/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) TransformationProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) TransformationProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
        (18) AND
            (19) QDP
                (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (21) YES
            (22) QDP
                (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (24) YES
    (25) QDP
        (26) TransformationProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) 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) ;

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

foldFM_LE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
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 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 a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

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

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

(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_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_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
"
"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) ;

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

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

(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"];114[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 6[label="",style="solid", color="burlywood", weight=3];
115[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 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="burlywood",shape="box"];116[label="wz30/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 12[label="",style="solid", color="burlywood", weight=3];
117[label="wz30/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 13[label="",style="solid", color="burlywood", weight=3];
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118 -> 14[label="",style="solid", color="burlywood", weight=3];
119[label="wz4/True",fontsize=10,color="white",style="solid",shape="box"];12 -> 119[label="",style="solid", color="burlywood", weight=9];
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120 -> 16[label="",style="solid", color="burlywood", weight=3];
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32 -> 5[label="",style="dashed", color="red", weight=0];
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32 -> 43[label="",style="dashed", color="magenta", weight=3];
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48[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 False wz31 wz6) True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];48 -> 56[label="",style="solid", color="black", weight=3];
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56 -> 54[label="",style="dashed", color="red", weight=0];
56[label="FiniteMap.keysFM_LE0 False wz31 wz6",fontsize=16,color="magenta"];56 -> 63[label="",style="dashed", color="magenta", weight=3];
57 -> 64[label="",style="dashed", color="red", weight=0];
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60[label="False : wz5",fontsize=16,color="green",shape="box"];62 -> 54[label="",style="dashed", color="red", weight=0];
62[label="FiniteMap.keysFM_LE0 False wz31 wz5",fontsize=16,color="magenta"];61[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False wz340 wz341 wz342 wz343 wz344 (wz340 <= False)",fontsize=16,color="burlywood",shape="triangle"];128[label="wz340/False",fontsize=10,color="white",style="solid",shape="box"];61 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 68[label="",style="solid", color="burlywood", weight=3];
129[label="wz340/True",fontsize=10,color="white",style="solid",shape="box"];61 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 69[label="",style="solid", color="burlywood", weight=3];
63[label="wz6",fontsize=16,color="green",shape="box"];65 -> 54[label="",style="dashed", color="red", weight=0];
65[label="FiniteMap.keysFM_LE0 False wz31 wz6",fontsize=16,color="magenta"];65 -> 70[label="",style="dashed", color="magenta", weight=3];
64[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True wz340 wz341 wz342 wz343 wz344 (wz340 <= True)",fontsize=16,color="burlywood",shape="triangle"];130[label="wz340/False",fontsize=10,color="white",style="solid",shape="box"];64 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 71[label="",style="solid", color="burlywood", weight=3];
131[label="wz340/True",fontsize=10,color="white",style="solid",shape="box"];64 -> 131[label="",style="solid", color="burlywood", weight=9];
131 -> 72[label="",style="solid", color="burlywood", weight=3];
67[label="True : wz7",fontsize=16,color="green",shape="box"];66 -> 58[label="",style="dashed", color="red", weight=0];
66[label="FiniteMap.keysFM_LE0 True wz31 wz7",fontsize=16,color="magenta"];68[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False False wz341 wz342 wz343 wz344 (False <= False)",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3];
69[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False True wz341 wz342 wz343 wz344 (True <= False)",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3];
70[label="wz6",fontsize=16,color="green",shape="box"];71[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True False wz341 wz342 wz343 wz344 (False <= True)",fontsize=16,color="black",shape="box"];71 -> 75[label="",style="solid", color="black", weight=3];
72[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True True wz341 wz342 wz343 wz344 (True <= True)",fontsize=16,color="black",shape="box"];72 -> 76[label="",style="solid", color="black", weight=3];
73[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False False wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];73 -> 77[label="",style="solid", color="black", weight=3];
74[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False True wz341 wz342 wz343 wz344 False",fontsize=16,color="black",shape="box"];74 -> 78[label="",style="solid", color="black", weight=3];
75[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True False wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];75 -> 79[label="",style="solid", color="black", weight=3];
76[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True True wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];76 -> 80[label="",style="solid", color="black", weight=3];
77 -> 26[label="",style="dashed", color="red", weight=0];
77[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 False wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz8 False wz343)) False wz344",fontsize=16,color="magenta"];77 -> 81[label="",style="dashed", color="magenta", weight=3];
77 -> 82[label="",style="dashed", color="magenta", weight=3];
77 -> 83[label="",style="dashed", color="magenta", weight=3];
78[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz8 False True wz341 wz342 wz343 wz344 otherwise",fontsize=16,color="black",shape="box"];78 -> 84[label="",style="solid", color="black", weight=3];
79 -> 28[label="",style="dashed", color="red", weight=0];
79[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 False wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True wz343)) True wz344",fontsize=16,color="magenta"];79 -> 85[label="",style="dashed", color="magenta", weight=3];
79 -> 86[label="",style="dashed", color="magenta", weight=3];
79 -> 87[label="",style="dashed", color="magenta", weight=3];
80 -> 31[label="",style="dashed", color="red", weight=0];
80[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 True wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True wz343)) True wz344",fontsize=16,color="magenta"];80 -> 88[label="",style="dashed", color="magenta", weight=3];
80 -> 89[label="",style="dashed", color="magenta", weight=3];
80 -> 90[label="",style="dashed", color="magenta", weight=3];
81[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz8 False wz343",fontsize=16,color="burlywood",shape="triangle"];132[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];81 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 91[label="",style="solid", color="burlywood", weight=3];
133[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];81 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 92[label="",style="solid", color="burlywood", weight=3];
82[label="wz344",fontsize=16,color="green",shape="box"];83[label="wz341",fontsize=16,color="green",shape="box"];84[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz8 False True wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];84 -> 93[label="",style="solid", color="black", weight=3];
85[label="wz344",fontsize=16,color="green",shape="box"];86[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True wz343",fontsize=16,color="burlywood",shape="triangle"];134[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];86 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 94[label="",style="solid", color="burlywood", weight=3];
135[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];86 -> 135[label="",style="solid", color="burlywood", weight=9];
135 -> 95[label="",style="solid", color="burlywood", weight=3];
87[label="wz341",fontsize=16,color="green",shape="box"];88 -> 86[label="",style="dashed", color="red", weight=0];
88[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True wz343",fontsize=16,color="magenta"];89[label="wz344",fontsize=16,color="green",shape="box"];90[label="wz341",fontsize=16,color="green",shape="box"];91[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz8 False FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];91 -> 96[label="",style="solid", color="black", weight=3];
92[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz8 False (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];92 -> 97[label="",style="solid", color="black", weight=3];
93 -> 81[label="",style="dashed", color="red", weight=0];
93[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz8 False wz343",fontsize=16,color="magenta"];94[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];94 -> 98[label="",style="solid", color="black", weight=3];
95[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz9 True (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3];
96[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz8 False FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
97[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz8 False (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
98[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz9 True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];98 -> 102[label="",style="solid", color="black", weight=3];
99[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz9 True (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];99 -> 103[label="",style="solid", color="black", weight=3];
100[label="wz8",fontsize=16,color="green",shape="box"];101 -> 61[label="",style="dashed", color="red", weight=0];
101[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz8 False wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= False)",fontsize=16,color="magenta"];101 -> 104[label="",style="dashed", color="magenta", weight=3];
101 -> 105[label="",style="dashed", color="magenta", weight=3];
101 -> 106[label="",style="dashed", color="magenta", weight=3];
101 -> 107[label="",style="dashed", color="magenta", weight=3];
101 -> 108[label="",style="dashed", color="magenta", weight=3];
102[label="wz9",fontsize=16,color="green",shape="box"];103 -> 64[label="",style="dashed", color="red", weight=0];
103[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz9 True wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= True)",fontsize=16,color="magenta"];103 -> 109[label="",style="dashed", color="magenta", weight=3];
103 -> 110[label="",style="dashed", color="magenta", weight=3];
103 -> 111[label="",style="dashed", color="magenta", weight=3];
103 -> 112[label="",style="dashed", color="magenta", weight=3];
103 -> 113[label="",style="dashed", color="magenta", weight=3];
104[label="wz3430",fontsize=16,color="green",shape="box"];105[label="wz3432",fontsize=16,color="green",shape="box"];106[label="wz3434",fontsize=16,color="green",shape="box"];107[label="wz3433",fontsize=16,color="green",shape="box"];108[label="wz3431",fontsize=16,color="green",shape="box"];109[label="wz3430",fontsize=16,color="green",shape="box"];110[label="wz3432",fontsize=16,color="green",shape="box"];111[label="wz3434",fontsize=16,color="green",shape="box"];112[label="wz3433",fontsize=16,color="green",shape="box"];113[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_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz9, wz343, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE0(wz9, EmptyFM, h) -> wz9
   new_keysFM_LE00(wz31, wz7, h) -> :(True, wz7)
   new_foldFM_LE4(wz31, wz7, EmptyFM, h) -> new_keysFM_LE00(wz31, wz7, h)
   new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(False, wz5)
   new_foldFM_LE10(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE5(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE10(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE10(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, False, x1, x2, x3, x4, x5)
   new_foldFM_LE4(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE4(x0, x1, EmptyFM, x2)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)

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

(10) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(False, wz6), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(False, wz6), wz340, wz341, wz342, wz343, wz344, h))


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz9, wz343, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(False, wz6), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE0(wz9, EmptyFM, h) -> wz9
   new_keysFM_LE00(wz31, wz7, h) -> :(True, wz7)
   new_foldFM_LE4(wz31, wz7, EmptyFM, h) -> new_keysFM_LE00(wz31, wz7, h)
   new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(False, wz5)
   new_foldFM_LE10(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE5(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE10(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE10(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, False, x1, x2, x3, x4, x5)
   new_foldFM_LE4(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE4(x0, x1, EmptyFM, x2)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)

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

(12) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(True, wz7), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(True, wz7), wz340, wz341, wz342, wz343, wz344, h))


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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz9, wz343, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(False, wz6), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(True, wz7), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE0(wz9, EmptyFM, h) -> wz9
   new_keysFM_LE00(wz31, wz7, h) -> :(True, wz7)
   new_foldFM_LE4(wz31, wz7, EmptyFM, h) -> new_keysFM_LE00(wz31, wz7, h)
   new_foldFM_LE4(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(False, wz5)
   new_foldFM_LE10(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE5(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE10(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE10(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, False, x1, x2, x3, x4, x5)
   new_foldFM_LE4(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE4(x0, x1, EmptyFM, x2)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)

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

(14) 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_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz9, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


*new_foldFM_LE1(wz9, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz9, 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_LE3(wz9, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz9, wz3430, wz3431, wz3432, wz3433, wz3434, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7


*new_foldFM_LE(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(False, wz6), 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_LE2(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(True, wz7), 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(wz9, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE1(wz9, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz9, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


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

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)
   new_foldFM_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)
   new_foldFM_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)
   new_foldFM_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)

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

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

(18)
Complex Obligation (AND)

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

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

   new_foldFM_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)
   new_foldFM_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)

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

(20) 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_LE13(False, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


*new_foldFM_LE13(False, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(False, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(21)
YES

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)
   new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)

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

(23) 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_LE13(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


*new_foldFM_LE13(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE13(True, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(24)
YES

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE11(wz8, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz8, wz343, h)
   new_foldFM_LE8(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE11(wz8, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
   new_foldFM_LE11(wz8, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)

The TRS R consists of the following rules:

   new_foldFM_LE9(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE12(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE7(wz8, EmptyFM, h) -> wz8
   new_foldFM_LE7(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE12(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE9(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE12(wz8, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz8, wz343, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(False, wz5)
   new_foldFM_LE12(wz8, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE9(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE9(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE12(x0, False, x1, x2, x3, x4, x5)
   new_foldFM_LE9(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE12(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_LE7(x0, EmptyFM, x1)
   new_foldFM_LE7(x0, Branch(x1, x2, x3, x4, x5), x6)

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

(26) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(False, wz5), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(False, wz5), wz340, wz341, wz342, wz343, wz344, h))


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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE11(wz8, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz8, wz343, h)
   new_foldFM_LE8(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz8, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
   new_foldFM_LE11(wz8, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(False, wz5), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE9(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE12(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE7(wz8, EmptyFM, h) -> wz8
   new_foldFM_LE7(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE12(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE9(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE12(wz8, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz8, wz343, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(False, wz5)
   new_foldFM_LE12(wz8, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE9(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)

The set Q consists of the following terms:

   new_foldFM_LE9(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE12(x0, False, x1, x2, x3, x4, x5)
   new_foldFM_LE9(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE12(x0, True, x1, x2, x3, x4, x5)
   new_foldFM_LE7(x0, EmptyFM, x1)
   new_foldFM_LE7(x0, Branch(x1, x2, x3, x4, x5), x6)

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

(28) 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_LE8(wz8, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz8, wz3430, wz3431, wz3432, wz3433, wz3434, h)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7


*new_foldFM_LE11(wz8, False, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz8, 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_LE6(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(False, 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_LE11(wz8, True, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz8, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


*new_foldFM_LE11(wz8, False, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE7(wz8, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


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

(29)
YES