/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a 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;

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

(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 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 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;
}

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

(3) 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);
"
"compare0 x y True = GT;
"
"compare1 x y True = LT;
compare1 x y False = compare0 x y otherwise;
"
"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;
"

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

(4)
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 a b where { 
}
foldFM_GE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
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;
}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.foldFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.foldFM_GE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.foldFM_GE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="FiniteMap.foldFM_GE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];77[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 7[label="",style="solid", color="burlywood", weight=3];
78[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="FiniteMap.foldFM_GE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.foldFM_GE wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.foldFM_GE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.foldFM_GE2 wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="wz4",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 >= wz5)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= LT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare3 wz60 wz5 == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare2 wz60 wz5 (wz60 == wz5) == LT))",fontsize=16,color="burlywood",shape="box"];79[label="wz60/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 17[label="",style="solid", color="burlywood", weight=3];
80[label="wz60/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 False wz61 wz62 wz63 wz64 (not (compare2 False wz5 (False == wz5) == LT))",fontsize=16,color="burlywood",shape="box"];81[label="wz5/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 19[label="",style="solid", color="burlywood", weight=3];
82[label="wz5/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="FiniteMap.foldFM_GE1 wz3 wz4 wz5 True wz61 wz62 wz63 wz64 (not (compare2 True wz5 (True == wz5) == LT))",fontsize=16,color="burlywood",shape="box"];83[label="wz5/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 21[label="",style="solid", color="burlywood", weight=3];
84[label="wz5/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="FiniteMap.foldFM_GE1 wz3 wz4 False False wz61 wz62 wz63 wz64 (not (compare2 False False (False == False) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.foldFM_GE1 wz3 wz4 True True wz61 wz62 wz63 wz64 (not (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="FiniteMap.foldFM_GE1 wz3 wz4 False False wz61 wz62 wz63 wz64 (not (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="FiniteMap.foldFM_GE1 wz3 wz4 True True wz61 wz62 wz63 wz64 (not (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.foldFM_GE1 wz3 wz4 False False wz61 wz62 wz63 wz64 (not (EQ == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="FiniteMap.foldFM_GE1 wz3 wz4 True True wz61 wz62 wz63 wz64 (not (EQ == LT))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="FiniteMap.foldFM_GE1 wz3 wz4 False False wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not (compare1 False True True == LT))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare1 True False False == LT))",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="FiniteMap.foldFM_GE1 wz3 wz4 True True wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="FiniteMap.foldFM_GE1 wz3 wz4 False False wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not (LT == LT))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare0 True False otherwise == LT))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="FiniteMap.foldFM_GE1 wz3 wz4 True True wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39 -> 6[label="",style="dashed", color="red", weight=0];
39[label="FiniteMap.foldFM_GE wz3 (wz3 False wz61 (FiniteMap.foldFM_GE wz3 wz4 False wz64)) False wz63",fontsize=16,color="magenta"];39 -> 43[label="",style="dashed", color="magenta", weight=3];
39 -> 44[label="",style="dashed", color="magenta", weight=3];
39 -> 45[label="",style="dashed", color="magenta", weight=3];
40[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];40 -> 46[label="",style="solid", color="black", weight=3];
41[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (compare0 True False True == LT))",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42 -> 6[label="",style="dashed", color="red", weight=0];
42[label="FiniteMap.foldFM_GE wz3 (wz3 True wz61 (FiniteMap.foldFM_GE wz3 wz4 True wz64)) True wz63",fontsize=16,color="magenta"];42 -> 48[label="",style="dashed", color="magenta", weight=3];
42 -> 49[label="",style="dashed", color="magenta", weight=3];
42 -> 50[label="",style="dashed", color="magenta", weight=3];
43[label="False",fontsize=16,color="green",shape="box"];44[label="wz3 False wz61 (FiniteMap.foldFM_GE wz3 wz4 False wz64)",fontsize=16,color="green",shape="box"];44 -> 51[label="",style="dashed", color="green", weight=3];
44 -> 52[label="",style="dashed", color="green", weight=3];
44 -> 53[label="",style="dashed", color="green", weight=3];
45[label="wz63",fontsize=16,color="green",shape="box"];46[label="FiniteMap.foldFM_GE1 wz3 wz4 True False wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];46 -> 54[label="",style="solid", color="black", weight=3];
47[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not (GT == LT))",fontsize=16,color="black",shape="box"];47 -> 55[label="",style="solid", color="black", weight=3];
48[label="True",fontsize=16,color="green",shape="box"];49[label="wz3 True wz61 (FiniteMap.foldFM_GE wz3 wz4 True wz64)",fontsize=16,color="green",shape="box"];49 -> 56[label="",style="dashed", color="green", weight=3];
49 -> 57[label="",style="dashed", color="green", weight=3];
49 -> 58[label="",style="dashed", color="green", weight=3];
50[label="wz63",fontsize=16,color="green",shape="box"];51[label="False",fontsize=16,color="green",shape="box"];52[label="wz61",fontsize=16,color="green",shape="box"];53 -> 6[label="",style="dashed", color="red", weight=0];
53[label="FiniteMap.foldFM_GE wz3 wz4 False wz64",fontsize=16,color="magenta"];53 -> 59[label="",style="dashed", color="magenta", weight=3];
53 -> 60[label="",style="dashed", color="magenta", weight=3];
54[label="FiniteMap.foldFM_GE0 wz3 wz4 True False wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];54 -> 61[label="",style="solid", color="black", weight=3];
55[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];55 -> 62[label="",style="solid", color="black", weight=3];
56[label="True",fontsize=16,color="green",shape="box"];57[label="wz61",fontsize=16,color="green",shape="box"];58 -> 6[label="",style="dashed", color="red", weight=0];
58[label="FiniteMap.foldFM_GE wz3 wz4 True wz64",fontsize=16,color="magenta"];58 -> 63[label="",style="dashed", color="magenta", weight=3];
58 -> 64[label="",style="dashed", color="magenta", weight=3];
59[label="False",fontsize=16,color="green",shape="box"];60[label="wz64",fontsize=16,color="green",shape="box"];61[label="FiniteMap.foldFM_GE0 wz3 wz4 True False wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];61 -> 65[label="",style="solid", color="black", weight=3];
62[label="FiniteMap.foldFM_GE1 wz3 wz4 False True wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63[label="True",fontsize=16,color="green",shape="box"];64[label="wz64",fontsize=16,color="green",shape="box"];65 -> 6[label="",style="dashed", color="red", weight=0];
65[label="FiniteMap.foldFM_GE wz3 wz4 True wz64",fontsize=16,color="magenta"];65 -> 67[label="",style="dashed", color="magenta", weight=3];
65 -> 68[label="",style="dashed", color="magenta", weight=3];
66 -> 6[label="",style="dashed", color="red", weight=0];
66[label="FiniteMap.foldFM_GE wz3 (wz3 True wz61 (FiniteMap.foldFM_GE wz3 wz4 False wz64)) False wz63",fontsize=16,color="magenta"];66 -> 69[label="",style="dashed", color="magenta", weight=3];
66 -> 70[label="",style="dashed", color="magenta", weight=3];
66 -> 71[label="",style="dashed", color="magenta", weight=3];
67[label="True",fontsize=16,color="green",shape="box"];68[label="wz64",fontsize=16,color="green",shape="box"];69[label="False",fontsize=16,color="green",shape="box"];70[label="wz3 True wz61 (FiniteMap.foldFM_GE wz3 wz4 False wz64)",fontsize=16,color="green",shape="box"];70 -> 72[label="",style="dashed", color="green", weight=3];
70 -> 73[label="",style="dashed", color="green", weight=3];
70 -> 74[label="",style="dashed", color="green", weight=3];
71[label="wz63",fontsize=16,color="green",shape="box"];72[label="True",fontsize=16,color="green",shape="box"];73[label="wz61",fontsize=16,color="green",shape="box"];74 -> 6[label="",style="dashed", color="red", weight=0];
74[label="FiniteMap.foldFM_GE wz3 wz4 False wz64",fontsize=16,color="magenta"];74 -> 75[label="",style="dashed", color="magenta", weight=3];
74 -> 76[label="",style="dashed", color="magenta", weight=3];
75[label="False",fontsize=16,color="green",shape="box"];76[label="wz64",fontsize=16,color="green",shape="box"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
   new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
   new_foldFM_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
   new_foldFM_GE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
   new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz63, h, ba)
   new_foldFM_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)
   new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   new_foldFM_GE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
   new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
   new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz63, h, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_GE(wz3, True, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, True, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, True, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


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(11)
YES

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

   new_foldFM_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
   new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
   new_foldFM_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)
   new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)

R is empty.
Q is empty.
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_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz63, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, False, Branch(False, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_GE(wz3, False, Branch(True, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, False, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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(14)
YES