/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) 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 a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
foldFM_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt _ fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = foldFM_LE k z fr fm_l;

keysFM_LE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_LE fm fr = foldFM_LE (\key elt rest ->key : rest) [] fr fm;

}
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 : rest"
is transformed to
"keysFM_LE0 key elt rest = key : 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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
foldFM_LE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
foldFM_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt _ fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
 | otherwise = foldFM_LE k z fr fm_l;

keysFM_LE :: Ord b => FiniteMap b a  ->  b  ->  [b];
keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm;

keysFM_LE0 key elt rest = key : rest;

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

keysFM_LE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm;

keysFM_LE0 key elt rest = key : rest;

}
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_LE k z fr EmptyFM = z;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r)|key <= frfoldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r|otherwisefoldFM_LE k z fr fm_l;
"
is transformed to
"foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);
"
"foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
"
"foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;
"
"foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);
"
"foldFM_LE3 k z fr EmptyFM = z;
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
"

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

(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) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
foldFM_LE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);

foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;

foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;

foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);

foldFM_LE3 k z fr EmptyFM = z;
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;

keysFM_LE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_LE fm fr = foldFM_LE keysFM_LE0 [] fr fm;

keysFM_LE0 key elt rest = key : rest;

}
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.keysFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.keysFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.keysFM_LE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] 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_LE FiniteMap.keysFM_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] 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_LE3 FiniteMap.keysFM_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 [] 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_LE1 FiniteMap.keysFM_LE0 [] 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_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= GT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == GT))",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_LE1 FiniteMap.keysFM_LE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == GT))",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_LE1 FiniteMap.keysFM_LE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == GT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 [] () () 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_LE1 FiniteMap.keysFM_LE0 [] () () 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_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 [] () wz33)) () wz34",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_LE FiniteMap.keysFM_LE0 [] () wz33",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_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () wz34",fontsize=16,color="burlywood",shape="triangle"];58[label="wz34/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="wz34/FiniteMap.Branch wz340 wz341 wz342 wz343 wz344",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="wz33",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
26[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.keysFM_LE0 () 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_LE1 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz31 wz5) () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
30[label="() : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0];
32[label="FiniteMap.keysFM_LE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (compare wz340 () /= GT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
34[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (not (compare wz340 () == GT))",fontsize=16,color="burlywood",shape="box"];60[label="wz340/()",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_LE1 FiniteMap.keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (EQ == GT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
38[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz6 () () wz341 wz342 wz343 wz344 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_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 () wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz6 () wz343)) () wz344",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="wz341",fontsize=16,color="green",shape="box"];41[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz6 () wz343",fontsize=16,color="burlywood",shape="box"];61[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];41 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 43[label="",style="solid", color="burlywood", weight=3];
62[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];41 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 44[label="",style="solid", color="burlywood", weight=3];
42[label="wz344",fontsize=16,color="green",shape="box"];43[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz6 () (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz6 () (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",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_LE1 FiniteMap.keysFM_LE0 wz6 () wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= ())",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="wz3430",fontsize=16,color="green",shape="box"];50[label="wz3434",fontsize=16,color="green",shape="box"];51[label="wz3433",fontsize=16,color="green",shape="box"];52[label="wz3432",fontsize=16,color="green",shape="box"];53[label="wz3431",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_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE3(@0, wz33, 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_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE3(@0, wz33, 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_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
   new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)

The TRS R consists of the following rules:

   new_foldFM_LE2(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE0(wz6, EmptyFM, h) -> wz6
   new_foldFM_LE2(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(@0, wz5)
   new_foldFM_LE0(wz6, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE10(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE2(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, @0, x1, x2, x3, x4, x5)

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

(13) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@0, wz5), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@0, wz5), wz340, wz341, wz342, wz343, wz344, h))


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

   new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
   new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@0, wz5), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE2(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE0(wz6, EmptyFM, h) -> wz6
   new_foldFM_LE2(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(@0, wz5)
   new_foldFM_LE0(wz6, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE10(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE2(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, @0, x1, x2, x3, x4, x5)

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_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@0, wz5), wz340, wz341, wz342, wz343, wz344, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7


*new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
The graph contains the following edges 1 >= 1, 5 > 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 7 >= 7


*new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


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(16)
YES