19) 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(:(vy90, vy91), vy8, h) -> new_psPs(vy91, vy8, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(20)
YES

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) -> new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd)

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

(22) 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_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) -> new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10


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

(23)
YES