/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) TransformationProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) TransformationProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) 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 a b where { 
}
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 _ 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;

keysFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [b];
keysFM_GE fm fr = foldFM_GE (\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_GE0 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_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;

keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;

keysFM_GE0 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 b a where { 
}
foldFM_GE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  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;

keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;

keysFM_GE0 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
"compare x y|x == yEQ|x <= yLT|otherwiseGT;
"
is transformed to
"compare x y = compare3 x y;
"
"compare2 x y True = EQ;
compare2 x y False = compare1 x y (x <= y);
"
"compare1 x y True = LT;
compare1 x y False = compare0 x y otherwise;
"
"compare0 x y True = GT;
"
"compare3 x y = compare2 x y (x == y);
"
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_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;
"

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

(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_GE :: Ord c => (c  ->  a  ->  b  ->  b)  ->  b  ->  c  ->  FiniteMap c a  ->  b;
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;

keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;

keysFM_GE0 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_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.keysFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.keysFM_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];170[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 6[label="",style="solid", color="burlywood", weight=3];
171[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="FiniteMap.foldFM_GE FiniteMap.keysFM_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.keysFM_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.keysFM_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.keysFM_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.keysFM_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.keysFM_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.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare3 wz30 wz4 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare2 wz30 wz4 (wz30 == wz4) == LT))",fontsize=16,color="burlywood",shape="box"];172[label="wz30/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 16[label="",style="solid", color="burlywood", weight=3];
173[label="wz30/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 False wz31 wz32 wz33 wz34 (not (compare2 False wz4 (False == wz4) == LT))",fontsize=16,color="burlywood",shape="box"];174[label="wz4/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 18[label="",style="solid", color="burlywood", weight=3];
175[label="wz4/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 175[label="",style="solid", color="burlywood", weight=9];
175 -> 19[label="",style="solid", color="burlywood", weight=3];
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176 -> 20[label="",style="solid", color="burlywood", weight=3];
177[label="wz4/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 177[label="",style="solid", color="burlywood", weight=9];
177 -> 21[label="",style="solid", color="burlywood", weight=3];
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19[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare1 False True True == LT))",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare1 True False False == LT))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (LT == LT))",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare0 True False otherwise == LT))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38 -> 83[label="",style="dashed", color="red", weight=0];
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39[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not True)",fontsize=16,color="black",shape="box"];39 -> 44[label="",style="solid", color="black", weight=3];
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41 -> 46[label="",style="dashed", color="red", weight=0];
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84 -> 92[label="",style="dashed", color="red", weight=0];
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83[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False wz33",fontsize=16,color="burlywood",shape="triangle"];178[label="wz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];83 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 94[label="",style="solid", color="burlywood", weight=3];
179[label="wz33/FiniteMap.Branch wz330 wz331 wz332 wz333 wz334",fontsize=10,color="white",style="solid",shape="box"];83 -> 179[label="",style="solid", color="burlywood", weight=9];
179 -> 95[label="",style="solid", color="burlywood", weight=3];
44[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 False",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (GT == LT))",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
47 -> 5[label="",style="dashed", color="red", weight=0];
47[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] True wz34",fontsize=16,color="magenta"];47 -> 54[label="",style="dashed", color="magenta", weight=3];
47 -> 55[label="",style="dashed", color="magenta", weight=3];
46[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 True wz31 wz6) True wz33",fontsize=16,color="burlywood",shape="triangle"];180[label="wz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];46 -> 180[label="",style="solid", color="burlywood", weight=9];
180 -> 56[label="",style="solid", color="burlywood", weight=3];
181[label="wz33/FiniteMap.Branch wz330 wz331 wz332 wz333 wz334",fontsize=10,color="white",style="solid",shape="box"];46 -> 181[label="",style="solid", color="burlywood", weight=9];
181 -> 57[label="",style="solid", color="burlywood", weight=3];
93 -> 83[label="",style="dashed", color="red", weight=0];
93[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] False wz34",fontsize=16,color="magenta"];93 -> 96[label="",style="dashed", color="magenta", weight=3];
93 -> 97[label="",style="dashed", color="magenta", weight=3];
92[label="FiniteMap.keysFM_GE0 False wz31 wz11",fontsize=16,color="black",shape="triangle"];92 -> 98[label="",style="solid", color="black", weight=3];
94[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];94 -> 101[label="",style="solid", color="black", weight=3];
95[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];95 -> 102[label="",style="solid", color="black", weight=3];
52[label="FiniteMap.foldFM_GE0 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 otherwise",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
53[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
54[label="wz34",fontsize=16,color="green",shape="box"];55[label="True",fontsize=16,color="green",shape="box"];56[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 True wz31 wz6) True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3];
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----------------------------------------

(8)
Complex Obligation (AND)

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

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

   new_foldFM_GE6(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
   new_foldFM_GE6(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, 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_GE6(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


*new_foldFM_GE6(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
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_GE1(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE1(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
   new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)

The TRS R consists of the following rules:

   new_foldFM_GE2(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE3(wz31, wz6, EmptyFM, h) -> new_keysFM_GE0(wz31, wz6, h)
   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
   new_foldFM_GE10(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE3(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
   new_foldFM_GE10(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz9, wz334, h)
   new_foldFM_GE2(wz9, EmptyFM, h) -> wz9
   new_foldFM_GE3(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)

The set Q consists of the following terms:

   new_foldFM_GE10(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_GE2(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_GE10(x0, False, x1, x2, x3, x4, x5)
   new_keysFM_GE0(x0, x1, x2)
   new_foldFM_GE3(x0, x1, EmptyFM, x2)
   new_foldFM_GE2(x0, EmptyFM, x1)
   new_foldFM_GE3(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

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

(13) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h),new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h))


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

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

   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE1(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
   new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h)

The TRS R consists of the following rules:

   new_foldFM_GE2(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
   new_foldFM_GE3(wz31, wz6, EmptyFM, h) -> new_keysFM_GE0(wz31, wz6, h)
   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
   new_foldFM_GE10(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE3(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
   new_foldFM_GE10(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz9, wz334, h)
   new_foldFM_GE2(wz9, EmptyFM, h) -> wz9
   new_foldFM_GE3(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)

The set Q consists of the following terms:

   new_foldFM_GE10(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_GE2(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_GE10(x0, False, x1, x2, x3, x4, x5)
   new_keysFM_GE0(x0, x1, x2)
   new_foldFM_GE3(x0, x1, EmptyFM, x2)
   new_foldFM_GE2(x0, EmptyFM, x1)
   new_foldFM_GE3(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

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

(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(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, 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(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3


*new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4


*new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7


*new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h)
The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7


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

(16)
YES

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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)

The TRS R consists of the following rules:

   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10

The set Q consists of the following terms:

   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
   new_keysFM_GE0(x0, x1, x2)
   new_foldFM_GE5(x0, EmptyFM, x1)
   new_keysFM_GE00(x0, x1, x2)

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

(18) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h),new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h))


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

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)

The TRS R consists of the following rules:

   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10

The set Q consists of the following terms:

   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
   new_keysFM_GE0(x0, x1, x2)
   new_foldFM_GE5(x0, EmptyFM, x1)
   new_keysFM_GE00(x0, x1, x2)

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

(20) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h),new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h))


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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h)

The TRS R consists of the following rules:

   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10

The set Q consists of the following terms:

   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
   new_keysFM_GE0(x0, x1, x2)
   new_foldFM_GE5(x0, EmptyFM, x1)
   new_keysFM_GE00(x0, x1, x2)

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

(22) 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_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3


*new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
The graph contains the following edges 2 > 2, 3 >= 3


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

(23)
YES