/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) AND
    (7) QDP
        (8) DependencyGraphProof [EQUIVALENT, 0 ms]
        (9) AND
            (10) QDP
                (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (12) YES
            (13) QDP
                (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (15) YES
    (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) 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;
class Monad a => MonadPlus a where {
mplus :: MonadPlus a => a b  ->  a b  ->  a b;
mzero :: MonadPlus a => a b;
}
instance MonadPlus Maybe where { 
mplus Nothing ys = ys;
mplus xs _ys = xs;
mzero = Nothing;
}
instance MonadPlus [] where { 
mplus = (++);
mzero = [];
}
msum :: MonadPlus b => [b a]  ->  b a;
msum = foldr mplus mzero;

}

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

(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;
class Monad a => MonadPlus a where {
mplus :: MonadPlus a => a b  ->  a b  ->  a b;
mzero :: MonadPlus a => a b;
}
instance MonadPlus Maybe where { 
mplus Nothing ys = ys;
mplus xs _ys = xs;
mzero = Nothing;
}
instance MonadPlus [] where { 
mplus = (++);
mzero = [];
}
msum :: MonadPlus b => [b a]  ->  b a;
msum = foldr mplus mzero;

}

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

(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;
class Monad a => MonadPlus a where {
mplus :: MonadPlus a => a b  ->  a b  ->  a b;
mzero :: MonadPlus a => a b;
}
instance MonadPlus Maybe where { 
mplus Nothing ys = ys;
mplus xs _ys = xs;
mzero = Nothing;
}
instance MonadPlus [] where { 
mplus = (++);
mzero = [];
}
msum :: MonadPlus a => [a b]  ->  a b;
msum = foldr mplus mzero;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

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

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

(8) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(9)
Complex Obligation (AND)

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

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

   new_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, 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_foldr(:(vy30, vy31), ty_Maybe, h) -> new_foldr(vy31, ty_Maybe, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(12)
YES

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)

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

(14) 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_foldr(:(vy30, vy31), ty_[], h) -> new_foldr(vy31, ty_[], h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(15)
YES

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

(16)
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.
----------------------------------------

(17) 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


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

(18)
YES