10.86/4.51	YES
13.04/5.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
13.04/5.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.04/5.07	
13.04/5.07	
13.04/5.07	H-Termination with start terms of the given HASKELL could be proven:
13.04/5.07	
13.04/5.07	(0) HASKELL
13.04/5.07	(1) CR [EQUIVALENT, 0 ms]
13.04/5.07	(2) HASKELL
13.04/5.07	(3) BR [EQUIVALENT, 0 ms]
13.04/5.07	(4) HASKELL
13.04/5.07	(5) COR [EQUIVALENT, 26 ms]
13.04/5.07	(6) HASKELL
13.04/5.07	(7) Narrow [SOUND, 0 ms]
13.04/5.07	(8) QDP
13.04/5.07	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
13.04/5.07	(10) AND
13.04/5.07	    (11) QDP
13.04/5.07	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.04/5.07	        (13) YES
13.04/5.07	    (14) QDP
13.04/5.07	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.04/5.07	        (16) YES
13.04/5.07	
13.04/5.07	
13.04/5.07	----------------------------------------
13.04/5.07	
13.04/5.07	(0)
13.04/5.07	Obligation:
13.04/5.07	mainModule Main 
13.04/5.07	module Maybe where {
13.04/5.07	import qualified List;
13.04/5.07	import qualified Main;
13.04/5.07	import qualified Prelude;
13.04/5.07	}
13.04/5.07	module List where {
13.04/5.07	import qualified Main;
13.04/5.07	import qualified Maybe;
13.04/5.07	import qualified Prelude;
13.04/5.07	insert :: Ord a => a  ->  [a]  ->  [a];
13.04/5.07	insert e ls = insertBy compare e ls;
13.04/5.07	
13.04/5.07	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
13.04/5.07	insertBy _ x [] = x : [];
13.04/5.08	insertBy cmp x ys@(y : ys') = case cmp x y of { 
13.04/5.08	GT-> y : insertBy cmp x ys'; 
13.04/5.08	_-> x : ys; 
13.04/5.08	 } ;
13.04/5.08	
13.04/5.08	}
13.04/5.08	module Main where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(1) CR (EQUIVALENT)
13.04/5.08	Case Reductions:
13.04/5.08	The following Case expression
13.04/5.08	"case cmp x y of {
13.04/5.08	GT  -> y : insertBy cmp x ys';
13.04/5.08	_  -> x : ys}
13.04/5.08	"
13.04/5.08	is transformed to
13.04/5.08	"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
13.04/5.08	insertBy0 y cmp x ys' ys _ = x : ys;
13.04/5.08	"
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(2)
13.04/5.08	Obligation:
13.04/5.08	mainModule Main 
13.04/5.08	module Maybe where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	module List where {
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	insert :: Ord a => a  ->  [a]  ->  [a];
13.04/5.08	insert e ls = insertBy compare e ls;
13.04/5.08	
13.04/5.08	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
13.04/5.08	insertBy _ x [] = x : [];
13.04/5.08	insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);
13.04/5.08	
13.04/5.08	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
13.04/5.08	insertBy0 y cmp x ys' ys _ = x : ys;
13.04/5.08	
13.04/5.08	}
13.04/5.08	module Main where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(3) BR (EQUIVALENT)
13.04/5.08	Replaced joker patterns by fresh variables and removed binding patterns.
13.04/5.08	
13.04/5.08	Binding Reductions:
13.04/5.08	The bind variable of the following binding Pattern
13.04/5.08	"ys@(wu : wv)"
13.04/5.08	is replaced by the following term
13.04/5.08	"wu : wv"
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(4)
13.04/5.08	Obligation:
13.04/5.08	mainModule Main 
13.04/5.08	module Maybe where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	module List where {
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	insert :: Ord a => a  ->  [a]  ->  [a];
13.04/5.08	insert e ls = insertBy compare e ls;
13.04/5.08	
13.04/5.08	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
13.04/5.08	insertBy vz x [] = x : [];
13.04/5.08	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
13.04/5.08	
13.04/5.08	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
13.04/5.08	insertBy0 y cmp x ys' ys vy = x : ys;
13.04/5.08	
13.04/5.08	}
13.04/5.08	module Main where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(5) COR (EQUIVALENT)
13.04/5.08	Cond Reductions:
13.04/5.08	The following Function with conditions
13.04/5.08	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
13.04/5.08	"
13.04/5.08	is transformed to
13.04/5.08	"compare x y = compare3 x y;
13.04/5.08	"
13.04/5.08	"compare0 x y True = GT;
13.04/5.08	"
13.04/5.08	"compare1 x y True = LT;
13.04/5.08	compare1 x y False = compare0 x y otherwise;
13.04/5.08	"
13.04/5.08	"compare2 x y True = EQ;
13.04/5.08	compare2 x y False = compare1 x y (x <= y);
13.04/5.08	"
13.04/5.08	"compare3 x y = compare2 x y (x == y);
13.04/5.08	"
13.04/5.08	The following Function with conditions
13.04/5.08	"undefined |Falseundefined;
13.04/5.08	"
13.04/5.08	is transformed to
13.04/5.08	"undefined  = undefined1;
13.04/5.08	"
13.04/5.08	"undefined0 True = undefined;
13.04/5.08	"
13.04/5.08	"undefined1  = undefined0 False;
13.04/5.08	"
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(6)
13.04/5.08	Obligation:
13.04/5.08	mainModule Main 
13.04/5.08	module Maybe where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	module List where {
13.04/5.08	import qualified Main;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	insert :: Ord a => a  ->  [a]  ->  [a];
13.04/5.08	insert e ls = insertBy compare e ls;
13.04/5.08	
13.04/5.08	insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
13.04/5.08	insertBy vz x [] = x : [];
13.04/5.08	insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);
13.04/5.08	
13.04/5.08	insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
13.04/5.08	insertBy0 y cmp x ys' ys vy = x : ys;
13.04/5.08	
13.04/5.08	}
13.04/5.08	module Main where {
13.04/5.08	import qualified List;
13.04/5.08	import qualified Maybe;
13.04/5.08	import qualified Prelude;
13.04/5.08	}
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(7) Narrow (SOUND)
13.04/5.08	Haskell To QDPs
13.04/5.08	
13.04/5.08	digraph dp_graph {
13.04/5.08	node [outthreshold=100, inthreshold=100];1[label="List.insert",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
13.04/5.08	3[label="List.insert ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
13.04/5.08	4[label="List.insert ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
13.04/5.08	5[label="List.insertBy compare ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];78[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 78[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	78 -> 6[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	79[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 79[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	79 -> 7[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	6[label="List.insertBy compare ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
13.04/5.08	7[label="List.insertBy compare ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
13.04/5.08	8[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
13.04/5.08	9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare3 ww3 ww40)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
13.04/5.08	11[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare2 ww3 ww40 (ww3 == ww40))",fontsize=16,color="burlywood",shape="box"];80[label="ww3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 80[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	80 -> 12[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	81[label="ww3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	81 -> 13[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	82[label="ww3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 82[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	82 -> 14[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	12[label="List.insertBy0 ww40 compare LT ww41 (ww40 : ww41) (compare2 LT ww40 (LT == ww40))",fontsize=16,color="burlywood",shape="box"];83[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	83 -> 15[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	84[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 84[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	84 -> 16[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	85[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 85[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	85 -> 17[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	13[label="List.insertBy0 ww40 compare EQ ww41 (ww40 : ww41) (compare2 EQ ww40 (EQ == ww40))",fontsize=16,color="burlywood",shape="box"];86[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	86 -> 18[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	87[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 87[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	87 -> 19[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	88[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 88[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	88 -> 20[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	14[label="List.insertBy0 ww40 compare GT ww41 (ww40 : ww41) (compare2 GT ww40 (GT == ww40))",fontsize=16,color="burlywood",shape="box"];89[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 89[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	89 -> 21[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	90[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 90[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	90 -> 22[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	91[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 91[label="",style="solid", color="burlywood", weight=9];
13.04/5.08	91 -> 23[label="",style="solid", color="burlywood", weight=3];
13.04/5.08	15[label="List.insertBy0 LT compare LT ww41 (LT : ww41) (compare2 LT LT (LT == LT))",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
13.04/5.08	16[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare2 LT EQ (LT == EQ))",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
13.04/5.08	17[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare2 LT GT (LT == GT))",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
13.04/5.08	18[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare2 EQ LT (EQ == LT))",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
13.04/5.08	19[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) (compare2 EQ EQ (EQ == EQ))",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
13.04/5.08	20[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare2 EQ GT (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
13.04/5.08	21[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare2 GT LT (GT == LT))",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
13.04/5.08	22[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare2 GT EQ (GT == EQ))",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
13.04/5.08	23[label="List.insertBy0 GT compare GT ww41 (GT : ww41) (compare2 GT GT (GT == GT))",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
13.04/5.08	24[label="List.insertBy0 LT compare LT ww41 (LT : ww41) (compare2 LT LT True)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
13.04/5.08	25[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare2 LT EQ False)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
13.04/5.08	26[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare2 LT GT False)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
13.04/5.08	27[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare2 EQ LT False)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
13.04/5.08	28[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) (compare2 EQ EQ True)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
13.04/5.08	29[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare2 EQ GT False)",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
13.04/5.08	30[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare2 GT LT False)",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
13.04/5.08	31[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare2 GT EQ False)",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
13.04/5.08	32[label="List.insertBy0 GT compare GT ww41 (GT : ww41) (compare2 GT GT True)",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
13.04/5.08	33[label="List.insertBy0 LT compare LT ww41 (LT : ww41) EQ",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
13.04/5.08	34[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare1 LT EQ (LT <= EQ))",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
13.04/5.08	35[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare1 LT GT (LT <= GT))",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
13.04/5.08	36[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare1 EQ LT (EQ <= LT))",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
13.04/5.08	37[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) EQ",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
13.04/5.08	38[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare1 EQ GT (EQ <= GT))",fontsize=16,color="black",shape="box"];38 -> 47[label="",style="solid", color="black", weight=3];
13.04/5.08	39[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare1 GT LT (GT <= LT))",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3];
13.04/5.08	40[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare1 GT EQ (GT <= EQ))",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3];
13.04/5.08	41[label="List.insertBy0 GT compare GT ww41 (GT : ww41) EQ",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3];
13.04/5.08	42[label="LT : LT : ww41",fontsize=16,color="green",shape="box"];43[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare1 LT EQ True)",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
13.04/5.08	44[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare1 LT GT True)",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
13.04/5.08	45[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare1 EQ LT False)",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
13.04/5.08	46[label="EQ : EQ : ww41",fontsize=16,color="green",shape="box"];47[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare1 EQ GT True)",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3];
13.04/5.08	48[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare1 GT LT False)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
13.04/5.08	49[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare1 GT EQ False)",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3];
13.04/5.08	50[label="GT : GT : ww41",fontsize=16,color="green",shape="box"];51[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) LT",fontsize=16,color="black",shape="box"];51 -> 57[label="",style="solid", color="black", weight=3];
13.04/5.08	52[label="List.insertBy0 GT compare LT ww41 (GT : ww41) LT",fontsize=16,color="black",shape="box"];52 -> 58[label="",style="solid", color="black", weight=3];
13.04/5.08	53[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare0 EQ LT otherwise)",fontsize=16,color="black",shape="box"];53 -> 59[label="",style="solid", color="black", weight=3];
13.04/5.08	54[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) LT",fontsize=16,color="black",shape="box"];54 -> 60[label="",style="solid", color="black", weight=3];
13.04/5.08	55[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare0 GT LT otherwise)",fontsize=16,color="black",shape="box"];55 -> 61[label="",style="solid", color="black", weight=3];
13.04/5.08	56[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare0 GT EQ otherwise)",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3];
13.04/5.08	57[label="LT : EQ : ww41",fontsize=16,color="green",shape="box"];58[label="LT : GT : ww41",fontsize=16,color="green",shape="box"];59[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare0 EQ LT True)",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3];
13.04/5.08	60[label="EQ : GT : ww41",fontsize=16,color="green",shape="box"];61[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare0 GT LT True)",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
13.04/5.08	62[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare0 GT EQ True)",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
13.04/5.08	63[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) GT",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
13.04/5.08	64[label="List.insertBy0 LT compare GT ww41 (LT : ww41) GT",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3];
13.04/5.08	65[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) GT",fontsize=16,color="black",shape="box"];65 -> 68[label="",style="solid", color="black", weight=3];
13.04/5.08	66[label="LT : List.insertBy compare EQ ww41",fontsize=16,color="green",shape="box"];66 -> 69[label="",style="dashed", color="green", weight=3];
13.04/5.08	67[label="LT : List.insertBy compare GT ww41",fontsize=16,color="green",shape="box"];67 -> 70[label="",style="dashed", color="green", weight=3];
13.04/5.08	68[label="EQ : List.insertBy compare GT ww41",fontsize=16,color="green",shape="box"];68 -> 71[label="",style="dashed", color="green", weight=3];
13.04/5.08	69 -> 5[label="",style="dashed", color="red", weight=0];
13.04/5.08	69[label="List.insertBy compare EQ ww41",fontsize=16,color="magenta"];69 -> 72[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	69 -> 73[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	70 -> 5[label="",style="dashed", color="red", weight=0];
13.04/5.08	70[label="List.insertBy compare GT ww41",fontsize=16,color="magenta"];70 -> 74[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	70 -> 75[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	71 -> 5[label="",style="dashed", color="red", weight=0];
13.04/5.08	71[label="List.insertBy compare GT ww41",fontsize=16,color="magenta"];71 -> 76[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	71 -> 77[label="",style="dashed", color="magenta", weight=3];
13.04/5.08	72[label="ww41",fontsize=16,color="green",shape="box"];73[label="EQ",fontsize=16,color="green",shape="box"];74[label="ww41",fontsize=16,color="green",shape="box"];75[label="GT",fontsize=16,color="green",shape="box"];76[label="ww41",fontsize=16,color="green",shape="box"];77[label="GT",fontsize=16,color="green",shape="box"];}
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(8)
13.04/5.08	Obligation:
13.04/5.08	Q DP problem:
13.04/5.08	The TRS P consists of the following rules:
13.04/5.08	
13.04/5.08	   new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	   new_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	   new_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)
13.04/5.08	
13.04/5.08	R is empty.
13.04/5.08	Q is empty.
13.04/5.08	We have to consider all minimal (P,Q,R)-chains.
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(9) DependencyGraphProof (EQUIVALENT)
13.04/5.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(10)
13.04/5.08	Complex Obligation (AND)
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(11)
13.04/5.08	Obligation:
13.04/5.08	Q DP problem:
13.04/5.08	The TRS P consists of the following rules:
13.04/5.08	
13.04/5.08	   new_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)
13.04/5.08	
13.04/5.08	R is empty.
13.04/5.08	Q is empty.
13.04/5.08	We have to consider all minimal (P,Q,R)-chains.
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(12) QDPSizeChangeProof (EQUIVALENT)
13.04/5.08	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. 
13.04/5.08	
13.04/5.08	From the DPs we obtained the following set of size-change graphs:
13.04/5.08	*new_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)
13.04/5.08	The graph contains the following edges 1 >= 1, 2 > 2
13.04/5.08	
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(13)
13.04/5.08	YES
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(14)
13.04/5.08	Obligation:
13.04/5.08	Q DP problem:
13.04/5.08	The TRS P consists of the following rules:
13.04/5.08	
13.04/5.08	   new_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	   new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	
13.04/5.08	R is empty.
13.04/5.08	Q is empty.
13.04/5.08	We have to consider all minimal (P,Q,R)-chains.
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(15) QDPSizeChangeProof (EQUIVALENT)
13.04/5.08	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. 
13.04/5.08	
13.04/5.08	From the DPs we obtained the following set of size-change graphs:
13.04/5.08	*new_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	The graph contains the following edges 1 >= 1, 2 > 2
13.04/5.08	
13.04/5.08	
13.04/5.08	*new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)
13.04/5.08	The graph contains the following edges 1 >= 1, 2 > 2
13.04/5.08	
13.04/5.08	
13.04/5.08	----------------------------------------
13.04/5.08	
13.04/5.08	(16)
13.04/5.08	YES
13.24/6.46	EOF