/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) TransformationProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) TransformationProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) TransformationProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) TransformationProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) DependencyGraphProof [EQUIVALENT, 0 ms]
        (25) AND
            (26) QDP
                (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (28) YES
            (29) QDP
                (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (31) YES
            (32) QDP
                (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (34) YES
    (35) QDP
        (36) TransformationProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) TransformationProof [EQUIVALENT, 0 ms]
        (39) QDP
        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (41) 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) ;

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 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 a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;

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

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

keysFM_LE0 key elt rest = key : rest;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

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

(4)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

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 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_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
"
"foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;
"
"foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);
"
"foldFM_LE3 k z fr EmptyFM = z;
foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
"

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

(6)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap 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 = foldFM_LE3 k z fr EmptyFM;
foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);

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

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

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

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

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

keysFM_LE0 key elt rest = key : rest;

}
module Maybe where {
import qualified FiniteMap;
import qualified Main;
import qualified Prelude;
}
module Main where {
import qualified FiniteMap;
import qualified Maybe;
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
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216 -> 6[label="",style="solid", color="burlywood", weight=3];
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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];
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218 -> 12[label="",style="solid", color="burlywood", weight=3];
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219 -> 13[label="",style="solid", color="burlywood", weight=3];
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220 -> 14[label="",style="solid", color="burlywood", weight=3];
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153 -> 44[label="",style="dashed", color="red", weight=0];
153[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 LT wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ wz343)) EQ wz344",fontsize=16,color="magenta"];153 -> 164[label="",style="dashed", color="magenta", weight=3];
153 -> 165[label="",style="dashed", color="magenta", weight=3];
153 -> 166[label="",style="dashed", color="magenta", weight=3];
154 -> 49[label="",style="dashed", color="red", weight=0];
154[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 EQ wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ wz343)) EQ wz344",fontsize=16,color="magenta"];154 -> 167[label="",style="dashed", color="magenta", weight=3];
154 -> 168[label="",style="dashed", color="magenta", weight=3];
154 -> 169[label="",style="dashed", color="magenta", weight=3];
155[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz12 EQ GT wz341 wz342 wz343 wz344 otherwise",fontsize=16,color="black",shape="box"];155 -> 170[label="",style="solid", color="black", weight=3];
156 -> 46[label="",style="dashed", color="red", weight=0];
156[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 LT wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343)) GT wz344",fontsize=16,color="magenta"];156 -> 171[label="",style="dashed", color="magenta", weight=3];
156 -> 172[label="",style="dashed", color="magenta", weight=3];
156 -> 173[label="",style="dashed", color="magenta", weight=3];
157 -> 51[label="",style="dashed", color="red", weight=0];
157[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 EQ wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343)) GT wz344",fontsize=16,color="magenta"];157 -> 174[label="",style="dashed", color="magenta", weight=3];
157 -> 175[label="",style="dashed", color="magenta", weight=3];
157 -> 176[label="",style="dashed", color="magenta", weight=3];
158 -> 55[label="",style="dashed", color="red", weight=0];
158[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 (FiniteMap.keysFM_LE0 GT wz341 (FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343)) GT wz344",fontsize=16,color="magenta"];158 -> 177[label="",style="dashed", color="magenta", weight=3];
158 -> 178[label="",style="dashed", color="magenta", weight=3];
158 -> 179[label="",style="dashed", color="magenta", weight=3];
159[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz11 LT wz343",fontsize=16,color="burlywood",shape="triangle"];251[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];159 -> 251[label="",style="solid", color="burlywood", weight=9];
251 -> 180[label="",style="solid", color="burlywood", weight=3];
252[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];159 -> 252[label="",style="solid", color="burlywood", weight=9];
252 -> 181[label="",style="solid", color="burlywood", weight=3];
160[label="wz344",fontsize=16,color="green",shape="box"];161[label="wz341",fontsize=16,color="green",shape="box"];162[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz11 LT EQ wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];162 -> 182[label="",style="solid", color="black", weight=3];
163[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz11 LT GT wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];163 -> 183[label="",style="solid", color="black", weight=3];
164[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ wz343",fontsize=16,color="burlywood",shape="triangle"];253[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];164 -> 253[label="",style="solid", color="burlywood", weight=9];
253 -> 184[label="",style="solid", color="burlywood", weight=3];
254[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];164 -> 254[label="",style="solid", color="burlywood", weight=9];
254 -> 185[label="",style="solid", color="burlywood", weight=3];
165[label="wz344",fontsize=16,color="green",shape="box"];166[label="wz341",fontsize=16,color="green",shape="box"];167[label="wz344",fontsize=16,color="green",shape="box"];168[label="wz341",fontsize=16,color="green",shape="box"];169 -> 164[label="",style="dashed", color="red", weight=0];
169[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ wz343",fontsize=16,color="magenta"];170[label="FiniteMap.foldFM_LE0 FiniteMap.keysFM_LE0 wz12 EQ GT wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];170 -> 186[label="",style="solid", color="black", weight=3];
171[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343",fontsize=16,color="burlywood",shape="triangle"];255[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];171 -> 255[label="",style="solid", color="burlywood", weight=9];
255 -> 187[label="",style="solid", color="burlywood", weight=3];
256[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];171 -> 256[label="",style="solid", color="burlywood", weight=9];
256 -> 188[label="",style="solid", color="burlywood", weight=3];
172[label="wz344",fontsize=16,color="green",shape="box"];173[label="wz341",fontsize=16,color="green",shape="box"];174[label="wz344",fontsize=16,color="green",shape="box"];175 -> 171[label="",style="dashed", color="red", weight=0];
175[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343",fontsize=16,color="magenta"];176[label="wz341",fontsize=16,color="green",shape="box"];177 -> 171[label="",style="dashed", color="red", weight=0];
177[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT wz343",fontsize=16,color="magenta"];178[label="wz344",fontsize=16,color="green",shape="box"];179[label="wz341",fontsize=16,color="green",shape="box"];180[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz11 LT FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];180 -> 189[label="",style="solid", color="black", weight=3];
181[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz11 LT (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];181 -> 190[label="",style="solid", color="black", weight=3];
182 -> 159[label="",style="dashed", color="red", weight=0];
182[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz11 LT wz343",fontsize=16,color="magenta"];183 -> 159[label="",style="dashed", color="red", weight=0];
183[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz11 LT wz343",fontsize=16,color="magenta"];184[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];184 -> 191[label="",style="solid", color="black", weight=3];
185[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];185 -> 192[label="",style="solid", color="black", weight=3];
186 -> 164[label="",style="dashed", color="red", weight=0];
186[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz12 EQ wz343",fontsize=16,color="magenta"];187[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];187 -> 193[label="",style="solid", color="black", weight=3];
188[label="FiniteMap.foldFM_LE FiniteMap.keysFM_LE0 wz13 GT (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];188 -> 194[label="",style="solid", color="black", weight=3];
189[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz11 LT FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];189 -> 195[label="",style="solid", color="black", weight=3];
190[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz11 LT (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];190 -> 196[label="",style="solid", color="black", weight=3];
191[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz12 EQ FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];191 -> 197[label="",style="solid", color="black", weight=3];
192[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz12 EQ (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];192 -> 198[label="",style="solid", color="black", weight=3];
193[label="FiniteMap.foldFM_LE3 FiniteMap.keysFM_LE0 wz13 GT FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];193 -> 199[label="",style="solid", color="black", weight=3];
194[label="FiniteMap.foldFM_LE2 FiniteMap.keysFM_LE0 wz13 GT (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];194 -> 200[label="",style="solid", color="black", weight=3];
195[label="wz11",fontsize=16,color="green",shape="box"];196 -> 115[label="",style="dashed", color="red", weight=0];
196[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz11 LT wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= LT)",fontsize=16,color="magenta"];196 -> 201[label="",style="dashed", color="magenta", weight=3];
196 -> 202[label="",style="dashed", color="magenta", weight=3];
196 -> 203[label="",style="dashed", color="magenta", weight=3];
196 -> 204[label="",style="dashed", color="magenta", weight=3];
196 -> 205[label="",style="dashed", color="magenta", weight=3];
197[label="wz12",fontsize=16,color="green",shape="box"];198 -> 118[label="",style="dashed", color="red", weight=0];
198[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz12 EQ wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= EQ)",fontsize=16,color="magenta"];198 -> 206[label="",style="dashed", color="magenta", weight=3];
198 -> 207[label="",style="dashed", color="magenta", weight=3];
198 -> 208[label="",style="dashed", color="magenta", weight=3];
198 -> 209[label="",style="dashed", color="magenta", weight=3];
198 -> 210[label="",style="dashed", color="magenta", weight=3];
199[label="wz13",fontsize=16,color="green",shape="box"];200 -> 122[label="",style="dashed", color="red", weight=0];
200[label="FiniteMap.foldFM_LE1 FiniteMap.keysFM_LE0 wz13 GT wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= GT)",fontsize=16,color="magenta"];200 -> 211[label="",style="dashed", color="magenta", weight=3];
200 -> 212[label="",style="dashed", color="magenta", weight=3];
200 -> 213[label="",style="dashed", color="magenta", weight=3];
200 -> 214[label="",style="dashed", color="magenta", weight=3];
200 -> 215[label="",style="dashed", color="magenta", weight=3];
201[label="wz3432",fontsize=16,color="green",shape="box"];202[label="wz3434",fontsize=16,color="green",shape="box"];203[label="wz3431",fontsize=16,color="green",shape="box"];204[label="wz3430",fontsize=16,color="green",shape="box"];205[label="wz3433",fontsize=16,color="green",shape="box"];206[label="wz3432",fontsize=16,color="green",shape="box"];207[label="wz3434",fontsize=16,color="green",shape="box"];208[label="wz3431",fontsize=16,color="green",shape="box"];209[label="wz3430",fontsize=16,color="green",shape="box"];210[label="wz3433",fontsize=16,color="green",shape="box"];211[label="wz3432",fontsize=16,color="green",shape="box"];212[label="wz3434",fontsize=16,color="green",shape="box"];213[label="wz3431",fontsize=16,color="green",shape="box"];214[label="wz3430",fontsize=16,color="green",shape="box"];215[label="wz3433",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_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE17(wz11, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)
   new_foldFM_LE17(wz11, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE18(wz341, new_foldFM_LE19(wz11, wz343, h), wz344, h)
   new_foldFM_LE20(wz11, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE17(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE17(wz11, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE17(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE17(wz11, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)

The TRS R consists of the following rules:

   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE110(wz11, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE21(wz341, new_foldFM_LE19(wz11, wz343, h), wz344, h)
   new_foldFM_LE19(wz11, EmptyFM, h) -> wz11
   new_foldFM_LE110(wz11, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE19(wz11, wz343, h)
   new_foldFM_LE19(wz11, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE110(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE21(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE21(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE110(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE110(wz11, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE19(wz11, wz343, h)

The set Q consists of the following terms:

   new_foldFM_LE19(x0, EmptyFM, x1)
   new_foldFM_LE110(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE110(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE110(x0, GT, x1, x2, x3, x4, x5)
   new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE19(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE21(x0, x1, EmptyFM, x2)

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

(10) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(:(LT, wz5), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(:(LT, wz5), wz340, wz341, wz342, wz343, wz344, h))


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

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

   new_foldFM_LE17(wz11, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)
   new_foldFM_LE17(wz11, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE18(wz341, new_foldFM_LE19(wz11, wz343, h), wz344, h)
   new_foldFM_LE20(wz11, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE17(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE17(wz11, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE17(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE17(wz11, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)
   new_foldFM_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(:(LT, wz5), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE110(wz11, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE21(wz341, new_foldFM_LE19(wz11, wz343, h), wz344, h)
   new_foldFM_LE19(wz11, EmptyFM, h) -> wz11
   new_foldFM_LE110(wz11, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE19(wz11, wz343, h)
   new_foldFM_LE19(wz11, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE110(wz11, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE21(wz31, wz5, EmptyFM, h) -> new_keysFM_LE0(wz31, wz5, h)
   new_foldFM_LE21(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE110(new_keysFM_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE110(wz11, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE19(wz11, wz343, h)

The set Q consists of the following terms:

   new_foldFM_LE19(x0, EmptyFM, x1)
   new_foldFM_LE110(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE110(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE110(x0, GT, x1, x2, x3, x4, x5)
   new_foldFM_LE21(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE19(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE21(x0, x1, EmptyFM, x2)

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

(12) 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_LE20(wz11, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE17(wz11, 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_LE18(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE17(:(LT, 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_LE17(wz11, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE17(wz11, 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_LE17(wz11, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE18(wz341, new_foldFM_LE19(wz11, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE17(wz11, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


*new_foldFM_LE17(wz11, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE20(wz11, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


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

(13)
YES

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

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

   new_foldFM_LE1(wz13, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE3(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE10(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz9, EmptyFM, h) -> new_keysFM_LE00(wz31, wz9, h)
   new_foldFM_LE7(wz31, wz7, EmptyFM, h) -> new_keysFM_LE0(wz31, wz7, h)
   new_keysFM_LE01(wz31, wz10, h) -> :(GT, wz10)
   new_foldFM_LE10(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE7(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE5(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz13, EmptyFM, h) -> wz13
   new_foldFM_LE10(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE0(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz10, EmptyFM, h) -> new_keysFM_LE01(wz31, wz10, h)

The set Q consists of the following terms:

   new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE7(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE10(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE6(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE10(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)
   new_keysFM_LE01(x0, x1, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_foldFM_LE6(x0, x1, EmptyFM, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, GT, x1, x2, x3, x4, x5)

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

(15) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

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


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

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

   new_foldFM_LE1(wz13, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE3(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(LT, wz7), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE10(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz9, EmptyFM, h) -> new_keysFM_LE00(wz31, wz9, h)
   new_foldFM_LE7(wz31, wz7, EmptyFM, h) -> new_keysFM_LE0(wz31, wz7, h)
   new_keysFM_LE01(wz31, wz10, h) -> :(GT, wz10)
   new_foldFM_LE10(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE7(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE5(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz13, EmptyFM, h) -> wz13
   new_foldFM_LE10(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE0(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz10, EmptyFM, h) -> new_keysFM_LE01(wz31, wz10, h)

The set Q consists of the following terms:

   new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE7(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE10(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE6(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE10(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)
   new_keysFM_LE01(x0, x1, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_foldFM_LE6(x0, x1, EmptyFM, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, GT, x1, x2, x3, x4, x5)

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

(17) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(GT, wz10), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(GT, wz10), wz340, wz341, wz342, wz343, wz344, h))


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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE1(wz13, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE3(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(LT, wz7), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(GT, wz10), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE10(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz9, EmptyFM, h) -> new_keysFM_LE00(wz31, wz9, h)
   new_foldFM_LE7(wz31, wz7, EmptyFM, h) -> new_keysFM_LE0(wz31, wz7, h)
   new_keysFM_LE01(wz31, wz10, h) -> :(GT, wz10)
   new_foldFM_LE10(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE7(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE5(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz13, EmptyFM, h) -> wz13
   new_foldFM_LE10(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE0(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz10, EmptyFM, h) -> new_keysFM_LE01(wz31, wz10, h)

The set Q consists of the following terms:

   new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE7(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE10(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE6(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE10(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)
   new_keysFM_LE01(x0, x1, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_foldFM_LE6(x0, x1, EmptyFM, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, GT, x1, x2, x3, x4, x5)

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

(19) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(EQ, wz9), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(EQ, wz9), wz340, wz341, wz342, wz343, wz344, h))


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE1(wz13, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
   new_foldFM_LE3(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE1(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(LT, wz7), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(GT, wz10), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(EQ, wz9), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE6(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE01(wz31, wz10, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE10(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE6(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE5(wz31, wz9, EmptyFM, h) -> new_keysFM_LE00(wz31, wz9, h)
   new_foldFM_LE7(wz31, wz7, EmptyFM, h) -> new_keysFM_LE0(wz31, wz7, h)
   new_keysFM_LE01(wz31, wz10, h) -> :(GT, wz10)
   new_foldFM_LE10(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE5(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE7(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE0(wz31, wz7, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE5(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_keysFM_LE00(wz31, wz9, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE0(wz13, EmptyFM, h) -> wz13
   new_foldFM_LE10(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE7(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
   new_foldFM_LE0(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz13, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE6(wz31, wz10, EmptyFM, h) -> new_keysFM_LE01(wz31, wz10, h)

The set Q consists of the following terms:

   new_foldFM_LE7(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE7(x0, x1, EmptyFM, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE10(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE5(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE6(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE10(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE5(x0, x1, EmptyFM, x2)
   new_keysFM_LE01(x0, x1, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_foldFM_LE6(x0, x1, EmptyFM, x2)
   new_foldFM_LE0(x0, EmptyFM, x1)
   new_foldFM_LE10(x0, GT, x1, x2, x3, x4, x5)

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

(21) 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(wz13, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz13, 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(wz13, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE1(wz13, 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_LE2(wz31, wz9, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(EQ, wz9), 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_LE4(wz31, wz10, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(GT, wz10), 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_LE(wz31, wz7, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(LT, 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(wz13, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE4(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz13, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE1(wz13, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


*new_foldFM_LE1(wz13, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE3(wz13, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


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

(22)
YES

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE22(EQ, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
   new_foldFM_LE22(EQ, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
   new_foldFM_LE22(LT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
   new_foldFM_LE22(LT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
   new_foldFM_LE22(GT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
   new_foldFM_LE22(GT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
   new_foldFM_LE22(EQ, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
   new_foldFM_LE22(LT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
   new_foldFM_LE22(GT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)

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

(24) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
----------------------------------------

(25)
Complex Obligation (AND)

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

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE22(GT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
   new_foldFM_LE22(GT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
   new_foldFM_LE22(GT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)

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

(27) 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_LE22(GT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(GT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(GT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(GT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(28)
YES

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

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE22(LT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
   new_foldFM_LE22(LT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
   new_foldFM_LE22(LT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)

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

(30) 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_LE22(LT, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(LT, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(LT, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(LT, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(31)
YES

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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE22(EQ, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
   new_foldFM_LE22(EQ, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
   new_foldFM_LE22(EQ, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)

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

(33) 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_LE22(EQ, Branch(LT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(EQ, Branch(GT, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_foldFM_LE22(EQ, Branch(EQ, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE22(EQ, wz33, h)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3


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

(34)
YES

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

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE12(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE11(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE13(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE16(wz31, wz8, EmptyFM, h) -> new_keysFM_LE00(wz31, wz8, h)
   new_foldFM_LE15(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_foldFM_LE14(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE9(wz12, wz343, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE9(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE14(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE15(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE14(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE16(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE16(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE14(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE15(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE9(wz12, EmptyFM, h) -> wz12

The set Q consists of the following terms:

   new_foldFM_LE9(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE16(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE14(x0, GT, x1, x2, x3, x4, x5)
   new_foldFM_LE14(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE16(x0, x1, EmptyFM, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE14(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE15(x0, x1, EmptyFM, x2)
   new_foldFM_LE9(x0, EmptyFM, x1)
   new_foldFM_LE15(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

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

(36) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(LT, wz6), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(LT, wz6), wz340, wz341, wz342, wz343, wz344, h))


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

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE12(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE11(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE13(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(LT, wz6), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE16(wz31, wz8, EmptyFM, h) -> new_keysFM_LE00(wz31, wz8, h)
   new_foldFM_LE15(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_foldFM_LE14(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE9(wz12, wz343, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE9(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE14(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE15(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE14(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE16(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE16(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE14(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE15(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE9(wz12, EmptyFM, h) -> wz12

The set Q consists of the following terms:

   new_foldFM_LE9(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE16(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE14(x0, GT, x1, x2, x3, x4, x5)
   new_foldFM_LE14(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE16(x0, x1, EmptyFM, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE14(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE15(x0, x1, EmptyFM, x2)
   new_foldFM_LE9(x0, EmptyFM, x1)
   new_foldFM_LE15(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

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

(38) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(EQ, wz8), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(EQ, wz8), wz340, wz341, wz342, wz343, wz344, h))


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

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE12(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE11(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
   new_foldFM_LE13(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE11(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(LT, wz6), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(EQ, wz8), wz340, wz341, wz342, wz343, wz344, h)

The TRS R consists of the following rules:

   new_foldFM_LE16(wz31, wz8, EmptyFM, h) -> new_keysFM_LE00(wz31, wz8, h)
   new_foldFM_LE15(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE0(wz31, wz6, h), wz340, wz341, wz342, wz343, wz344, h)
   new_keysFM_LE00(wz31, wz8, h) -> :(EQ, wz8)
   new_foldFM_LE14(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE9(wz12, wz343, h)
   new_keysFM_LE0(wz31, wz5, h) -> :(LT, wz5)
   new_foldFM_LE9(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE14(wz12, wz3430, wz3431, wz3432, wz3433, wz3434, h)
   new_foldFM_LE15(wz31, wz6, EmptyFM, h) -> new_keysFM_LE0(wz31, wz6, h)
   new_foldFM_LE14(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE16(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE16(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE14(new_keysFM_LE00(wz31, wz8, h), wz340, wz341, wz342, wz343, wz344, h)
   new_foldFM_LE14(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE15(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
   new_foldFM_LE9(wz12, EmptyFM, h) -> wz12

The set Q consists of the following terms:

   new_foldFM_LE9(x0, Branch(x1, x2, x3, x4, x5), x6)
   new_foldFM_LE16(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
   new_foldFM_LE14(x0, GT, x1, x2, x3, x4, x5)
   new_foldFM_LE14(x0, EQ, x1, x2, x3, x4, x5)
   new_foldFM_LE16(x0, x1, EmptyFM, x2)
   new_keysFM_LE00(x0, x1, x2)
   new_keysFM_LE0(x0, x1, x2)
   new_foldFM_LE14(x0, LT, x1, x2, x3, x4, x5)
   new_foldFM_LE15(x0, x1, EmptyFM, x2)
   new_foldFM_LE9(x0, EmptyFM, x1)
   new_foldFM_LE15(x0, x1, Branch(x2, x3, x4, x5, x6), x7)

We have to consider all minimal (P,Q,R)-chains.
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(40) 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(wz12, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE11(wz12, 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_LE12(wz31, wz8, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(EQ, wz8), 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(wz12, LT, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE11(wz12, 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_LE8(wz31, wz6, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE11(:(LT, 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_LE11(wz12, LT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE8(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE12(wz341, new_foldFM_LE9(wz12, wz343, h), wz344, h)
The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4


*new_foldFM_LE11(wz12, EQ, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


*new_foldFM_LE11(wz12, GT, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE13(wz12, wz343, h)
The graph contains the following edges 1 >= 1, 5 >= 2, 7 >= 3


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(41)
YES