10.02/4.58	YES
12.02/5.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.02/5.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.02/5.08	
12.02/5.08	
12.02/5.08	H-Termination with start terms of the given HASKELL could be proven:
12.02/5.08	
12.02/5.08	(0) HASKELL
12.02/5.08	(1) BR [EQUIVALENT, 0 ms]
12.02/5.08	(2) HASKELL
12.02/5.08	(3) COR [EQUIVALENT, 0 ms]
12.02/5.08	(4) HASKELL
12.02/5.08	(5) Narrow [SOUND, 0 ms]
12.02/5.08	(6) QDP
12.02/5.08	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
12.02/5.08	(8) AND
12.02/5.08	    (9) QDP
12.02/5.08	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.02/5.08	        (11) YES
12.02/5.08	    (12) QDP
12.02/5.08	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.02/5.08	        (14) YES
12.02/5.08	
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(0)
12.02/5.08	Obligation:
12.02/5.08	mainModule Main 
12.02/5.08	module FiniteMap where {
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.02/5.08	
12.02/5.08	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.02/5.08	}
12.02/5.08	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
12.02/5.08	lookupFM EmptyFM key = Nothing;
12.02/5.08	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.02/5.08	 | key_to_find > key = lookupFM fm_r key_to_find
12.02/5.08	 | otherwise = Just elt;
12.02/5.08	
12.02/5.08	}
12.02/5.08	module Maybe where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	module Main where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(1) BR (EQUIVALENT)
12.02/5.08	Replaced joker patterns by fresh variables and removed binding patterns.
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(2)
12.02/5.08	Obligation:
12.02/5.08	mainModule Main 
12.02/5.08	module FiniteMap where {
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.02/5.08	
12.02/5.08	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.02/5.08	}
12.02/5.08	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
12.02/5.08	lookupFM EmptyFM key = Nothing;
12.02/5.08	lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.02/5.08	 | key_to_find > key = lookupFM fm_r key_to_find
12.02/5.08	 | otherwise = Just elt;
12.02/5.08	
12.02/5.08	}
12.02/5.08	module Maybe where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	module Main where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(3) COR (EQUIVALENT)
12.02/5.08	Cond Reductions:
12.02/5.08	The following Function with conditions
12.02/5.08	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
12.02/5.08	"
12.02/5.08	is transformed to
12.02/5.08	"compare x y = compare3 x y;
12.02/5.08	"
12.02/5.08	"compare2 x y True = EQ;
12.02/5.08	compare2 x y False = compare1 x y (x <= y);
12.02/5.08	"
12.02/5.08	"compare1 x y True = LT;
12.02/5.08	compare1 x y False = compare0 x y otherwise;
12.02/5.08	"
12.02/5.08	"compare0 x y True = GT;
12.02/5.08	"
12.02/5.08	"compare3 x y = compare2 x y (x == y);
12.02/5.08	"
12.02/5.08	The following Function with conditions
12.02/5.08	"undefined |Falseundefined;
12.02/5.08	"
12.02/5.08	is transformed to
12.02/5.08	"undefined  = undefined1;
12.02/5.08	"
12.02/5.08	"undefined0 True = undefined;
12.02/5.08	"
12.02/5.08	"undefined1  = undefined0 False;
12.02/5.08	"
12.02/5.08	The following Function with conditions
12.02/5.08	"lookupFM EmptyFM key = Nothing;
12.02/5.08	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;
12.02/5.08	"
12.02/5.08	is transformed to
12.02/5.08	"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.02/5.08	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.02/5.08	"
12.02/5.08	"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.02/5.08	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.02/5.08	"
12.02/5.08	"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.02/5.08	"
12.02/5.08	"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.02/5.08	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);
12.02/5.08	"
12.02/5.08	"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);
12.02/5.08	"
12.02/5.08	"lookupFM4 EmptyFM key = Nothing;
12.02/5.08	lookupFM4 wv ww = lookupFM3 wv ww;
12.02/5.08	"
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(4)
12.02/5.08	Obligation:
12.02/5.08	mainModule Main 
12.02/5.08	module FiniteMap where {
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.02/5.08	
12.02/5.08	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.02/5.08	}
12.02/5.08	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.02/5.08	lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.02/5.08	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.02/5.08	
12.02/5.08	lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.02/5.08	
12.02/5.08	lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.02/5.08	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.02/5.08	
12.02/5.08	lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.02/5.08	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);
12.02/5.08	
12.02/5.08	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);
12.02/5.08	
12.02/5.08	lookupFM4 EmptyFM key = Nothing;
12.02/5.08	lookupFM4 wv ww = lookupFM3 wv ww;
12.02/5.08	
12.02/5.08	}
12.02/5.08	module Maybe where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Main;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	module Main where {
12.02/5.08	import qualified FiniteMap;
12.02/5.08	import qualified Maybe;
12.02/5.08	import qualified Prelude;
12.02/5.08	}
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(5) Narrow (SOUND)
12.02/5.08	Haskell To QDPs
12.02/5.08	
12.02/5.08	digraph dp_graph {
12.02/5.08	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.lookupFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.02/5.08	3[label="FiniteMap.lookupFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.02/5.08	4[label="FiniteMap.lookupFM wx3 wx4",fontsize=16,color="burlywood",shape="triangle"];75[label="wx3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 75[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	75 -> 5[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	76[label="wx3/FiniteMap.Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];4 -> 76[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	76 -> 6[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	5[label="FiniteMap.lookupFM FiniteMap.EmptyFM wx4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.02/5.08	6[label="FiniteMap.lookupFM (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.02/5.08	7[label="FiniteMap.lookupFM4 FiniteMap.EmptyFM wx4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.02/5.08	8[label="FiniteMap.lookupFM3 (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.02/5.08	9[label="Nothing",fontsize=16,color="green",shape="box"];10[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (wx4 < wx30)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
12.02/5.08	11[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare wx4 wx30 == LT)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.02/5.08	12[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare3 wx4 wx30 == LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.02/5.08	13[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx4 (compare2 wx4 wx30 (wx4 == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];77[label="wx4/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	77 -> 14[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	78[label="wx4/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 78[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	78 -> 15[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	14[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 False (compare2 False wx30 (False == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];79[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	79 -> 16[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	80[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	80 -> 17[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	15[label="FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 True (compare2 True wx30 (True == wx30) == LT)",fontsize=16,color="burlywood",shape="box"];81[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 81[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	81 -> 18[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	82[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 82[label="",style="solid", color="burlywood", weight=9];
12.02/5.08	82 -> 19[label="",style="solid", color="burlywood", weight=3];
12.02/5.08	16[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == LT)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
12.02/5.08	17[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True (False == True) == LT)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
12.02/5.08	18[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == LT)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.02/5.08	19[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == LT)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.02/5.08	20[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (compare2 False False True == LT)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.02/5.08	21[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare2 False True False == LT)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.02/5.08	22[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare2 True False False == LT)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.02/5.08	23[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (compare2 True True True == LT)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.02/5.08	24[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False (EQ == LT)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.02/5.08	25[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True (False <= True) == LT)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.02/5.08	26[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == LT)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.02/5.08	27[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True (EQ == LT)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.02/5.08	28[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 False False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
12.02/5.08	29[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (compare1 False True True == LT)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
12.02/5.08	30[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare1 True False False == LT)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
12.02/5.08	31[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 True False",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
12.02/5.08	32[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (False > False)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
12.02/5.08	33[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False (LT == LT)",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
12.02/5.08	34[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == LT)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
12.02/5.08	35[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (True > True)",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
12.02/5.08	36[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare False False == GT)",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
12.02/5.08	37[label="FiniteMap.lookupFM2 True wx31 wx32 wx33 wx34 False True",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
12.02/5.08	38[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (compare0 True False True == LT)",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
12.02/5.08	39[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare True True == GT)",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
12.02/5.08	40[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare3 False False == GT)",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
12.02/5.08	41 -> 4[label="",style="dashed", color="red", weight=0];
12.02/5.08	41[label="FiniteMap.lookupFM wx33 False",fontsize=16,color="magenta"];41 -> 45[label="",style="dashed", color="magenta", weight=3];
12.02/5.08	41 -> 46[label="",style="dashed", color="magenta", weight=3];
12.02/5.08	42[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True (GT == LT)",fontsize=16,color="black",shape="box"];42 -> 47[label="",style="solid", color="black", weight=3];
12.02/5.08	43[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare3 True True == GT)",fontsize=16,color="black",shape="box"];43 -> 48[label="",style="solid", color="black", weight=3];
12.02/5.08	44[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False (False == False) == GT)",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
12.02/5.08	45[label="wx33",fontsize=16,color="green",shape="box"];46[label="False",fontsize=16,color="green",shape="box"];47[label="FiniteMap.lookupFM2 False wx31 wx32 wx33 wx34 True False",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
12.02/5.08	48[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True (True == True) == GT)",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
12.02/5.08	49[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (compare2 False False True == GT)",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
12.02/5.08	50[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (True > False)",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3];
12.02/5.08	51[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (compare2 True True True == GT)",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
12.02/5.08	52[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False (EQ == GT)",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
12.02/5.08	53[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare True False == GT)",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3];
12.02/5.08	54[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True (EQ == GT)",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
12.02/5.08	55[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 False False",fontsize=16,color="black",shape="box"];55 -> 58[label="",style="solid", color="black", weight=3];
12.02/5.08	56[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare3 True False == GT)",fontsize=16,color="black",shape="box"];56 -> 59[label="",style="solid", color="black", weight=3];
12.02/5.08	57[label="FiniteMap.lookupFM1 True wx31 wx32 wx33 wx34 True False",fontsize=16,color="black",shape="box"];57 -> 60[label="",style="solid", color="black", weight=3];
12.02/5.08	58[label="FiniteMap.lookupFM0 False wx31 wx32 wx33 wx34 False otherwise",fontsize=16,color="black",shape="box"];58 -> 61[label="",style="solid", color="black", weight=3];
12.02/5.08	59[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False (True == False) == GT)",fontsize=16,color="black",shape="box"];59 -> 62[label="",style="solid", color="black", weight=3];
12.02/5.08	60[label="FiniteMap.lookupFM0 True wx31 wx32 wx33 wx34 True otherwise",fontsize=16,color="black",shape="box"];60 -> 63[label="",style="solid", color="black", weight=3];
12.02/5.08	61[label="FiniteMap.lookupFM0 False wx31 wx32 wx33 wx34 False True",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
12.02/5.08	62[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare2 True False False == GT)",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
12.02/5.08	63[label="FiniteMap.lookupFM0 True wx31 wx32 wx33 wx34 True True",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
12.02/5.08	64[label="Just wx31",fontsize=16,color="green",shape="box"];65[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False (True <= False) == GT)",fontsize=16,color="black",shape="box"];65 -> 67[label="",style="solid", color="black", weight=3];
12.02/5.08	66[label="Just wx31",fontsize=16,color="green",shape="box"];67[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare1 True False False == GT)",fontsize=16,color="black",shape="box"];67 -> 68[label="",style="solid", color="black", weight=3];
12.02/5.08	68[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False otherwise == GT)",fontsize=16,color="black",shape="box"];68 -> 69[label="",style="solid", color="black", weight=3];
12.02/5.08	69[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (compare0 True False True == GT)",fontsize=16,color="black",shape="box"];69 -> 70[label="",style="solid", color="black", weight=3];
12.02/5.08	70[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True (GT == GT)",fontsize=16,color="black",shape="box"];70 -> 71[label="",style="solid", color="black", weight=3];
12.02/5.08	71[label="FiniteMap.lookupFM1 False wx31 wx32 wx33 wx34 True True",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
12.02/5.08	72 -> 4[label="",style="dashed", color="red", weight=0];
12.02/5.08	72[label="FiniteMap.lookupFM wx34 True",fontsize=16,color="magenta"];72 -> 73[label="",style="dashed", color="magenta", weight=3];
12.02/5.08	72 -> 74[label="",style="dashed", color="magenta", weight=3];
12.02/5.08	73[label="wx34",fontsize=16,color="green",shape="box"];74[label="True",fontsize=16,color="green",shape="box"];}
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(6)
12.02/5.08	Obligation:
12.02/5.08	Q DP problem:
12.02/5.08	The TRS P consists of the following rules:
12.02/5.08	
12.02/5.08	   new_lookupFM(Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupFM(wx33, False, h)
12.02/5.08	   new_lookupFM(Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupFM(wx34, True, h)
12.02/5.08	
12.02/5.08	R is empty.
12.02/5.08	Q is empty.
12.02/5.08	We have to consider all minimal (P,Q,R)-chains.
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(7) DependencyGraphProof (EQUIVALENT)
12.02/5.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(8)
12.02/5.08	Complex Obligation (AND)
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(9)
12.02/5.08	Obligation:
12.02/5.08	Q DP problem:
12.02/5.08	The TRS P consists of the following rules:
12.02/5.08	
12.02/5.08	   new_lookupFM(Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupFM(wx34, True, h)
12.02/5.08	
12.02/5.08	R is empty.
12.02/5.08	Q is empty.
12.02/5.08	We have to consider all minimal (P,Q,R)-chains.
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(10) QDPSizeChangeProof (EQUIVALENT)
12.02/5.08	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. 
12.02/5.08	
12.02/5.08	From the DPs we obtained the following set of size-change graphs:
12.02/5.08	*new_lookupFM(Branch(False, wx31, wx32, wx33, wx34), True, h) -> new_lookupFM(wx34, True, h)
12.02/5.08	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.02/5.08	
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(11)
12.02/5.08	YES
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(12)
12.02/5.08	Obligation:
12.02/5.08	Q DP problem:
12.02/5.08	The TRS P consists of the following rules:
12.02/5.08	
12.02/5.08	   new_lookupFM(Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupFM(wx33, False, h)
12.02/5.08	
12.02/5.08	R is empty.
12.02/5.08	Q is empty.
12.02/5.08	We have to consider all minimal (P,Q,R)-chains.
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(13) QDPSizeChangeProof (EQUIVALENT)
12.02/5.08	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. 
12.02/5.08	
12.02/5.08	From the DPs we obtained the following set of size-change graphs:
12.02/5.08	*new_lookupFM(Branch(True, wx31, wx32, wx33, wx34), False, h) -> new_lookupFM(wx33, False, h)
12.02/5.08	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.02/5.08	
12.02/5.08	
12.02/5.08	----------------------------------------
12.02/5.08	
12.02/5.08	(14)
12.02/5.08	YES
12.07/5.13	EOF