/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = case lookupFM fm key of { 
Nothing-> False; 
Just elt-> True; 
 } ;

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;

}
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  -> False;
Just elt  -> True}
"
is transformed to
"elemFM0 Nothing = False;
elemFM0 (Just elt) = True;
"

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

(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) ;

instance (Eq a, Eq b) => Eq FiniteMap b a where { 
}
elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

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;

}
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 { 
}
elemFM :: Ord b => b  ->  FiniteMap b a  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
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;

}
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
"compare x y|x == yEQ|x <= yLT|otherwiseGT;
"
is transformed to
"compare x y = compare3 x y;
"
"compare0 x y True = GT;
"
"compare2 x y True = EQ;
compare2 x y False = compare1 x y (x <= y);
"
"compare1 x y True = LT;
compare1 x y False = compare0 x y otherwise;
"
"compare3 x y = compare2 x y (x == y);
"
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"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;
"
"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;
"
"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
"
"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 a b where { 
}
elemFM :: Ord b => b  ->  FiniteMap b a  ->  Bool;
elemFM key fm = elemFM0 (lookupFM fm key);

elemFM0 Nothing = False;
elemFM0 (Just elt) = True;

lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
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;

}
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.elemFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="FiniteMap.elemFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="FiniteMap.elemFM wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx4 wx3)",fontsize=16,color="burlywood",shape="triangle"];78[label="wx4/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 6[label="",style="solid", color="burlywood", weight=3];
79[label="wx4/FiniteMap.Branch wx40 wx41 wx42 wx43 wx44",fontsize=10,color="white",style="solid",shape="box"];5 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="FiniteMap.elemFM0 (FiniteMap.lookupFM FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="FiniteMap.elemFM0 (FiniteMap.lookupFM (FiniteMap.Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="FiniteMap.elemFM0 (FiniteMap.lookupFM4 FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="FiniteMap.elemFM0 (FiniteMap.lookupFM3 (FiniteMap.Branch wx40 wx41 wx42 wx43 wx44) wx3)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="FiniteMap.elemFM0 Nothing",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (wx3 < wx40))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="False",fontsize=16,color="green",shape="box"];13[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (compare wx3 wx40 == LT))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (compare3 wx3 wx40 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 wx3 (compare2 wx3 wx40 (wx3 == wx40) == LT))",fontsize=16,color="burlywood",shape="box"];80[label="wx3/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 16[label="",style="solid", color="burlywood", weight=3];
81[label="wx3/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 False (compare2 False wx40 (False == wx40) == LT))",fontsize=16,color="burlywood",shape="box"];82[label="wx40/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 18[label="",style="solid", color="burlywood", weight=3];
83[label="wx40/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 wx40 wx41 wx42 wx43 wx44 True (compare2 True wx40 (True == wx40) == LT))",fontsize=16,color="burlywood",shape="box"];84[label="wx40/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 20[label="",style="solid", color="burlywood", weight=3];
85[label="wx40/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 21[label="",style="solid", color="burlywood", weight=3];
18[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 False (compare2 False False (False == False) == LT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 True (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 False (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 True (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 False (EQ == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 True (EQ == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 False False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False (compare1 False True True == LT))",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare1 True False False == LT))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 True False)",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (False > False))",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False (LT == LT))",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare0 True False otherwise == LT))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (True > True))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (compare False False == GT))",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 True wx41 wx42 wx43 wx44 False True)",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
40[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (compare0 True False True == LT))",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
41[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (compare True True == GT))",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
42[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (compare3 False False == GT))",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
43 -> 5[label="",style="dashed", color="red", weight=0];
43[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx43 False)",fontsize=16,color="magenta"];43 -> 47[label="",style="dashed", color="magenta", weight=3];
43 -> 48[label="",style="dashed", color="magenta", weight=3];
44[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True (GT == LT))",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
45[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (compare3 True True == GT))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (compare2 False False (False == False) == GT))",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47[label="wx43",fontsize=16,color="green",shape="box"];48[label="False",fontsize=16,color="green",shape="box"];49[label="FiniteMap.elemFM0 (FiniteMap.lookupFM2 False wx41 wx42 wx43 wx44 True False)",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
50[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (compare2 True True (True == True) == GT))",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
51[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (compare2 False False True == GT))",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
52[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (True > False))",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
53[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (compare2 True True True == GT))",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3];
54[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False (EQ == GT))",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare True False == GT))",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
56[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True (EQ == GT))",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
57[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 False False)",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
58[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare3 True False == GT))",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
59[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 True wx41 wx42 wx43 wx44 True False)",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
60[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 False wx41 wx42 wx43 wx44 False otherwise)",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
61[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare2 True False (True == False) == GT))",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
62[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 True wx41 wx42 wx43 wx44 True otherwise)",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
63[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 False wx41 wx42 wx43 wx44 False True)",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
64[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare2 True False False == GT))",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3];
65[label="FiniteMap.elemFM0 (FiniteMap.lookupFM0 True wx41 wx42 wx43 wx44 True True)",fontsize=16,color="black",shape="box"];65 -> 68[label="",style="solid", color="black", weight=3];
66[label="FiniteMap.elemFM0 (Just wx41)",fontsize=16,color="black",shape="triangle"];66 -> 69[label="",style="solid", color="black", weight=3];
67[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare1 True False (True <= False) == GT))",fontsize=16,color="black",shape="box"];67 -> 70[label="",style="solid", color="black", weight=3];
68 -> 66[label="",style="dashed", color="red", weight=0];
68[label="FiniteMap.elemFM0 (Just wx41)",fontsize=16,color="magenta"];69[label="True",fontsize=16,color="green",shape="box"];70[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare1 True False False == GT))",fontsize=16,color="black",shape="box"];70 -> 71[label="",style="solid", color="black", weight=3];
71[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare0 True False otherwise == GT))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
72[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (compare0 True False True == GT))",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
73[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True (GT == GT))",fontsize=16,color="black",shape="box"];73 -> 74[label="",style="solid", color="black", weight=3];
74[label="FiniteMap.elemFM0 (FiniteMap.lookupFM1 False wx41 wx42 wx43 wx44 True True)",fontsize=16,color="black",shape="box"];74 -> 75[label="",style="solid", color="black", weight=3];
75 -> 5[label="",style="dashed", color="red", weight=0];
75[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx44 True)",fontsize=16,color="magenta"];75 -> 76[label="",style="dashed", color="magenta", weight=3];
75 -> 77[label="",style="dashed", color="magenta", weight=3];
76[label="wx44",fontsize=16,color="green",shape="box"];77[label="True",fontsize=16,color="green",shape="box"];}

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

   new_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, h)
   new_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
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(10)
Complex Obligation (AND)

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

   new_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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(13)
YES

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

   new_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, 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_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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(16)
YES