/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) 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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
fmToList_GE :: Ord a => FiniteMap a b  ->  a  ->  [(a,b)];
fmToList_GE fm fr = foldFM_GE (\key elt rest ->(key,elt) : rest) [] fr fm;

foldFM_GE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt _ fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = foldFM_GE k z fr fm_r;

}
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
"fmToList_GE0 key elt rest = (key,elt) : rest;
"

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

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

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
fmToList_GE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
fmToList_GE fm fr = foldFM_GE fmToList_GE0 [] fr fm;

fmToList_GE0 key elt rest = (key,elt) : rest;

foldFM_GE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt _ fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = foldFM_GE k z fr fm_r;

}
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) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
fmToList_GE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
fmToList_GE fm fr = foldFM_GE fmToList_GE0 [] fr fm;

fmToList_GE0 key elt rest = (key,elt) : rest;

foldFM_GE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
 | otherwise = foldFM_GE k z fr fm_r;

}
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;
"
The following Function with conditions
"foldFM_GE k z fr EmptyFM = z;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r)|key >= frfoldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l|otherwisefoldFM_GE k z fr fm_r;
"
is transformed to
"foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);
"
"foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;
"
"foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
"
"foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);
"
"foldFM_GE3 k z fr EmptyFM = z;
foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
"

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

(6)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
fmToList_GE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
fmToList_GE fm fr = foldFM_GE fmToList_GE0 [] fr fm;

fmToList_GE0 key elt rest = (key,elt) : rest;

foldFM_GE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);

foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;

foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;

foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);

foldFM_GE3 k z fr EmptyFM = z;
foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;

}
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_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.fmToList_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.fmToList_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];54[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 6[label="",style="solid", color="burlywood", weight=3];
55[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.foldFM_GE3 FiniteMap.fmToList_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.foldFM_GE2 FiniteMap.fmToList_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 >= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",fontsize=16,color="burlywood",shape="box"];56[label="wz30/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 14[label="",style="solid", color="burlywood", weight=3];
14[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == LT))",fontsize=16,color="burlywood",shape="box"];57[label="wz4/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 15[label="",style="solid", color="burlywood", weight=3];
15[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 [] () () wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
19 -> 20[label="",style="dashed", color="red", weight=0];
19[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 (FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 [] () wz34)) () wz33",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
21 -> 5[label="",style="dashed", color="red", weight=0];
21[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 [] () wz34",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
21 -> 23[label="",style="dashed", color="magenta", weight=3];
20[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () wz33",fontsize=16,color="burlywood",shape="triangle"];58[label="wz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];20 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 24[label="",style="solid", color="burlywood", weight=3];
59[label="wz33/FiniteMap.Branch wz330 wz331 wz332 wz333 wz334",fontsize=10,color="white",style="solid",shape="box"];20 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="wz34",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="FiniteMap.foldFM_GE3 FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.foldFM_GE2 FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.fmToList_GE0 () wz31 wz5",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
29 -> 31[label="",style="dashed", color="red", weight=0];
29[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz31 wz5) () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
30[label="((),wz31) : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="FiniteMap.fmToList_GE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (compare wz330 () /= LT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (not (compare wz330 () == LT))",fontsize=16,color="burlywood",shape="box"];60[label="wz330/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 35[label="",style="solid", color="burlywood", weight=3];
35[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (EQ == LT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () () wz331 wz332 wz333 wz334 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () () wz331 wz332 wz333 wz334 True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39 -> 20[label="",style="dashed", color="red", weight=0];
39[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 (FiniteMap.fmToList_GE0 () wz331 (FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 wz6 () wz334)) () wz333",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
39 -> 41[label="",style="dashed", color="magenta", weight=3];
39 -> 42[label="",style="dashed", color="magenta", weight=3];
40[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 wz6 () wz334",fontsize=16,color="burlywood",shape="box"];61[label="wz334/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];40 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 43[label="",style="solid", color="burlywood", weight=3];
62[label="wz334/FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344",fontsize=10,color="white",style="solid",shape="box"];40 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 44[label="",style="solid", color="burlywood", weight=3];
41[label="wz333",fontsize=16,color="green",shape="box"];42[label="wz331",fontsize=16,color="green",shape="box"];43[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="FiniteMap.foldFM_GE FiniteMap.fmToList_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="FiniteMap.foldFM_GE3 FiniteMap.fmToList_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="FiniteMap.foldFM_GE2 FiniteMap.fmToList_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="wz6",fontsize=16,color="green",shape="box"];48 -> 31[label="",style="dashed", color="red", weight=0];
48[label="FiniteMap.foldFM_GE1 FiniteMap.fmToList_GE0 wz6 () wz3340 wz3341 wz3342 wz3343 wz3344 (wz3340 >= ())",fontsize=16,color="magenta"];48 -> 49[label="",style="dashed", color="magenta", weight=3];
48 -> 50[label="",style="dashed", color="magenta", weight=3];
48 -> 51[label="",style="dashed", color="magenta", weight=3];
48 -> 52[label="",style="dashed", color="magenta", weight=3];
48 -> 53[label="",style="dashed", color="magenta", weight=3];
49[label="wz3343",fontsize=16,color="green",shape="box"];50[label="wz3340",fontsize=16,color="green",shape="box"];51[label="wz3344",fontsize=16,color="green",shape="box"];52[label="wz3342",fontsize=16,color="green",shape="box"];53[label="wz3341",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h)

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_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


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

(11)
YES

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

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

   new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_fmToList_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)

The TRS R consists of the following rules:

   new_fmToList_GE0(wz31, wz5, h) -> :(@2(@0, wz31), wz5)
   new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_fmToList_GE0(wz31, wz5, h)
   new_foldFM_GE0(wz6, EmptyFM, h) -> wz6
   new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
   new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_fmToList_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)

The set Q consists of the following terms:

   new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_GE2(x0, x1, EmptyFM, x2)
   new_foldFM_GE0(x0, EmptyFM, x1)
   new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5)
   new_fmToList_GE0(x0, x1, x2)
   new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

We have to consider all minimal (P,Q,R)-chains.
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(13) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_fmToList_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@2(@0, wz31), wz5), wz330, wz331, wz332, wz333, wz334, h),new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@2(@0, wz31), wz5), wz330, wz331, wz332, wz333, wz334, h))


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(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
   new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@2(@0, wz31), wz5), wz330, wz331, wz332, wz333, wz334, h)

The TRS R consists of the following rules:

   new_fmToList_GE0(wz31, wz5, h) -> :(@2(@0, wz31), wz5)
   new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_fmToList_GE0(wz31, wz5, h)
   new_foldFM_GE0(wz6, EmptyFM, h) -> wz6
   new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
   new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_fmToList_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)

The set Q consists of the following terms:

   new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_GE2(x0, x1, EmptyFM, x2)
   new_foldFM_GE0(x0, EmptyFM, x1)
   new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5)
   new_fmToList_GE0(x0, x1, x2)
   new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

We have to consider all minimal (P,Q,R)-chains.
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(15) 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_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7


*new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4


*new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(@2(@0, wz31), wz5), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7


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(16)
YES