10.18/4.49	YES
12.32/5.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.32/5.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.32/5.10	
12.32/5.10	
12.32/5.10	H-Termination with start terms of the given HASKELL could be proven:
12.32/5.10	
12.32/5.10	(0) HASKELL
12.32/5.10	(1) CR [EQUIVALENT, 0 ms]
12.32/5.10	(2) HASKELL
12.32/5.10	(3) BR [EQUIVALENT, 0 ms]
12.32/5.10	(4) HASKELL
12.32/5.10	(5) COR [EQUIVALENT, 0 ms]
12.32/5.10	(6) HASKELL
12.32/5.10	(7) Narrow [SOUND, 0 ms]
12.32/5.10	(8) QDP
12.32/5.10	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.32/5.10	(10) AND
12.32/5.10	    (11) QDP
12.32/5.10	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.32/5.10	        (13) YES
12.32/5.10	    (14) QDP
12.32/5.10	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.32/5.10	        (16) YES
12.32/5.10	
12.32/5.10	
12.32/5.10	----------------------------------------
12.32/5.10	
12.32/5.10	(0)
12.32/5.10	Obligation:
12.32/5.10	mainModule Main 
12.32/5.10	module FiniteMap where {
12.32/5.10	import qualified Main;
12.32/5.10	import qualified Maybe;
12.32/5.10	import qualified Prelude;
12.32/5.10	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.32/5.10	
12.32/5.10	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.32/5.10	}
12.32/5.10	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.32/5.10	lookupFM EmptyFM key = Nothing;
12.32/5.10	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.50/5.12	 | key_to_find > key = lookupFM fm_r key_to_find
12.50/5.12	 | otherwise = Just elt;
12.50/5.12	
12.50/5.12	lookupWithDefaultFM :: Ord a => FiniteMap a b  ->  b  ->  a  ->  b;
12.50/5.12	lookupWithDefaultFM fm deflt key = case lookupFM fm key of { 
12.50/5.12	Nothing-> deflt; 
12.50/5.12	Just elt-> elt; 
12.50/5.12	 } ;
12.50/5.12	
12.50/5.12	}
12.50/5.12	module Maybe where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	module Main where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(1) CR (EQUIVALENT)
12.50/5.12	Case Reductions:
12.50/5.12	The following Case expression
12.50/5.12	"case lookupFM fm key of {
12.50/5.12	Nothing  -> deflt;
12.50/5.12	Just elt  -> elt}
12.50/5.12	"
12.50/5.12	is transformed to
12.50/5.12	"lookupWithDefaultFM0 deflt Nothing = deflt;
12.50/5.12	lookupWithDefaultFM0 deflt (Just elt) = elt;
12.50/5.12	"
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(2)
12.50/5.12	Obligation:
12.50/5.12	mainModule Main 
12.50/5.12	module FiniteMap where {
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.50/5.12	
12.50/5.12	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.50/5.12	}
12.50/5.12	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.50/5.12	lookupFM EmptyFM key = Nothing;
12.50/5.12	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.50/5.12	 | key_to_find > key = lookupFM fm_r key_to_find
12.50/5.12	 | otherwise = Just elt;
12.50/5.12	
12.50/5.12	lookupWithDefaultFM :: Ord a => FiniteMap a b  ->  b  ->  a  ->  b;
12.50/5.12	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
12.50/5.12	
12.50/5.12	lookupWithDefaultFM0 deflt Nothing = deflt;
12.50/5.12	lookupWithDefaultFM0 deflt (Just elt) = elt;
12.50/5.12	
12.50/5.12	}
12.50/5.12	module Maybe where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	module Main where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(3) BR (EQUIVALENT)
12.50/5.12	Replaced joker patterns by fresh variables and removed binding patterns.
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(4)
12.50/5.12	Obligation:
12.50/5.12	mainModule Main 
12.50/5.12	module FiniteMap where {
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.50/5.12	
12.50/5.12	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.50/5.12	}
12.50/5.12	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
12.50/5.12	lookupFM EmptyFM key = Nothing;
12.50/5.12	lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.50/5.12	 | key_to_find > key = lookupFM fm_r key_to_find
12.50/5.12	 | otherwise = Just elt;
12.50/5.12	
12.50/5.12	lookupWithDefaultFM :: Ord a => FiniteMap a b  ->  b  ->  a  ->  b;
12.50/5.12	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
12.50/5.12	
12.50/5.12	lookupWithDefaultFM0 deflt Nothing = deflt;
12.50/5.12	lookupWithDefaultFM0 deflt (Just elt) = elt;
12.50/5.12	
12.50/5.12	}
12.50/5.12	module Maybe where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	module Main where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(5) COR (EQUIVALENT)
12.50/5.12	Cond Reductions:
12.50/5.12	The following Function with conditions
12.50/5.12	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
12.50/5.12	"
12.50/5.12	is transformed to
12.50/5.12	"compare x y = compare3 x y;
12.50/5.12	"
12.50/5.12	"compare1 x y True = LT;
12.50/5.12	compare1 x y False = compare0 x y otherwise;
12.50/5.12	"
12.50/5.12	"compare2 x y True = EQ;
12.50/5.12	compare2 x y False = compare1 x y (x <= y);
12.50/5.12	"
12.50/5.12	"compare0 x y True = GT;
12.50/5.12	"
12.50/5.12	"compare3 x y = compare2 x y (x == y);
12.50/5.12	"
12.50/5.12	The following Function with conditions
12.50/5.12	"undefined |Falseundefined;
12.50/5.12	"
12.50/5.12	is transformed to
12.50/5.12	"undefined  = undefined1;
12.50/5.12	"
12.50/5.12	"undefined0 True = undefined;
12.50/5.12	"
12.50/5.12	"undefined1  = undefined0 False;
12.50/5.12	"
12.50/5.12	The following Function with conditions
12.50/5.12	"lookupFM EmptyFM key = Nothing;
12.50/5.12	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;
12.50/5.12	"
12.50/5.12	is transformed to
12.50/5.12	"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.50/5.12	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.50/5.12	"
12.50/5.12	"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.50/5.12	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.50/5.12	"
12.50/5.12	"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.50/5.12	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);
12.50/5.12	"
12.50/5.12	"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.50/5.12	"
12.50/5.12	"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);
12.50/5.12	"
12.50/5.12	"lookupFM4 EmptyFM key = Nothing;
12.50/5.12	lookupFM4 wv ww = lookupFM3 wv ww;
12.50/5.12	"
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(6)
12.50/5.12	Obligation:
12.50/5.12	mainModule Main 
12.50/5.12	module FiniteMap where {
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.50/5.12	
12.50/5.12	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.50/5.12	}
12.50/5.12	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.50/5.12	lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.50/5.12	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.50/5.12	
12.50/5.12	lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.50/5.12	
12.50/5.12	lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.50/5.12	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.50/5.12	
12.50/5.12	lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.50/5.12	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);
12.50/5.12	
12.50/5.12	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);
12.50/5.12	
12.50/5.12	lookupFM4 EmptyFM key = Nothing;
12.50/5.12	lookupFM4 wv ww = lookupFM3 wv ww;
12.50/5.12	
12.50/5.12	lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
12.50/5.12	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
12.50/5.12	
12.50/5.12	lookupWithDefaultFM0 deflt Nothing = deflt;
12.50/5.12	lookupWithDefaultFM0 deflt (Just elt) = elt;
12.50/5.12	
12.50/5.12	}
12.50/5.12	module Maybe where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Main;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	module Main where {
12.50/5.12	import qualified FiniteMap;
12.50/5.12	import qualified Maybe;
12.50/5.12	import qualified Prelude;
12.50/5.12	}
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(7) Narrow (SOUND)
12.50/5.12	Haskell To QDPs
12.50/5.12	
12.50/5.12	digraph dp_graph {
12.50/5.12	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.lookupWithDefaultFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.50/5.12	3[label="FiniteMap.lookupWithDefaultFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.50/5.12	4[label="FiniteMap.lookupWithDefaultFM wx3 wx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
12.50/5.12	5[label="FiniteMap.lookupWithDefaultFM wx3 wx4 wx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
12.50/5.12	6[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx3 wx5)",fontsize=16,color="burlywood",shape="triangle"];79[label="wx3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 79[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	79 -> 7[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	80[label="wx3/FiniteMap.Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];6 -> 80[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	80 -> 8[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	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];
12.50/5.12	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];
12.50/5.12	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];
12.50/5.12	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];
12.50/5.12	11[label="FiniteMap.lookupWithDefaultFM0 wx4 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12.50/5.12	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];
12.50/5.12	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];
12.50/5.12	15[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare3 wx5 wx30 == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
12.50/5.12	16[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare2 wx5 wx30 (wx5 == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];81[label="wx5/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 81[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	81 -> 17[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	82[label="wx5/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 82[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	82 -> 18[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	17[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 False (compare2 False wx30 (False == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];83[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 83[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	83 -> 19[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	84[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	84 -> 20[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	18[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 True (compare2 True wx30 (True == wx30) == LT))",fontsize=16,color="burlywood",shape="box"];85[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 85[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	85 -> 21[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	86[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 86[label="",style="solid", color="burlywood", weight=9];
12.50/5.12	86 -> 22[label="",style="solid", color="burlywood", weight=3];
12.50/5.12	19[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.50/5.12	20[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.50/5.12	21[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.50/5.12	22[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.50/5.12	23[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.50/5.12	24[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.50/5.12	25[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.50/5.12	26[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.50/5.12	27[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (EQ == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.50/5.12	28[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
12.50/5.12	29[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
12.50/5.12	30[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (EQ == LT))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
12.50/5.12	31[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
12.50/5.12	32[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True True == LT))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
12.50/5.12	33[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False False == LT))",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
12.50/5.12	34[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
12.50/5.12	35[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (False > False))",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
12.50/5.12	36[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (LT == LT))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
12.50/5.12	37[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == LT))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
12.50/5.12	38[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (True > True))",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
12.50/5.12	39[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare False False == GT))",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
12.50/5.12	40[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False True)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
12.50/5.12	41[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False True == LT))",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
12.50/5.12	42[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare True True == GT))",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
12.50/5.12	43[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare3 False False == GT))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
12.50/5.12	44 -> 6[label="",style="dashed", color="red", weight=0];
12.50/5.12	44[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx33 False)",fontsize=16,color="magenta"];44 -> 48[label="",style="dashed", color="magenta", weight=3];
12.50/5.12	44 -> 49[label="",style="dashed", color="magenta", weight=3];
12.50/5.12	45[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (GT == LT))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
12.50/5.12	46[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare3 True True == GT))",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
12.50/5.12	47[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == GT))",fontsize=16,color="black",shape="box"];47 -> 52[label="",style="solid", color="black", weight=3];
12.50/5.12	48[label="False",fontsize=16,color="green",shape="box"];49[label="wx33",fontsize=16,color="green",shape="box"];50[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
12.50/5.12	51[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == GT))",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
12.50/5.12	52[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False True == GT))",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
12.50/5.12	53[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (True > False))",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3];
12.50/5.12	54[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True True == GT))",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
12.50/5.12	55[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (EQ == GT))",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
12.50/5.12	56[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare True False == GT))",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
12.50/5.12	57[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (EQ == GT))",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
12.50/5.12	58[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False False)",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
12.50/5.12	59[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare3 True False == GT))",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
12.50/5.12	60[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True False)",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
12.50/5.12	61[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 False wx31 wx32 wx33 wx34 False otherwise)",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
12.50/5.12	62[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == GT))",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
12.50/5.12	63[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 True wx31 wx32 wx33 wx34 True otherwise)",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
12.50/5.12	64[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 False wx31 wx32 wx33 wx34 False True)",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3];
12.50/5.12	65[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False False == GT))",fontsize=16,color="black",shape="box"];65 -> 68[label="",style="solid", color="black", weight=3];
12.50/5.12	66[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 True wx31 wx32 wx33 wx34 True True)",fontsize=16,color="black",shape="box"];66 -> 69[label="",style="solid", color="black", weight=3];
12.50/5.12	67[label="FiniteMap.lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="black",shape="triangle"];67 -> 70[label="",style="solid", color="black", weight=3];
12.50/5.12	68[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == GT))",fontsize=16,color="black",shape="box"];68 -> 71[label="",style="solid", color="black", weight=3];
12.50/5.12	69 -> 67[label="",style="dashed", color="red", weight=0];
12.50/5.12	69[label="FiniteMap.lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="magenta"];70[label="wx31",fontsize=16,color="green",shape="box"];71[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False False == GT))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
12.50/5.12	72[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == GT))",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
12.50/5.12	73[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False True == GT))",fontsize=16,color="black",shape="box"];73 -> 74[label="",style="solid", color="black", weight=3];
12.50/5.12	74[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (GT == GT))",fontsize=16,color="black",shape="box"];74 -> 75[label="",style="solid", color="black", weight=3];
12.50/5.12	75[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True True)",fontsize=16,color="black",shape="box"];75 -> 76[label="",style="solid", color="black", weight=3];
12.50/5.12	76 -> 6[label="",style="dashed", color="red", weight=0];
12.50/5.12	76[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx34 True)",fontsize=16,color="magenta"];76 -> 77[label="",style="dashed", color="magenta", weight=3];
12.50/5.12	76 -> 78[label="",style="dashed", color="magenta", weight=3];
12.50/5.12	77[label="True",fontsize=16,color="green",shape="box"];78[label="wx34",fontsize=16,color="green",shape="box"];}
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(8)
12.50/5.12	Obligation:
12.50/5.12	Q DP problem:
12.50/5.12	The TRS P consists of the following rules:
12.50/5.12	
12.50/5.12	   new_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupWithDefaultFM0(wx4, wx33, False, h)
12.50/5.12	   new_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupWithDefaultFM0(wx4, wx34, True, h)
12.50/5.12	
12.50/5.12	R is empty.
12.50/5.12	Q is empty.
12.50/5.12	We have to consider all minimal (P,Q,R)-chains.
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(9) DependencyGraphProof (EQUIVALENT)
12.50/5.12	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(10)
12.50/5.12	Complex Obligation (AND)
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(11)
12.50/5.12	Obligation:
12.50/5.12	Q DP problem:
12.50/5.12	The TRS P consists of the following rules:
12.50/5.12	
12.50/5.12	   new_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupWithDefaultFM0(wx4, wx34, True, h)
12.50/5.12	
12.50/5.12	R is empty.
12.50/5.12	Q is empty.
12.50/5.12	We have to consider all minimal (P,Q,R)-chains.
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(12) QDPSizeChangeProof (EQUIVALENT)
12.50/5.12	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. 
12.50/5.12	
12.50/5.12	From the DPs we obtained the following set of size-change graphs:
12.50/5.12	*new_lookupWithDefaultFM0(wx4, Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupWithDefaultFM0(wx4, wx34, True, h)
12.50/5.12	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
12.50/5.12	
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(13)
12.50/5.12	YES
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(14)
12.50/5.12	Obligation:
12.50/5.12	Q DP problem:
12.50/5.12	The TRS P consists of the following rules:
12.50/5.12	
12.50/5.12	   new_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupWithDefaultFM0(wx4, wx33, False, h)
12.50/5.12	
12.50/5.12	R is empty.
12.50/5.12	Q is empty.
12.50/5.12	We have to consider all minimal (P,Q,R)-chains.
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(15) QDPSizeChangeProof (EQUIVALENT)
12.50/5.12	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. 
12.50/5.12	
12.50/5.12	From the DPs we obtained the following set of size-change graphs:
12.50/5.12	*new_lookupWithDefaultFM0(wx4, Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupWithDefaultFM0(wx4, wx33, False, h)
12.50/5.12	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
12.50/5.12	
12.50/5.12	
12.50/5.12	----------------------------------------
12.50/5.12	
12.50/5.12	(16)
12.50/5.12	YES
12.51/5.16	EOF