/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) 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 a b 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;

}
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 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;

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

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

(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 a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  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;

}
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];
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6[label="FiniteMap.foldFM_GE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];152[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 7[label="",style="solid", color="burlywood", weight=3];
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153 -> 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];
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154 -> 17[label="",style="solid", color="burlywood", weight=3];
155[label="wz60/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 155[label="",style="solid", color="burlywood", weight=9];
155 -> 18[label="",style="solid", color="burlywood", weight=3];
156[label="wz60/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 156[label="",style="solid", color="burlywood", weight=9];
156 -> 19[label="",style="solid", color="burlywood", weight=3];
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157 -> 20[label="",style="solid", color="burlywood", weight=3];
158[label="wz5/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 158[label="",style="solid", color="burlywood", weight=9];
158 -> 21[label="",style="solid", color="burlywood", weight=3];
159[label="wz5/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 159[label="",style="solid", color="burlywood", weight=9];
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160 -> 23[label="",style="solid", color="burlywood", weight=3];
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161 -> 24[label="",style="solid", color="burlywood", weight=3];
162[label="wz5/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 162[label="",style="solid", color="burlywood", weight=9];
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52[label="FiniteMap.foldFM_GE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 (not (compare1 EQ GT True == LT))",fontsize=16,color="black",shape="box"];52 -> 61[label="",style="solid", color="black", weight=3];
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60[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];60 -> 69[label="",style="solid", color="black", weight=3];
61[label="FiniteMap.foldFM_GE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 (not (LT == LT))",fontsize=16,color="black",shape="box"];61 -> 70[label="",style="solid", color="black", weight=3];
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65 -> 6[label="",style="dashed", color="red", weight=0];
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65 -> 75[label="",style="dashed", color="magenta", weight=3];
65 -> 76[label="",style="dashed", color="magenta", weight=3];
66[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];66 -> 77[label="",style="solid", color="black", weight=3];
67[label="FiniteMap.foldFM_GE1 wz3 wz4 GT LT wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];67 -> 78[label="",style="solid", color="black", weight=3];
68[label="FiniteMap.foldFM_GE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 (not (compare0 EQ LT True == LT))",fontsize=16,color="black",shape="box"];68 -> 79[label="",style="solid", color="black", weight=3];
69 -> 6[label="",style="dashed", color="red", weight=0];
69[label="FiniteMap.foldFM_GE wz3 (wz3 EQ wz61 (FiniteMap.foldFM_GE wz3 wz4 EQ wz64)) EQ wz63",fontsize=16,color="magenta"];69 -> 80[label="",style="dashed", color="magenta", weight=3];
69 -> 81[label="",style="dashed", color="magenta", weight=3];
69 -> 82[label="",style="dashed", color="magenta", weight=3];
70[label="FiniteMap.foldFM_GE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 (not True)",fontsize=16,color="black",shape="box"];70 -> 83[label="",style="solid", color="black", weight=3];
71[label="FiniteMap.foldFM_GE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 (not (compare0 GT LT True == LT))",fontsize=16,color="black",shape="box"];71 -> 84[label="",style="solid", color="black", weight=3];
72[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 (not (compare0 GT EQ True == LT))",fontsize=16,color="black",shape="box"];72 -> 85[label="",style="solid", color="black", weight=3];
73 -> 6[label="",style="dashed", color="red", weight=0];
73[label="FiniteMap.foldFM_GE wz3 (wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 GT wz64)) GT wz63",fontsize=16,color="magenta"];73 -> 86[label="",style="dashed", color="magenta", weight=3];
73 -> 87[label="",style="dashed", color="magenta", weight=3];
73 -> 88[label="",style="dashed", color="magenta", weight=3];
74[label="wz3 LT wz61 (FiniteMap.foldFM_GE wz3 wz4 LT wz64)",fontsize=16,color="green",shape="box"];74 -> 89[label="",style="dashed", color="green", weight=3];
74 -> 90[label="",style="dashed", color="green", weight=3];
74 -> 91[label="",style="dashed", color="green", weight=3];
75[label="wz63",fontsize=16,color="green",shape="box"];76[label="LT",fontsize=16,color="green",shape="box"];77[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];77 -> 92[label="",style="solid", color="black", weight=3];
78[label="FiniteMap.foldFM_GE1 wz3 wz4 GT LT wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];78 -> 93[label="",style="solid", color="black", weight=3];
79[label="FiniteMap.foldFM_GE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 (not (GT == LT))",fontsize=16,color="black",shape="box"];79 -> 94[label="",style="solid", color="black", weight=3];
80[label="wz3 EQ wz61 (FiniteMap.foldFM_GE wz3 wz4 EQ wz64)",fontsize=16,color="green",shape="box"];80 -> 95[label="",style="dashed", color="green", weight=3];
80 -> 96[label="",style="dashed", color="green", weight=3];
80 -> 97[label="",style="dashed", color="green", weight=3];
81[label="wz63",fontsize=16,color="green",shape="box"];82[label="EQ",fontsize=16,color="green",shape="box"];83[label="FiniteMap.foldFM_GE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];83 -> 98[label="",style="solid", color="black", weight=3];
84[label="FiniteMap.foldFM_GE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 (not (GT == LT))",fontsize=16,color="black",shape="box"];84 -> 99[label="",style="solid", color="black", weight=3];
85[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 (not (GT == LT))",fontsize=16,color="black",shape="box"];85 -> 100[label="",style="solid", color="black", weight=3];
86[label="wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 GT wz64)",fontsize=16,color="green",shape="box"];86 -> 101[label="",style="dashed", color="green", weight=3];
86 -> 102[label="",style="dashed", color="green", weight=3];
86 -> 103[label="",style="dashed", color="green", weight=3];
87[label="wz63",fontsize=16,color="green",shape="box"];88[label="GT",fontsize=16,color="green",shape="box"];89[label="LT",fontsize=16,color="green",shape="box"];90[label="wz61",fontsize=16,color="green",shape="box"];91 -> 6[label="",style="dashed", color="red", weight=0];
91[label="FiniteMap.foldFM_GE wz3 wz4 LT wz64",fontsize=16,color="magenta"];91 -> 104[label="",style="dashed", color="magenta", weight=3];
91 -> 105[label="",style="dashed", color="magenta", weight=3];
92[label="FiniteMap.foldFM_GE0 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];92 -> 106[label="",style="solid", color="black", weight=3];
93[label="FiniteMap.foldFM_GE0 wz3 wz4 GT LT wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];93 -> 107[label="",style="solid", color="black", weight=3];
94[label="FiniteMap.foldFM_GE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];94 -> 108[label="",style="solid", color="black", weight=3];
95[label="EQ",fontsize=16,color="green",shape="box"];96[label="wz61",fontsize=16,color="green",shape="box"];97 -> 6[label="",style="dashed", color="red", weight=0];
97[label="FiniteMap.foldFM_GE wz3 wz4 EQ wz64",fontsize=16,color="magenta"];97 -> 109[label="",style="dashed", color="magenta", weight=3];
97 -> 110[label="",style="dashed", color="magenta", weight=3];
98[label="FiniteMap.foldFM_GE0 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];98 -> 111[label="",style="solid", color="black", weight=3];
99[label="FiniteMap.foldFM_GE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];99 -> 112[label="",style="solid", color="black", weight=3];
100[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];100 -> 113[label="",style="solid", color="black", weight=3];
101[label="GT",fontsize=16,color="green",shape="box"];102[label="wz61",fontsize=16,color="green",shape="box"];103 -> 6[label="",style="dashed", color="red", weight=0];
103[label="FiniteMap.foldFM_GE wz3 wz4 GT wz64",fontsize=16,color="magenta"];103 -> 114[label="",style="dashed", color="magenta", weight=3];
103 -> 115[label="",style="dashed", color="magenta", weight=3];
104[label="wz64",fontsize=16,color="green",shape="box"];105[label="LT",fontsize=16,color="green",shape="box"];106[label="FiniteMap.foldFM_GE0 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];106 -> 116[label="",style="solid", color="black", weight=3];
107[label="FiniteMap.foldFM_GE0 wz3 wz4 GT LT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];107 -> 117[label="",style="solid", color="black", weight=3];
108[label="FiniteMap.foldFM_GE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];108 -> 118[label="",style="solid", color="black", weight=3];
109[label="wz64",fontsize=16,color="green",shape="box"];110[label="EQ",fontsize=16,color="green",shape="box"];111[label="FiniteMap.foldFM_GE0 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];111 -> 119[label="",style="solid", color="black", weight=3];
112[label="FiniteMap.foldFM_GE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];112 -> 120[label="",style="solid", color="black", weight=3];
113[label="FiniteMap.foldFM_GE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];113 -> 121[label="",style="solid", color="black", weight=3];
114[label="wz64",fontsize=16,color="green",shape="box"];115[label="GT",fontsize=16,color="green",shape="box"];116 -> 6[label="",style="dashed", color="red", weight=0];
116[label="FiniteMap.foldFM_GE wz3 wz4 EQ wz64",fontsize=16,color="magenta"];116 -> 122[label="",style="dashed", color="magenta", weight=3];
116 -> 123[label="",style="dashed", color="magenta", weight=3];
117 -> 6[label="",style="dashed", color="red", weight=0];
117[label="FiniteMap.foldFM_GE wz3 wz4 GT wz64",fontsize=16,color="magenta"];117 -> 124[label="",style="dashed", color="magenta", weight=3];
117 -> 125[label="",style="dashed", color="magenta", weight=3];
118 -> 6[label="",style="dashed", color="red", weight=0];
118[label="FiniteMap.foldFM_GE wz3 (wz3 EQ wz61 (FiniteMap.foldFM_GE wz3 wz4 LT wz64)) LT wz63",fontsize=16,color="magenta"];118 -> 126[label="",style="dashed", color="magenta", weight=3];
118 -> 127[label="",style="dashed", color="magenta", weight=3];
118 -> 128[label="",style="dashed", color="magenta", weight=3];
119 -> 6[label="",style="dashed", color="red", weight=0];
119[label="FiniteMap.foldFM_GE wz3 wz4 GT wz64",fontsize=16,color="magenta"];119 -> 129[label="",style="dashed", color="magenta", weight=3];
119 -> 130[label="",style="dashed", color="magenta", weight=3];
120 -> 6[label="",style="dashed", color="red", weight=0];
120[label="FiniteMap.foldFM_GE wz3 (wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 LT wz64)) LT wz63",fontsize=16,color="magenta"];120 -> 131[label="",style="dashed", color="magenta", weight=3];
120 -> 132[label="",style="dashed", color="magenta", weight=3];
120 -> 133[label="",style="dashed", color="magenta", weight=3];
121 -> 6[label="",style="dashed", color="red", weight=0];
121[label="FiniteMap.foldFM_GE wz3 (wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 EQ wz64)) EQ wz63",fontsize=16,color="magenta"];121 -> 134[label="",style="dashed", color="magenta", weight=3];
121 -> 135[label="",style="dashed", color="magenta", weight=3];
121 -> 136[label="",style="dashed", color="magenta", weight=3];
122[label="wz64",fontsize=16,color="green",shape="box"];123[label="EQ",fontsize=16,color="green",shape="box"];124[label="wz64",fontsize=16,color="green",shape="box"];125[label="GT",fontsize=16,color="green",shape="box"];126[label="wz3 EQ wz61 (FiniteMap.foldFM_GE wz3 wz4 LT wz64)",fontsize=16,color="green",shape="box"];126 -> 137[label="",style="dashed", color="green", weight=3];
126 -> 138[label="",style="dashed", color="green", weight=3];
126 -> 139[label="",style="dashed", color="green", weight=3];
127[label="wz63",fontsize=16,color="green",shape="box"];128[label="LT",fontsize=16,color="green",shape="box"];129[label="wz64",fontsize=16,color="green",shape="box"];130[label="GT",fontsize=16,color="green",shape="box"];131[label="wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 LT wz64)",fontsize=16,color="green",shape="box"];131 -> 140[label="",style="dashed", color="green", weight=3];
131 -> 141[label="",style="dashed", color="green", weight=3];
131 -> 142[label="",style="dashed", color="green", weight=3];
132[label="wz63",fontsize=16,color="green",shape="box"];133[label="LT",fontsize=16,color="green",shape="box"];134[label="wz3 GT wz61 (FiniteMap.foldFM_GE wz3 wz4 EQ wz64)",fontsize=16,color="green",shape="box"];134 -> 143[label="",style="dashed", color="green", weight=3];
134 -> 144[label="",style="dashed", color="green", weight=3];
134 -> 145[label="",style="dashed", color="green", weight=3];
135[label="wz63",fontsize=16,color="green",shape="box"];136[label="EQ",fontsize=16,color="green",shape="box"];137[label="EQ",fontsize=16,color="green",shape="box"];138[label="wz61",fontsize=16,color="green",shape="box"];139 -> 6[label="",style="dashed", color="red", weight=0];
139[label="FiniteMap.foldFM_GE wz3 wz4 LT wz64",fontsize=16,color="magenta"];139 -> 146[label="",style="dashed", color="magenta", weight=3];
139 -> 147[label="",style="dashed", color="magenta", weight=3];
140[label="GT",fontsize=16,color="green",shape="box"];141[label="wz61",fontsize=16,color="green",shape="box"];142 -> 6[label="",style="dashed", color="red", weight=0];
142[label="FiniteMap.foldFM_GE wz3 wz4 LT wz64",fontsize=16,color="magenta"];142 -> 148[label="",style="dashed", color="magenta", weight=3];
142 -> 149[label="",style="dashed", color="magenta", weight=3];
143[label="GT",fontsize=16,color="green",shape="box"];144[label="wz61",fontsize=16,color="green",shape="box"];145 -> 6[label="",style="dashed", color="red", weight=0];
145[label="FiniteMap.foldFM_GE wz3 wz4 EQ wz64",fontsize=16,color="magenta"];145 -> 150[label="",style="dashed", color="magenta", weight=3];
145 -> 151[label="",style="dashed", color="magenta", weight=3];
146[label="wz64",fontsize=16,color="green",shape="box"];147[label="LT",fontsize=16,color="green",shape="box"];148[label="wz64",fontsize=16,color="green",shape="box"];149[label="LT",fontsize=16,color="green",shape="box"];150[label="wz64",fontsize=16,color="green",shape="box"];151[label="EQ",fontsize=16,color="green",shape="box"];}

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

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

   new_foldFM_GE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz63, h, ba)
   new_foldFM_GE(wz3, LT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz63, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz64, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz64, h, ba)
   new_foldFM_GE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz63, h, ba)
   new_foldFM_GE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz63, h, ba)
   new_foldFM_GE(wz3, LT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz63, h, ba)
   new_foldFM_GE(wz3, LT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz64, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz63, h, ba)
   new_foldFM_GE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
   new_foldFM_GE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz64, h, ba)
   new_foldFM_GE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
   new_foldFM_GE(wz3, LT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, LT, wz64, h, ba)
   new_foldFM_GE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, 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 3 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   new_foldFM_GE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
   new_foldFM_GE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz63, h, ba)
   new_foldFM_GE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
   new_foldFM_GE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)

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

(10) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_foldFM_GE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, GT, wz64, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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


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

(11)
YES

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

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

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


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


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


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


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_GE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz64, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz63, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz64, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz63, h, ba)
   new_foldFM_GE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_GE(wz3, EQ, wz64, h, ba)

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


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


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


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


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(17)
YES