/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES


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

(0)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
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 _ 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;

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

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

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

(2)
Obligation:
mainModule Main 
module FiniteMap where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;

instance (Eq a, Eq b) => Eq FiniteMap a b where { 
}
foldFM_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;

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

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"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;
"

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

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

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
foldFM_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;

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

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="FiniteMap.foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];345[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 345[label="",style="solid", color="burlywood", weight=9];
345 -> 7[label="",style="solid", color="burlywood", weight=3];
346[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 346[label="",style="solid", color="burlywood", weight=9];
346 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="FiniteMap.foldFM_LE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.foldFM_LE wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.foldFM_LE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.foldFM_LE2 wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="wz4",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 <= wz5)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= GT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == GT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
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347 -> 16[label="",style="solid", color="burlywood", weight=3];
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348 -> 17[label="",style="solid", color="burlywood", weight=3];
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18[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char wz50) (Char wz600) wz61 wz62 wz63 wz64 (not (primCmpNat wz600 wz50 == GT))",fontsize=16,color="burlywood",shape="box"];349[label="wz600/Succ wz6000",fontsize=10,color="white",style="solid",shape="box"];18 -> 349[label="",style="solid", color="burlywood", weight=9];
349 -> 19[label="",style="solid", color="burlywood", weight=3];
350[label="wz600/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 350[label="",style="solid", color="burlywood", weight=9];
350 -> 20[label="",style="solid", color="burlywood", weight=3];
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351 -> 21[label="",style="solid", color="burlywood", weight=3];
352[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 352[label="",style="solid", color="burlywood", weight=9];
352 -> 22[label="",style="solid", color="burlywood", weight=3];
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353 -> 23[label="",style="solid", color="burlywood", weight=3];
354[label="wz50/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 354[label="",style="solid", color="burlywood", weight=9];
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355 -> 309[label="",style="solid", color="burlywood", weight=3];
356[label="wz42/Zero",fontsize=10,color="white",style="solid",shape="box"];238 -> 356[label="",style="solid", color="burlywood", weight=9];
356 -> 310[label="",style="solid", color="burlywood", weight=3];
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358[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];309 -> 358[label="",style="solid", color="burlywood", weight=9];
358 -> 312[label="",style="solid", color="burlywood", weight=3];
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359 -> 313[label="",style="solid", color="burlywood", weight=3];
360[label="wz43/Zero",fontsize=10,color="white",style="solid",shape="box"];310 -> 360[label="",style="solid", color="burlywood", weight=9];
360 -> 314[label="",style="solid", color="burlywood", weight=3];
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39[label="FiniteMap.foldFM_LE1 wz3 wz4 (Char (Succ wz500)) (Char Zero) wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
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311[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (primCmpNat (Succ wz420) (Succ wz430) == GT))",fontsize=16,color="black",shape="box"];311 -> 315[label="",style="solid", color="black", weight=3];
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46 -> 55[label="",style="dashed", color="magenta", weight=3];
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315 -> 320[label="",style="dashed", color="magenta", weight=3];
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317[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (LT == GT))",fontsize=16,color="black",shape="box"];317 -> 322[label="",style="solid", color="black", weight=3];
318[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not (EQ == GT))",fontsize=16,color="black",shape="box"];318 -> 323[label="",style="solid", color="black", weight=3];
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54[label="wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63)",fontsize=16,color="green",shape="box"];54 -> 68[label="",style="dashed", color="green", weight=3];
54 -> 69[label="",style="dashed", color="green", weight=3];
54 -> 70[label="",style="dashed", color="green", weight=3];
55[label="wz64",fontsize=16,color="green",shape="box"];56[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];57[label="wz3 (Char Zero) wz61 (FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63)",fontsize=16,color="green",shape="box"];57 -> 71[label="",style="dashed", color="green", weight=3];
57 -> 72[label="",style="dashed", color="green", weight=3];
57 -> 73[label="",style="dashed", color="green", weight=3];
58[label="wz64",fontsize=16,color="green",shape="box"];59[label="Char Zero",fontsize=16,color="green",shape="box"];319[label="wz430",fontsize=16,color="green",shape="box"];320[label="wz420",fontsize=16,color="green",shape="box"];321[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not True)",fontsize=16,color="black",shape="box"];321 -> 324[label="",style="solid", color="black", weight=3];
322[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="black",shape="triangle"];322 -> 325[label="",style="solid", color="black", weight=3];
323 -> 322[label="",style="dashed", color="red", weight=0];
323[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 (not False)",fontsize=16,color="magenta"];67 -> 6[label="",style="dashed", color="red", weight=0];
67[label="FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];67 -> 81[label="",style="dashed", color="magenta", weight=3];
67 -> 82[label="",style="dashed", color="magenta", weight=3];
68[label="Char Zero",fontsize=16,color="green",shape="box"];69[label="wz61",fontsize=16,color="green",shape="box"];70 -> 6[label="",style="dashed", color="red", weight=0];
70[label="FiniteMap.foldFM_LE wz3 wz4 (Char (Succ wz500)) wz63",fontsize=16,color="magenta"];70 -> 83[label="",style="dashed", color="magenta", weight=3];
70 -> 84[label="",style="dashed", color="magenta", weight=3];
71[label="Char Zero",fontsize=16,color="green",shape="box"];72[label="wz61",fontsize=16,color="green",shape="box"];73 -> 6[label="",style="dashed", color="red", weight=0];
73[label="FiniteMap.foldFM_LE wz3 wz4 (Char Zero) wz63",fontsize=16,color="magenta"];73 -> 85[label="",style="dashed", color="magenta", weight=3];
73 -> 86[label="",style="dashed", color="magenta", weight=3];
324[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 False",fontsize=16,color="black",shape="box"];324 -> 326[label="",style="solid", color="black", weight=3];
325[label="FiniteMap.foldFM_LE1 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];325 -> 327[label="",style="solid", color="black", weight=3];
81[label="wz63",fontsize=16,color="green",shape="box"];82[label="Char Zero",fontsize=16,color="green",shape="box"];83[label="wz63",fontsize=16,color="green",shape="box"];84[label="Char (Succ wz500)",fontsize=16,color="green",shape="box"];85[label="wz63",fontsize=16,color="green",shape="box"];86[label="Char Zero",fontsize=16,color="green",shape="box"];326[label="FiniteMap.foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 otherwise",fontsize=16,color="black",shape="box"];326 -> 328[label="",style="solid", color="black", weight=3];
327 -> 6[label="",style="dashed", color="red", weight=0];
327[label="FiniteMap.foldFM_LE wz34 (wz34 (Char (Succ wz37)) wz38 (FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)) (Char (Succ wz36)) wz41",fontsize=16,color="magenta"];327 -> 329[label="",style="dashed", color="magenta", weight=3];
327 -> 330[label="",style="dashed", color="magenta", weight=3];
327 -> 331[label="",style="dashed", color="magenta", weight=3];
327 -> 332[label="",style="dashed", color="magenta", weight=3];
328[label="FiniteMap.foldFM_LE0 wz34 wz35 (Char (Succ wz36)) (Char (Succ wz37)) wz38 wz39 wz40 wz41 True",fontsize=16,color="black",shape="box"];328 -> 333[label="",style="solid", color="black", weight=3];
329[label="wz34 (Char (Succ wz37)) wz38 (FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40)",fontsize=16,color="green",shape="box"];329 -> 334[label="",style="dashed", color="green", weight=3];
329 -> 335[label="",style="dashed", color="green", weight=3];
329 -> 336[label="",style="dashed", color="green", weight=3];
330[label="wz41",fontsize=16,color="green",shape="box"];331[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];332[label="wz34",fontsize=16,color="green",shape="box"];333 -> 6[label="",style="dashed", color="red", weight=0];
333[label="FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];333 -> 337[label="",style="dashed", color="magenta", weight=3];
333 -> 338[label="",style="dashed", color="magenta", weight=3];
333 -> 339[label="",style="dashed", color="magenta", weight=3];
333 -> 340[label="",style="dashed", color="magenta", weight=3];
334[label="Char (Succ wz37)",fontsize=16,color="green",shape="box"];335[label="wz38",fontsize=16,color="green",shape="box"];336 -> 6[label="",style="dashed", color="red", weight=0];
336[label="FiniteMap.foldFM_LE wz34 wz35 (Char (Succ wz36)) wz40",fontsize=16,color="magenta"];336 -> 341[label="",style="dashed", color="magenta", weight=3];
336 -> 342[label="",style="dashed", color="magenta", weight=3];
336 -> 343[label="",style="dashed", color="magenta", weight=3];
336 -> 344[label="",style="dashed", color="magenta", weight=3];
337[label="wz35",fontsize=16,color="green",shape="box"];338[label="wz40",fontsize=16,color="green",shape="box"];339[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];340[label="wz34",fontsize=16,color="green",shape="box"];341[label="wz35",fontsize=16,color="green",shape="box"];342[label="wz40",fontsize=16,color="green",shape="box"];343[label="Char (Succ wz36)",fontsize=16,color="green",shape="box"];344[label="wz34",fontsize=16,color="green",shape="box"];}

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

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

   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
   new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)
   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
   new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)

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

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

(8)
Complex Obligation (AND)

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

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

   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
   new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
   new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
   new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE1(wz3, wz500, wz6000, wz61, wz62, wz63, wz64, wz6000, wz500, bb, bc)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 3 > 7, 3 > 8, 2 > 9, 4 >= 10, 5 >= 11


*new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Succ(wz430), h, ba) -> new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, wz420, wz430, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 > 8, 9 > 9, 10 >= 10, 11 >= 11


*new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Zero, h, ba) -> new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8, 11 >= 9


*new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, Char(Succ(wz500)), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Succ(wz500)), wz64, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Succ(wz420), Zero, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5


*new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 10 >= 4, 11 >= 5


*new_foldFM_LE1(wz34, wz36, wz37, wz38, wz39, wz40, wz41, Zero, Succ(wz430), h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 10 >= 4, 11 >= 5


*new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz41, h, ba)
The graph contains the following edges 1 >= 1, 7 >= 3, 8 >= 4, 9 >= 5


*new_foldFM_LE10(wz34, wz36, wz37, wz38, wz39, wz40, wz41, h, ba) -> new_foldFM_LE(wz34, Char(Succ(wz36)), wz40, h, ba)
The graph contains the following edges 1 >= 1, 6 >= 3, 8 >= 4, 9 >= 5


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

(11)
YES

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

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

   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
   new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)

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

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

From the DPs we obtained the following set of size-change graphs:
*new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, Char(Zero), Branch(Char(Zero), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz64, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5


*new_foldFM_LE(wz3, Char(Zero), Branch(Char(Succ(wz6000)), wz61, wz62, wz63, wz64), bb, bc) -> new_foldFM_LE(wz3, Char(Zero), wz63, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5


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

(14)
YES