/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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


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

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

fmToList :: FiniteMap b a  ->  [(b,a)];
fmToList fm = foldFM (\key elt rest ->(key,elt) : rest) [] fm;

foldFM :: (c  ->  b  ->  a  ->  a)  ->  a  ->  FiniteMap c b  ->  a;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\keyeltrest->(key,elt) : rest"
is transformed to
"fmToList0 key elt rest = (key,elt) : rest;
"

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

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

fmToList :: FiniteMap b a  ->  [(b,a)];
fmToList fm = foldFM fmToList0 [] fm;

fmToList0 key elt rest = (key,elt) : rest;

foldFM :: (b  ->  a  ->  c  ->  c)  ->  c  ->  FiniteMap b a  ->  c;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

fmToList :: FiniteMap b a  ->  [(b,a)];
fmToList fm = foldFM fmToList0 [] fm;

fmToList0 key elt rest = (key,elt) : rest;

foldFM :: (c  ->  b  ->  a  ->  a)  ->  a  ->  FiniteMap c b  ->  a;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(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 FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

fmToList :: FiniteMap b a  ->  [(b,a)];
fmToList fm = foldFM fmToList0 [] fm;

fmToList0 key elt rest = (key,elt) : rest;

foldFM :: (a  ->  b  ->  c  ->  c)  ->  c  ->  FiniteMap a b  ->  c;
foldFM k z EmptyFM = z;
foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.fmToList",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.fmToList vz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="FiniteMap.foldFM FiniteMap.fmToList0 [] vz3",fontsize=16,color="burlywood",shape="triangle"];22[label="vz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 5[label="",style="solid", color="burlywood", weight=3];
23[label="vz3/FiniteMap.Branch vz30 vz31 vz32 vz33 vz34",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="FiniteMap.foldFM FiniteMap.fmToList0 [] FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="FiniteMap.foldFM FiniteMap.fmToList0 [] (FiniteMap.Branch vz30 vz31 vz32 vz33 vz34)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="[]",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="red", weight=0];
8[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 (FiniteMap.foldFM FiniteMap.fmToList0 [] vz34)) vz33",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="FiniteMap.foldFM FiniteMap.fmToList0 [] vz34",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 vz4) vz33",fontsize=16,color="burlywood",shape="triangle"];24[label="vz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 12[label="",style="solid", color="burlywood", weight=3];
25[label="vz33/FiniteMap.Branch vz330 vz331 vz332 vz333 vz334",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 13[label="",style="solid", color="burlywood", weight=3];
11[label="vz34",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 vz4) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 vz4) (FiniteMap.Branch vz330 vz331 vz332 vz333 vz334)",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.fmToList0 vz30 vz31 vz4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 9[label="",style="dashed", color="red", weight=0];
15[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz330 vz331 (FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 vz4) vz334)) vz333",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
15 -> 18[label="",style="dashed", color="magenta", weight=3];
15 -> 19[label="",style="dashed", color="magenta", weight=3];
15 -> 20[label="",style="dashed", color="magenta", weight=3];
16[label="(vz30,vz31) : vz4",fontsize=16,color="green",shape="box"];17[label="vz330",fontsize=16,color="green",shape="box"];18 -> 9[label="",style="dashed", color="red", weight=0];
18[label="FiniteMap.foldFM FiniteMap.fmToList0 (FiniteMap.fmToList0 vz30 vz31 vz4) vz334",fontsize=16,color="magenta"];18 -> 21[label="",style="dashed", color="magenta", weight=3];
19[label="vz331",fontsize=16,color="green",shape="box"];20[label="vz333",fontsize=16,color="green",shape="box"];21[label="vz334",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_foldFM1(Branch(vz30, vz31, vz32, vz33, vz34), h, ba) -> new_foldFM1(vz34, 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_foldFM1(Branch(vz30, vz31, vz32, vz33, vz34), h, ba) -> new_foldFM1(vz34, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(11)
YES

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

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

   new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz30, vz31, vz4, vz334, h, ba)
   new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)

The TRS R consists of the following rules:

   new_foldFM0(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM0(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)
   new_foldFM0(vz30, vz31, vz4, EmptyFM, h, ba) -> :(@2(vz30, vz31), vz4)

The set Q consists of the following terms:

   new_foldFM0(x0, x1, x2, Branch(x3, x4, x5, x6, x7), x8, x9)
   new_foldFM0(x0, x1, x2, EmptyFM, x3, x4)

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_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz30, vz31, vz4, vz334, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5, 6 >= 6


*new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)
The graph contains the following edges 4 > 1, 4 > 2, 4 > 4, 5 >= 5, 6 >= 6


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

(14)
YES