/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) 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;
foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\fax->foldM f fax xs"
is transformed to
"foldM0 f xs fax = foldM f fax xs;
"

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

(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;
foldM :: Monad b => (c  ->  a  ->  b c)  ->  c  ->  [a]  ->  b c;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(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;
foldM :: Monad a => (c  ->  b  ->  a c)  ->  c  ->  [b]  ->  a c;
foldM vy a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(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;
foldM :: Monad c => (a  ->  b  ->  c a)  ->  a  ->  [b]  ->  c a;
foldM vy a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];24[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 6[label="",style="solid", color="burlywood", weight=3];
25[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="return vz4",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
11[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];11 -> 17[label="",style="dashed", color="green", weight=3];
11 -> 18[label="",style="dashed", color="green", weight=3];
10[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];26[label="vz6/Nothing",fontsize=10,color="white",style="solid",shape="box"];10 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 15[label="",style="solid", color="burlywood", weight=3];
27[label="vz6/Just vz60",fontsize=10,color="white",style="solid",shape="box"];10 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 16[label="",style="solid", color="burlywood", weight=3];
12[label="Just vz4",fontsize=16,color="green",shape="box"];17[label="vz4",fontsize=16,color="green",shape="box"];18[label="vz50",fontsize=16,color="green",shape="box"];15[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="Just vz60 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21 -> 5[label="",style="dashed", color="red", weight=0];
21[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
21 -> 23[label="",style="dashed", color="magenta", weight=3];
22[label="vz51",fontsize=16,color="green",shape="box"];23[label="vz60",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldM(vz3, :(vz50, vz51), h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
   new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)

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

(9) 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(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4


*new_foldM(vz3, :(vz50, vz51), h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(10)
YES