10.24/4.51	YES
12.52/5.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.52/5.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.52/5.11	
12.52/5.11	
12.52/5.11	H-Termination with start terms of the given HASKELL could be proven:
12.52/5.11	
12.52/5.11	(0) HASKELL
12.52/5.11	(1) CR [EQUIVALENT, 0 ms]
12.52/5.11	(2) HASKELL
12.52/5.11	(3) BR [EQUIVALENT, 0 ms]
12.52/5.11	(4) HASKELL
12.52/5.11	(5) COR [EQUIVALENT, 0 ms]
12.52/5.11	(6) HASKELL
12.52/5.11	(7) Narrow [SOUND, 0 ms]
12.52/5.11	(8) QDP
12.52/5.11	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.52/5.11	(10) AND
12.52/5.11	    (11) QDP
12.52/5.11	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.52/5.11	        (13) YES
12.52/5.11	    (14) QDP
12.52/5.11	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.52/5.11	        (16) YES
12.52/5.11	
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(0)
12.52/5.11	Obligation:
12.52/5.11	mainModule Main 
12.52/5.11	module FiniteMap where {
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.52/5.11	
12.52/5.11	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.52/5.11	}
12.52/5.11	elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
12.52/5.11	elemFM key fm = case lookupFM fm key of { 
12.52/5.11	Nothing-> False; 
12.52/5.11	Just elt-> True; 
12.52/5.11	 } ;
12.52/5.11	
12.52/5.11	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.52/5.11	lookupFM EmptyFM key = Nothing;
12.52/5.11	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.52/5.11	 | key_to_find > key = lookupFM fm_r key_to_find
12.52/5.11	 | otherwise = Just elt;
12.52/5.11	
12.52/5.11	}
12.52/5.11	module Maybe where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	module Main where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(1) CR (EQUIVALENT)
12.52/5.11	Case Reductions:
12.52/5.11	The following Case expression
12.52/5.11	"case lookupFM fm key of {
12.52/5.11	Nothing  -> False;
12.52/5.11	Just elt  -> True}
12.52/5.11	"
12.52/5.11	is transformed to
12.52/5.11	"elemFM0 Nothing = False;
12.52/5.11	elemFM0 (Just elt) = True;
12.52/5.11	"
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(2)
12.52/5.11	Obligation:
12.52/5.11	mainModule Main 
12.52/5.11	module FiniteMap where {
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.52/5.11	
12.52/5.11	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.52/5.11	}
12.52/5.11	elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
12.52/5.11	elemFM key fm = elemFM0 (lookupFM fm key);
12.52/5.11	
12.52/5.11	elemFM0 Nothing = False;
12.52/5.11	elemFM0 (Just elt) = True;
12.52/5.11	
12.52/5.11	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.52/5.11	lookupFM EmptyFM key = Nothing;
12.52/5.11	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.52/5.11	 | key_to_find > key = lookupFM fm_r key_to_find
12.52/5.11	 | otherwise = Just elt;
12.52/5.11	
12.52/5.11	}
12.52/5.11	module Maybe where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	module Main where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(3) BR (EQUIVALENT)
12.52/5.11	Replaced joker patterns by fresh variables and removed binding patterns.
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(4)
12.52/5.11	Obligation:
12.52/5.11	mainModule Main 
12.52/5.11	module FiniteMap where {
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.52/5.11	
12.52/5.11	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.52/5.11	}
12.52/5.11	elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
12.52/5.11	elemFM key fm = elemFM0 (lookupFM fm key);
12.52/5.11	
12.52/5.11	elemFM0 Nothing = False;
12.52/5.11	elemFM0 (Just elt) = True;
12.52/5.11	
12.52/5.11	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
12.52/5.11	lookupFM EmptyFM key = Nothing;
12.52/5.11	lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
12.52/5.11	 | key_to_find > key = lookupFM fm_r key_to_find
12.52/5.11	 | otherwise = Just elt;
12.52/5.11	
12.52/5.11	}
12.52/5.11	module Maybe where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	module Main where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(5) COR (EQUIVALENT)
12.52/5.11	Cond Reductions:
12.52/5.11	The following Function with conditions
12.52/5.11	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
12.52/5.11	"
12.52/5.11	is transformed to
12.52/5.11	"compare x y = compare3 x y;
12.52/5.11	"
12.52/5.11	"compare0 x y True = GT;
12.52/5.11	"
12.52/5.11	"compare2 x y True = EQ;
12.52/5.11	compare2 x y False = compare1 x y (x <= y);
12.52/5.11	"
12.52/5.11	"compare1 x y True = LT;
12.52/5.11	compare1 x y False = compare0 x y otherwise;
12.52/5.11	"
12.52/5.11	"compare3 x y = compare2 x y (x == y);
12.52/5.11	"
12.52/5.11	The following Function with conditions
12.52/5.11	"undefined |Falseundefined;
12.52/5.11	"
12.52/5.11	is transformed to
12.52/5.11	"undefined  = undefined1;
12.52/5.11	"
12.52/5.11	"undefined0 True = undefined;
12.52/5.11	"
12.52/5.11	"undefined1  = undefined0 False;
12.52/5.11	"
12.52/5.11	The following Function with conditions
12.52/5.11	"lookupFM EmptyFM key = Nothing;
12.52/5.11	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;
12.52/5.11	"
12.52/5.11	is transformed to
12.52/5.11	"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.52/5.11	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.52/5.11	"
12.52/5.11	"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.52/5.11	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.52/5.11	"
12.52/5.11	"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.52/5.11	"
12.52/5.11	"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.52/5.11	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);
12.52/5.11	"
12.52/5.11	"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);
12.52/5.11	"
12.52/5.11	"lookupFM4 EmptyFM key = Nothing;
12.52/5.11	lookupFM4 wv ww = lookupFM3 wv ww;
12.52/5.11	"
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(6)
12.52/5.11	Obligation:
12.52/5.11	mainModule Main 
12.52/5.11	module FiniteMap where {
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.52/5.11	
12.52/5.11	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.52/5.11	}
12.52/5.11	elemFM :: Ord a => a  ->  FiniteMap a b  ->  Bool;
12.52/5.11	elemFM key fm = elemFM0 (lookupFM fm key);
12.52/5.11	
12.52/5.11	elemFM0 Nothing = False;
12.52/5.11	elemFM0 (Just elt) = True;
12.52/5.11	
12.52/5.11	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
12.52/5.11	lookupFM EmptyFM key = lookupFM4 EmptyFM key;
12.52/5.11	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
12.52/5.11	
12.52/5.11	lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
12.52/5.11	
12.52/5.11	lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
12.52/5.11	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
12.52/5.11	
12.52/5.11	lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
12.52/5.11	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);
12.52/5.11	
12.52/5.11	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);
12.52/5.11	
12.52/5.11	lookupFM4 EmptyFM key = Nothing;
12.52/5.11	lookupFM4 wv ww = lookupFM3 wv ww;
12.52/5.11	
12.52/5.11	}
12.52/5.11	module Maybe where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Main;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	module Main where {
12.52/5.11	import qualified FiniteMap;
12.52/5.11	import qualified Maybe;
12.52/5.11	import qualified Prelude;
12.52/5.11	}
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(7) Narrow (SOUND)
12.52/5.11	Haskell To QDPs
12.52/5.11	
12.52/5.11	digraph dp_graph {
12.52/5.11	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.elemFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.52/5.11	3[label="FiniteMap.elemFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.52/5.11	4[label="FiniteMap.elemFM wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.52/5.11	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];
12.52/5.11	78 -> 6[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	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];
12.52/5.11	79 -> 7[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	6[label="FiniteMap.elemFM0 (FiniteMap.lookupFM FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.52/5.11	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];
12.52/5.11	8[label="FiniteMap.elemFM0 (FiniteMap.lookupFM4 FiniteMap.EmptyFM wx3)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.52/5.11	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];
12.52/5.11	10[label="FiniteMap.elemFM0 Nothing",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	80 -> 16[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	81[label="wx3/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 81[label="",style="solid", color="burlywood", weight=9];
12.52/5.11	81 -> 17[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	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];
12.52/5.11	82 -> 18[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	83[label="wx40/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 83[label="",style="solid", color="burlywood", weight=9];
12.52/5.11	83 -> 19[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	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];
12.52/5.11	84 -> 20[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	85[label="wx40/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 85[label="",style="solid", color="burlywood", weight=9];
12.52/5.11	85 -> 21[label="",style="solid", color="burlywood", weight=3];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	43 -> 5[label="",style="dashed", color="red", weight=0];
12.52/5.11	43[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx43 False)",fontsize=16,color="magenta"];43 -> 47[label="",style="dashed", color="magenta", weight=3];
12.52/5.11	43 -> 48[label="",style="dashed", color="magenta", weight=3];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	47[label="False",fontsize=16,color="green",shape="box"];48[label="wx43",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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	66[label="FiniteMap.elemFM0 (Just wx41)",fontsize=16,color="black",shape="triangle"];66 -> 69[label="",style="solid", color="black", weight=3];
12.52/5.11	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];
12.52/5.11	68 -> 66[label="",style="dashed", color="red", weight=0];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	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];
12.52/5.11	75 -> 5[label="",style="dashed", color="red", weight=0];
12.52/5.11	75[label="FiniteMap.elemFM0 (FiniteMap.lookupFM wx44 True)",fontsize=16,color="magenta"];75 -> 76[label="",style="dashed", color="magenta", weight=3];
12.52/5.11	75 -> 77[label="",style="dashed", color="magenta", weight=3];
12.52/5.11	76[label="True",fontsize=16,color="green",shape="box"];77[label="wx44",fontsize=16,color="green",shape="box"];}
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(8)
12.52/5.11	Obligation:
12.52/5.11	Q DP problem:
12.52/5.11	The TRS P consists of the following rules:
12.52/5.11	
12.52/5.11	   new_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, h)
12.52/5.11	   new_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)
12.52/5.11	
12.52/5.11	R is empty.
12.52/5.11	Q is empty.
12.52/5.11	We have to consider all minimal (P,Q,R)-chains.
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(9) DependencyGraphProof (EQUIVALENT)
12.52/5.11	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(10)
12.52/5.11	Complex Obligation (AND)
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(11)
12.52/5.11	Obligation:
12.52/5.11	Q DP problem:
12.52/5.11	The TRS P consists of the following rules:
12.52/5.11	
12.52/5.11	   new_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)
12.52/5.11	
12.52/5.11	R is empty.
12.52/5.11	Q is empty.
12.52/5.11	We have to consider all minimal (P,Q,R)-chains.
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(12) QDPSizeChangeProof (EQUIVALENT)
12.52/5.11	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. 
12.52/5.11	
12.52/5.11	From the DPs we obtained the following set of size-change graphs:
12.52/5.11	*new_elemFM0(Branch(True, wx41, wx42, wx43, wx44), False, h) -> new_elemFM0(wx43, False, h)
12.52/5.11	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.52/5.11	
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(13)
12.52/5.11	YES
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(14)
12.52/5.11	Obligation:
12.52/5.11	Q DP problem:
12.52/5.11	The TRS P consists of the following rules:
12.52/5.11	
12.52/5.11	   new_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, h)
12.52/5.11	
12.52/5.11	R is empty.
12.52/5.11	Q is empty.
12.52/5.11	We have to consider all minimal (P,Q,R)-chains.
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(15) QDPSizeChangeProof (EQUIVALENT)
12.52/5.11	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. 
12.52/5.11	
12.52/5.11	From the DPs we obtained the following set of size-change graphs:
12.52/5.11	*new_elemFM0(Branch(False, wx41, wx42, wx43, wx44), True, h) -> new_elemFM0(wx44, True, h)
12.52/5.11	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.52/5.11	
12.52/5.11	
12.52/5.11	----------------------------------------
12.52/5.11	
12.52/5.11	(16)
12.52/5.11	YES
12.56/8.05	EOF