/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) 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;
liftM3 :: Monad e => (c  ->  b  ->  a  ->  d)  ->  e c  ->  e b  ->  e a  ->  e d;
liftM3 f m1 m2 m3 = m1 >>= (\x1 ->m2 >>= (\x2 ->m3 >>= (\x3 ->return (f x1 x2 x3))));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\x3->return (f x1 x2 x3)"
is transformed to
"liftM30 f x1 x2 x3 = return (f x1 x2 x3);
"
The following Lambda expression
"\x2->m3 >>= liftM30 f x1 x2"
is transformed to
"liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;
"
The following Lambda expression
"\x1->m2 >>= liftM31 m3 f x1"
is transformed to
"liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 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;
liftM3 :: Monad b => (e  ->  d  ->  c  ->  a)  ->  b e  ->  b d  ->  b c  ->  b a;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 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;
liftM3 :: Monad a => (b  ->  d  ->  c  ->  e)  ->  a b  ->  a d  ->  a c  ->  a e;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 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;
liftM3 :: Monad b => (c  ->  a  ->  d  ->  e)  ->  b c  ->  b a  ->  b d  ->  b e;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 f x1;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.liftM3",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.liftM3 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.liftM3 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.liftM3 vy3 vy4 vy5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="Monad.liftM3 vy3 vy4 vy5 vy6",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="vy4 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="burlywood",shape="triangle"];67[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 8[label="",style="solid", color="burlywood", weight=3];
68[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="vy40 : vy41 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="[] >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10 -> 27[label="",style="dashed", color="red", weight=0];
10[label="Monad.liftM32 vy5 vy6 vy3 vy40 ++ (vy41 >>= Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="magenta"];10 -> 28[label="",style="dashed", color="magenta", weight=3];
10 -> 29[label="",style="dashed", color="magenta", weight=3];
11[label="[]",fontsize=16,color="green",shape="box"];28[label="Monad.liftM32 vy5 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];28 -> 38[label="",style="solid", color="black", weight=3];
29 -> 7[label="",style="dashed", color="red", weight=0];
29[label="vy41 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="magenta"];29 -> 39[label="",style="dashed", color="magenta", weight=3];
27[label="vy8 ++ vy7",fontsize=16,color="burlywood",shape="triangle"];69[label="vy8/vy80 : vy81",fontsize=10,color="white",style="solid",shape="box"];27 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 40[label="",style="solid", color="burlywood", weight=3];
70[label="vy8/[]",fontsize=10,color="white",style="solid",shape="box"];27 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 41[label="",style="solid", color="burlywood", weight=3];
38[label="vy5 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="burlywood",shape="triangle"];71[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];38 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 42[label="",style="solid", color="burlywood", weight=3];
72[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];38 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 43[label="",style="solid", color="burlywood", weight=3];
39[label="vy41",fontsize=16,color="green",shape="box"];40[label="(vy80 : vy81) ++ vy7",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="[] ++ vy7",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="vy50 : vy51 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43[label="[] >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44[label="vy80 : vy81 ++ vy7",fontsize=16,color="green",shape="box"];44 -> 48[label="",style="dashed", color="green", weight=3];
45[label="vy7",fontsize=16,color="green",shape="box"];46 -> 27[label="",style="dashed", color="red", weight=0];
46[label="Monad.liftM31 vy6 vy3 vy40 vy50 ++ (vy51 >>= Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="magenta"];46 -> 49[label="",style="dashed", color="magenta", weight=3];
46 -> 50[label="",style="dashed", color="magenta", weight=3];
47[label="[]",fontsize=16,color="green",shape="box"];48 -> 27[label="",style="dashed", color="red", weight=0];
48[label="vy81 ++ vy7",fontsize=16,color="magenta"];48 -> 51[label="",style="dashed", color="magenta", weight=3];
49[label="Monad.liftM31 vy6 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
50 -> 38[label="",style="dashed", color="red", weight=0];
50[label="vy51 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="magenta"];50 -> 53[label="",style="dashed", color="magenta", weight=3];
51[label="vy81",fontsize=16,color="green",shape="box"];52[label="vy6 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="burlywood",shape="triangle"];73[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];52 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 54[label="",style="solid", color="burlywood", weight=3];
74[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];52 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 55[label="",style="solid", color="burlywood", weight=3];
53[label="vy51",fontsize=16,color="green",shape="box"];54[label="vy60 : vy61 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
55[label="[] >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
56 -> 27[label="",style="dashed", color="red", weight=0];
56[label="Monad.liftM30 vy3 vy40 vy50 vy60 ++ (vy61 >>= Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="magenta"];56 -> 58[label="",style="dashed", color="magenta", weight=3];
56 -> 59[label="",style="dashed", color="magenta", weight=3];
57[label="[]",fontsize=16,color="green",shape="box"];58[label="Monad.liftM30 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];58 -> 60[label="",style="solid", color="black", weight=3];
59 -> 52[label="",style="dashed", color="red", weight=0];
59[label="vy61 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="magenta"];59 -> 61[label="",style="dashed", color="magenta", weight=3];
60[label="return (vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="vy61",fontsize=16,color="green",shape="box"];62[label="vy3 vy40 vy50 vy60 : []",fontsize=16,color="green",shape="box"];62 -> 63[label="",style="dashed", color="green", weight=3];
63[label="vy3 vy40 vy50 vy60",fontsize=16,color="green",shape="box"];63 -> 64[label="",style="dashed", color="green", weight=3];
63 -> 65[label="",style="dashed", color="green", weight=3];
63 -> 66[label="",style="dashed", color="green", weight=3];
64[label="vy40",fontsize=16,color="green",shape="box"];65[label="vy50",fontsize=16,color="green",shape="box"];66[label="vy60",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_gtGtEs(:(vy60, vy61), vy3, vy40, vy50, h, ba, bb, bc) -> new_gtGtEs(vy61, vy3, vy40, vy50, h, ba, bb, bc)

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


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

(11)
YES

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

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

   new_gtGtEs1(:(vy40, vy41), vy5, vy6, vy3, h, ba, bb, bc) -> new_gtGtEs1(vy41, vy5, vy6, vy3, h, ba, bb, bc)

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(:(vy40, vy41), vy5, vy6, vy3, h, ba, bb, bc) -> new_gtGtEs1(vy41, vy5, vy6, vy3, h, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8


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

(14)
YES

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

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

   new_gtGtEs0(:(vy50, vy51), vy6, vy3, vy40, h, ba, bb, bc) -> new_gtGtEs0(vy51, vy6, vy3, vy40, h, ba, bb, bc)

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_gtGtEs0(:(vy50, vy51), vy6, vy3, vy40, h, ba, bb, bc) -> new_gtGtEs0(vy51, vy6, vy3, vy40, h, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8


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

(17)
YES

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

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

   new_psPs(:(vy80, vy81), vy7, h) -> new_psPs(vy81, vy7, 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(:(vy80, vy81), vy7, h) -> new_psPs(vy81, vy7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(20)
YES