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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) HASKELL
(1) CR [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) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES


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

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

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
lookupWithDefaultFM fm deflt key = case lookupFM fm key of { 
Nothing-> deflt; 
Just elt-> elt; 
 } ;

}
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) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case lookupFM fm key of {
Nothing  -> deflt;
Just elt  -> elt}
"
is transformed to
"lookupWithDefaultFM0 deflt Nothing = deflt;
lookupWithDefaultFM0 deflt (Just elt) = elt;
"

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

(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 { 
}
lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);

lookupWithDefaultFM0 deflt Nothing = deflt;
lookupWithDefaultFM0 deflt (Just elt) = elt;

}
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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
 | key_to_find > key = lookupFM fm_r key_to_find
 | otherwise = Just elt;

lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);

lookupWithDefaultFM0 deflt Nothing = deflt;
lookupWithDefaultFM0 deflt (Just elt) = elt;

}
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
"lookupFM EmptyFM key = Nothing;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find|key_to_find < keylookupFM fm_l key_to_find|key_to_find > keylookupFM fm_r key_to_find|otherwiseJust elt;
"
is transformed to
"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
"
"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
"
"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
"
"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);
"
"lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);
"
"lookupFM4 EmptyFM key = Nothing;
lookupFM4 wv ww = lookupFM3 wv ww;
"

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

(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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
lookupFM EmptyFM key = lookupFM4 EmptyFM key;
lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;

lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;

lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;

lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);

lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);

lookupFM4 EmptyFM key = Nothing;
lookupFM4 wv ww = lookupFM3 wv ww;

lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);

lookupWithDefaultFM0 deflt Nothing = deflt;
lookupWithDefaultFM0 deflt (Just elt) = elt;

}
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.lookupWithDefaultFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.lookupWithDefaultFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.lookupWithDefaultFM wx3 wx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="FiniteMap.lookupWithDefaultFM wx3 wx4 wx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx3 wx5)",fontsize=16,color="burlywood",shape="triangle"];813[label="wx3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 813[label="",style="solid", color="burlywood", weight=9];
813 -> 7[label="",style="solid", color="burlywood", weight=3];
814[label="wx3/FiniteMap.Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];6 -> 814[label="",style="solid", color="burlywood", weight=9];
814 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM FiniteMap.EmptyFM wx5)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM4 FiniteMap.EmptyFM wx5)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM3 (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="FiniteMap.lookupWithDefaultFM0 wx4 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (wx5 < wx30))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="wx4",fontsize=16,color="green",shape="box"];14[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare wx5 wx30 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (primCmpChar wx5 wx30 == LT))",fontsize=16,color="burlywood",shape="box"];815[label="wx5/Char wx50",fontsize=10,color="white",style="solid",shape="box"];15 -> 815[label="",style="solid", color="burlywood", weight=9];
815 -> 16[label="",style="solid", color="burlywood", weight=3];
16[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 (Char wx50) (primCmpChar (Char wx50) wx30 == LT))",fontsize=16,color="burlywood",shape="box"];816[label="wx30/Char wx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 816[label="",style="solid", color="burlywood", weight=9];
816 -> 17[label="",style="solid", color="burlywood", weight=3];
17[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char wx300) wx31 wx32 wx33 wx34 (Char wx50) (primCmpChar (Char wx50) (Char wx300) == LT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char wx300) wx31 wx32 wx33 wx34 (Char wx50) (primCmpNat wx50 wx300 == LT))",fontsize=16,color="burlywood",shape="box"];817[label="wx50/Succ wx500",fontsize=10,color="white",style="solid",shape="box"];18 -> 817[label="",style="solid", color="burlywood", weight=9];
817 -> 19[label="",style="solid", color="burlywood", weight=3];
818[label="wx50/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 818[label="",style="solid", color="burlywood", weight=9];
818 -> 20[label="",style="solid", color="burlywood", weight=3];
19[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char wx300) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpNat (Succ wx500) wx300 == LT))",fontsize=16,color="burlywood",shape="box"];819[label="wx300/Succ wx3000",fontsize=10,color="white",style="solid",shape="box"];19 -> 819[label="",style="solid", color="burlywood", weight=9];
819 -> 21[label="",style="solid", color="burlywood", weight=3];
820[label="wx300/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 820[label="",style="solid", color="burlywood", weight=9];
820 -> 22[label="",style="solid", color="burlywood", weight=3];
20[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char wx300) wx31 wx32 wx33 wx34 (Char Zero) (primCmpNat Zero wx300 == LT))",fontsize=16,color="burlywood",shape="box"];821[label="wx300/Succ wx3000",fontsize=10,color="white",style="solid",shape="box"];20 -> 821[label="",style="solid", color="burlywood", weight=9];
821 -> 23[label="",style="solid", color="burlywood", weight=3];
822[label="wx300/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 822[label="",style="solid", color="burlywood", weight=9];
822 -> 24[label="",style="solid", color="burlywood", weight=3];
21[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char (Succ wx3000)) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpNat (Succ wx500) (Succ wx3000) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpNat (Succ wx500) Zero == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char (Succ wx3000)) wx31 wx32 wx33 wx34 (Char Zero) (primCmpNat Zero (Succ wx3000) == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (primCmpNat Zero Zero == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25 -> 327[label="",style="dashed", color="red", weight=0];
25[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char (Succ wx3000)) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpNat wx500 wx3000 == LT))",fontsize=16,color="magenta"];25 -> 328[label="",style="dashed", color="magenta", weight=3];
25 -> 329[label="",style="dashed", color="magenta", weight=3];
25 -> 330[label="",style="dashed", color="magenta", weight=3];
25 -> 331[label="",style="dashed", color="magenta", weight=3];
25 -> 332[label="",style="dashed", color="magenta", weight=3];
25 -> 333[label="",style="dashed", color="magenta", weight=3];
25 -> 334[label="",style="dashed", color="magenta", weight=3];
25 -> 335[label="",style="dashed", color="magenta", weight=3];
25 -> 336[label="",style="dashed", color="magenta", weight=3];
26[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (GT == LT))",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char (Succ wx3000)) wx31 wx32 wx33 wx34 (Char Zero) (LT == LT))",fontsize=16,color="black",shape="box"];27 -> 32[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (EQ == LT))",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3];
328[label="wx500",fontsize=16,color="green",shape="box"];329[label="wx32",fontsize=16,color="green",shape="box"];330[label="wx4",fontsize=16,color="green",shape="box"];331[label="wx3000",fontsize=16,color="green",shape="box"];332[label="wx31",fontsize=16,color="green",shape="box"];333[label="wx34",fontsize=16,color="green",shape="box"];334[label="wx500",fontsize=16,color="green",shape="box"];335[label="wx3000",fontsize=16,color="green",shape="box"];336[label="wx33",fontsize=16,color="green",shape="box"];327[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat wx37 wx38 == LT))",fontsize=16,color="burlywood",shape="triangle"];823[label="wx37/Succ wx370",fontsize=10,color="white",style="solid",shape="box"];327 -> 823[label="",style="solid", color="burlywood", weight=9];
823 -> 418[label="",style="solid", color="burlywood", weight=3];
824[label="wx37/Zero",fontsize=10,color="white",style="solid",shape="box"];327 -> 824[label="",style="solid", color="burlywood", weight=9];
824 -> 419[label="",style="solid", color="burlywood", weight=3];
31[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) False)",fontsize=16,color="black",shape="box"];31 -> 38[label="",style="solid", color="black", weight=3];
32[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char (Succ wx3000)) wx31 wx32 wx33 wx34 (Char Zero) True)",fontsize=16,color="black",shape="box"];32 -> 39[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) False)",fontsize=16,color="black",shape="box"];33 -> 40[label="",style="solid", color="black", weight=3];
418[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat (Succ wx370) wx38 == LT))",fontsize=16,color="burlywood",shape="box"];825[label="wx38/Succ wx380",fontsize=10,color="white",style="solid",shape="box"];418 -> 825[label="",style="solid", color="burlywood", weight=9];
825 -> 420[label="",style="solid", color="burlywood", weight=3];
826[label="wx38/Zero",fontsize=10,color="white",style="solid",shape="box"];418 -> 826[label="",style="solid", color="burlywood", weight=9];
826 -> 421[label="",style="solid", color="burlywood", weight=3];
419[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat Zero wx38 == LT))",fontsize=16,color="burlywood",shape="box"];827[label="wx38/Succ wx380",fontsize=10,color="white",style="solid",shape="box"];419 -> 827[label="",style="solid", color="burlywood", weight=9];
827 -> 422[label="",style="solid", color="burlywood", weight=3];
828[label="wx38/Zero",fontsize=10,color="white",style="solid",shape="box"];419 -> 828[label="",style="solid", color="burlywood", weight=9];
828 -> 423[label="",style="solid", color="burlywood", weight=3];
38[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (Char (Succ wx500) > Char Zero))",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39 -> 6[label="",style="dashed", color="red", weight=0];
39[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx33 (Char Zero))",fontsize=16,color="magenta"];39 -> 46[label="",style="dashed", color="magenta", weight=3];
39 -> 47[label="",style="dashed", color="magenta", weight=3];
40[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (Char Zero > Char Zero))",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
420[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat (Succ wx370) (Succ wx380) == LT))",fontsize=16,color="black",shape="box"];420 -> 424[label="",style="solid", color="black", weight=3];
421[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat (Succ wx370) Zero == LT))",fontsize=16,color="black",shape="box"];421 -> 425[label="",style="solid", color="black", weight=3];
422[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat Zero (Succ wx380) == LT))",fontsize=16,color="black",shape="box"];422 -> 426[label="",style="solid", color="black", weight=3];
423[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat Zero Zero == LT))",fontsize=16,color="black",shape="box"];423 -> 427[label="",style="solid", color="black", weight=3];
45[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (compare (Char (Succ wx500)) (Char Zero) == GT))",fontsize=16,color="black",shape="box"];45 -> 54[label="",style="solid", color="black", weight=3];
46[label="wx33",fontsize=16,color="green",shape="box"];47[label="Char Zero",fontsize=16,color="green",shape="box"];48[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (compare (Char Zero) (Char Zero) == GT))",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
424 -> 327[label="",style="dashed", color="red", weight=0];
424[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat wx370 wx380 == LT))",fontsize=16,color="magenta"];424 -> 428[label="",style="dashed", color="magenta", weight=3];
424 -> 429[label="",style="dashed", color="magenta", weight=3];
425[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (GT == LT))",fontsize=16,color="black",shape="box"];425 -> 430[label="",style="solid", color="black", weight=3];
426[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (LT == LT))",fontsize=16,color="black",shape="box"];426 -> 431[label="",style="solid", color="black", weight=3];
427[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (EQ == LT))",fontsize=16,color="black",shape="box"];427 -> 432[label="",style="solid", color="black", weight=3];
54[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpChar (Char (Succ wx500)) (Char Zero) == GT))",fontsize=16,color="black",shape="box"];54 -> 63[label="",style="solid", color="black", weight=3];
55[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (primCmpChar (Char Zero) (Char Zero) == GT))",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
428[label="wx370",fontsize=16,color="green",shape="box"];429[label="wx380",fontsize=16,color="green",shape="box"];430[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) False)",fontsize=16,color="black",shape="triangle"];430 -> 433[label="",style="solid", color="black", weight=3];
431[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) True)",fontsize=16,color="black",shape="box"];431 -> 434[label="",style="solid", color="black", weight=3];
432 -> 430[label="",style="dashed", color="red", weight=0];
432[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM2 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) False)",fontsize=16,color="magenta"];63[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (primCmpNat (Succ wx500) Zero == GT))",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3];
64[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
433[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM1 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (Char (Succ wx36) > Char (Succ wx31)))",fontsize=16,color="black",shape="box"];433 -> 435[label="",style="solid", color="black", weight=3];
434 -> 6[label="",style="dashed", color="red", weight=0];
434[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM wx34 (Char (Succ wx36)))",fontsize=16,color="magenta"];434 -> 436[label="",style="dashed", color="magenta", weight=3];
434 -> 437[label="",style="dashed", color="magenta", weight=3];
434 -> 438[label="",style="dashed", color="magenta", weight=3];
73[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) (GT == GT))",fontsize=16,color="black",shape="box"];73 -> 82[label="",style="solid", color="black", weight=3];
74[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) (EQ == GT))",fontsize=16,color="black",shape="box"];74 -> 83[label="",style="solid", color="black", weight=3];
435[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM1 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (compare (Char (Succ wx36)) (Char (Succ wx31)) == GT))",fontsize=16,color="black",shape="box"];435 -> 439[label="",style="solid", color="black", weight=3];
436[label="wx30",fontsize=16,color="green",shape="box"];437[label="wx34",fontsize=16,color="green",shape="box"];438[label="Char (Succ wx36)",fontsize=16,color="green",shape="box"];82[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char (Succ wx500)) True)",fontsize=16,color="black",shape="box"];82 -> 93[label="",style="solid", color="black", weight=3];
83[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) False)",fontsize=16,color="black",shape="box"];83 -> 94[label="",style="solid", color="black", weight=3];
439[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM1 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpChar (Char (Succ wx36)) (Char (Succ wx31)) == GT))",fontsize=16,color="black",shape="box"];439 -> 440[label="",style="solid", color="black", weight=3];
93 -> 6[label="",style="dashed", color="red", weight=0];
93[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx34 (Char (Succ wx500)))",fontsize=16,color="magenta"];93 -> 105[label="",style="dashed", color="magenta", weight=3];
93 -> 106[label="",style="dashed", color="magenta", weight=3];
94[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) otherwise)",fontsize=16,color="black",shape="box"];94 -> 107[label="",style="solid", color="black", weight=3];
440 -> 689[label="",style="dashed", color="red", weight=0];
440[label="FiniteMap.lookupWithDefaultFM0 wx30 (FiniteMap.lookupFM1 (Char (Succ wx31)) wx32 wx33 wx34 wx35 (Char (Succ wx36)) (primCmpNat (Succ wx36) (Succ wx31) == GT))",fontsize=16,color="magenta"];440 -> 690[label="",style="dashed", color="magenta", weight=3];
440 -> 691[label="",style="dashed", color="magenta", weight=3];
440 -> 692[label="",style="dashed", color="magenta", weight=3];
440 -> 693[label="",style="dashed", color="magenta", weight=3];
440 -> 694[label="",style="dashed", color="magenta", weight=3];
440 -> 695[label="",style="dashed", color="magenta", weight=3];
440 -> 696[label="",style="dashed", color="magenta", weight=3];
440 -> 697[label="",style="dashed", color="magenta", weight=3];
440 -> 698[label="",style="dashed", color="magenta", weight=3];
105[label="wx34",fontsize=16,color="green",shape="box"];106[label="Char (Succ wx500)",fontsize=16,color="green",shape="box"];107[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 (Char Zero) wx31 wx32 wx33 wx34 (Char Zero) True)",fontsize=16,color="black",shape="box"];107 -> 117[label="",style="solid", color="black", weight=3];
690[label="wx34",fontsize=16,color="green",shape="box"];691[label="wx30",fontsize=16,color="green",shape="box"];692[label="Succ wx36",fontsize=16,color="green",shape="box"];693[label="wx36",fontsize=16,color="green",shape="box"];694[label="wx32",fontsize=16,color="green",shape="box"];695[label="wx33",fontsize=16,color="green",shape="box"];696[label="wx31",fontsize=16,color="green",shape="box"];697[label="Succ wx31",fontsize=16,color="green",shape="box"];698[label="wx35",fontsize=16,color="green",shape="box"];689[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat wx91 wx92 == GT))",fontsize=16,color="burlywood",shape="triangle"];829[label="wx91/Succ wx910",fontsize=10,color="white",style="solid",shape="box"];689 -> 829[label="",style="solid", color="burlywood", weight=9];
829 -> 789[label="",style="solid", color="burlywood", weight=3];
830[label="wx91/Zero",fontsize=10,color="white",style="solid",shape="box"];689 -> 830[label="",style="solid", color="burlywood", weight=9];
830 -> 790[label="",style="solid", color="burlywood", weight=3];
117[label="FiniteMap.lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="black",shape="triangle"];117 -> 129[label="",style="solid", color="black", weight=3];
789[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat (Succ wx910) wx92 == GT))",fontsize=16,color="burlywood",shape="box"];831[label="wx92/Succ wx920",fontsize=10,color="white",style="solid",shape="box"];789 -> 831[label="",style="solid", color="burlywood", weight=9];
831 -> 791[label="",style="solid", color="burlywood", weight=3];
832[label="wx92/Zero",fontsize=10,color="white",style="solid",shape="box"];789 -> 832[label="",style="solid", color="burlywood", weight=9];
832 -> 792[label="",style="solid", color="burlywood", weight=3];
790[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat Zero wx92 == GT))",fontsize=16,color="burlywood",shape="box"];833[label="wx92/Succ wx920",fontsize=10,color="white",style="solid",shape="box"];790 -> 833[label="",style="solid", color="burlywood", weight=9];
833 -> 793[label="",style="solid", color="burlywood", weight=3];
834[label="wx92/Zero",fontsize=10,color="white",style="solid",shape="box"];790 -> 834[label="",style="solid", color="burlywood", weight=9];
834 -> 794[label="",style="solid", color="burlywood", weight=3];
129[label="wx31",fontsize=16,color="green",shape="box"];791[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat (Succ wx910) (Succ wx920) == GT))",fontsize=16,color="black",shape="box"];791 -> 795[label="",style="solid", color="black", weight=3];
792[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat (Succ wx910) Zero == GT))",fontsize=16,color="black",shape="box"];792 -> 796[label="",style="solid", color="black", weight=3];
793[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat Zero (Succ wx920) == GT))",fontsize=16,color="black",shape="box"];793 -> 797[label="",style="solid", color="black", weight=3];
794[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat Zero Zero == GT))",fontsize=16,color="black",shape="box"];794 -> 798[label="",style="solid", color="black", weight=3];
795 -> 689[label="",style="dashed", color="red", weight=0];
795[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (primCmpNat wx910 wx920 == GT))",fontsize=16,color="magenta"];795 -> 799[label="",style="dashed", color="magenta", weight=3];
795 -> 800[label="",style="dashed", color="magenta", weight=3];
796[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (GT == GT))",fontsize=16,color="black",shape="box"];796 -> 801[label="",style="solid", color="black", weight=3];
797[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (LT == GT))",fontsize=16,color="black",shape="box"];797 -> 802[label="",style="solid", color="black", weight=3];
798[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) (EQ == GT))",fontsize=16,color="black",shape="box"];798 -> 803[label="",style="solid", color="black", weight=3];
799[label="wx910",fontsize=16,color="green",shape="box"];800[label="wx920",fontsize=16,color="green",shape="box"];801[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) True)",fontsize=16,color="black",shape="box"];801 -> 804[label="",style="solid", color="black", weight=3];
802[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) False)",fontsize=16,color="black",shape="triangle"];802 -> 805[label="",style="solid", color="black", weight=3];
803 -> 802[label="",style="dashed", color="red", weight=0];
803[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM1 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) False)",fontsize=16,color="magenta"];804 -> 6[label="",style="dashed", color="red", weight=0];
804[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM wx89 (Char (Succ wx90)))",fontsize=16,color="magenta"];804 -> 806[label="",style="dashed", color="magenta", weight=3];
804 -> 807[label="",style="dashed", color="magenta", weight=3];
804 -> 808[label="",style="dashed", color="magenta", weight=3];
805[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM0 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) otherwise)",fontsize=16,color="black",shape="box"];805 -> 809[label="",style="solid", color="black", weight=3];
806[label="wx84",fontsize=16,color="green",shape="box"];807[label="wx89",fontsize=16,color="green",shape="box"];808[label="Char (Succ wx90)",fontsize=16,color="green",shape="box"];809[label="FiniteMap.lookupWithDefaultFM0 wx84 (FiniteMap.lookupFM0 (Char (Succ wx85)) wx86 wx87 wx88 wx89 (Char (Succ wx90)) True)",fontsize=16,color="black",shape="box"];809 -> 810[label="",style="solid", color="black", weight=3];
810 -> 117[label="",style="dashed", color="red", weight=0];
810[label="FiniteMap.lookupWithDefaultFM0 wx84 (Just wx86)",fontsize=16,color="magenta"];810 -> 811[label="",style="dashed", color="magenta", weight=3];
810 -> 812[label="",style="dashed", color="magenta", weight=3];
811[label="wx84",fontsize=16,color="green",shape="box"];812[label="wx86",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Zero, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)
   new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Succ(wx920), ba) -> new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, wx910, wx920, ba)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Succ(wx380), h) -> new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, wx370, wx380, h)
   new_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Zero), bb) -> new_lookupWithDefaultFM01(wx4, wx33, Char(Zero), bb)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Succ(wx380), h) -> new_lookupWithDefaultFM01(wx30, wx34, Char(Succ(wx36)), h)
   new_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM0(wx4, wx3000, wx31, wx32, wx33, wx34, wx500, wx500, wx3000, bb)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Zero, h) -> new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h)
   new_lookupWithDefaultFM01(wx4, Branch(Char(Zero), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM01(wx4, wx34, Char(Succ(wx500)), bb)
   new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Zero, ba) -> new_lookupWithDefaultFM01(wx84, wx89, Char(Succ(wx90)), ba)

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

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

(10)
Complex Obligation (AND)

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

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

   new_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Zero), bb) -> new_lookupWithDefaultFM01(wx4, wx33, Char(Zero), bb)

R is empty.
Q is empty.
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_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Zero), bb) -> new_lookupWithDefaultFM01(wx4, wx33, Char(Zero), bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(13)
YES

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

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

   new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Succ(wx920), ba) -> new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, wx910, wx920, ba)
   new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Zero, ba) -> new_lookupWithDefaultFM01(wx84, wx89, Char(Succ(wx90)), ba)
   new_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM0(wx4, wx3000, wx31, wx32, wx33, wx34, wx500, wx500, wx3000, bb)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Zero, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Succ(wx380), h) -> new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, wx370, wx380, h)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Succ(wx380), h) -> new_lookupWithDefaultFM01(wx30, wx34, Char(Succ(wx36)), h)
   new_lookupWithDefaultFM01(wx4, Branch(Char(Zero), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM01(wx4, wx34, Char(Succ(wx500)), bb)
   new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Zero, h) -> new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h)
   new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)

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

(15) 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_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Succ(wx920), ba) -> new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, wx910, wx920, 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


*new_lookupWithDefaultFM00(wx84, wx85, wx86, wx87, wx88, wx89, wx90, Succ(wx910), Zero, ba) -> new_lookupWithDefaultFM01(wx84, wx89, Char(Succ(wx90)), ba)
The graph contains the following edges 1 >= 1, 6 >= 2, 10 >= 4


*new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Zero, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 10


*new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h) -> new_lookupWithDefaultFM00(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx36), Succ(wx31), h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 10


*new_lookupWithDefaultFM01(wx4, Branch(Char(Zero), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM01(wx4, wx34, Char(Succ(wx500)), bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*new_lookupWithDefaultFM01(wx4, Branch(Char(Succ(wx3000)), wx31, wx32, wx33, wx34), Char(Succ(wx500)), bb) -> new_lookupWithDefaultFM0(wx4, wx3000, wx31, wx32, wx33, wx34, wx500, wx500, wx3000, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 > 7, 3 > 8, 2 > 9, 4 >= 10


*new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Succ(wx380), h) -> new_lookupWithDefaultFM01(wx30, wx34, Char(Succ(wx36)), h)
The graph contains the following edges 1 >= 1, 5 >= 2, 10 >= 4


*new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Succ(wx370), Succ(wx380), h) -> new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, wx370, wx380, h)
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


*new_lookupWithDefaultFM0(wx30, wx31, wx32, wx33, wx34, wx35, wx36, Zero, Zero, h) -> new_lookupWithDefaultFM02(wx30, wx31, wx32, wx33, wx34, wx35, wx36, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 10 >= 8


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

(16)
YES