10.34/4.43	YES
12.58/4.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.58/4.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.58/4.98	
12.58/4.98	
12.58/4.98	H-Termination with start terms of the given HASKELL could be proven:
12.58/4.98	
12.58/4.98	(0) HASKELL
12.58/4.98	(1) IFR [EQUIVALENT, 0 ms]
12.58/4.98	(2) HASKELL
12.58/4.98	(3) BR [EQUIVALENT, 0 ms]
12.58/4.98	(4) HASKELL
12.58/4.98	(5) COR [EQUIVALENT, 28 ms]
12.58/4.98	(6) HASKELL
12.58/4.98	(7) Narrow [SOUND, 0 ms]
12.58/4.98	(8) AND
12.58/4.98	    (9) QDP
12.58/4.98	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.58/4.98	        (11) YES
12.58/4.98	    (12) QDP
12.58/4.98	        (13) DependencyGraphProof [EQUIVALENT, 0 ms]
12.58/4.98	        (14) AND
12.58/4.98	            (15) QDP
12.58/4.98	                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.58/4.98	                (17) YES
12.58/4.98	            (18) QDP
12.58/4.98	                (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.58/4.98	                (20) YES
12.58/4.98	            (21) QDP
12.58/4.98	                (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.58/4.98	                (23) YES
12.58/4.98	
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(0)
12.58/4.98	Obligation:
12.58/4.98	mainModule Main 
12.58/4.98	module Maybe where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	module List where {
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	infix 5 \\;
12.58/4.98	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.58/4.98	(\\) = foldl (flip delete);
12.58/4.98	
12.58/4.98	delete :: Eq a => a  ->  [a]  ->  [a];
12.58/4.98	delete = deleteBy (==);
12.58/4.98	
12.58/4.98	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.58/4.98	deleteBy _ _ [] = [];
12.58/4.98	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.58/4.98	
12.58/4.98	}
12.58/4.98	module Main where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(1) IFR (EQUIVALENT)
12.58/4.98	If Reductions:
12.58/4.98	The following If expression
12.58/4.98	"if eq x y then ys else y : deleteBy eq x ys"
12.58/4.98	is transformed to
12.58/4.98	"deleteBy0 ys y eq x True = ys;
12.58/4.98	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.58/4.98	"
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(2)
12.58/4.98	Obligation:
12.58/4.98	mainModule Main 
12.58/4.98	module Maybe where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	module List where {
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	infix 5 \\;
12.58/4.98	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.58/4.98	(\\) = foldl (flip delete);
12.58/4.98	
12.58/4.98	delete :: Eq a => a  ->  [a]  ->  [a];
12.58/4.98	delete = deleteBy (==);
12.58/4.98	
12.58/4.98	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.58/4.98	deleteBy _ _ [] = [];
12.58/4.98	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.58/4.98	
12.58/4.98	deleteBy0 ys y eq x True = ys;
12.58/4.98	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.58/4.98	
12.58/4.98	}
12.58/4.98	module Main where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(3) BR (EQUIVALENT)
12.58/4.98	Replaced joker patterns by fresh variables and removed binding patterns.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(4)
12.58/4.98	Obligation:
12.58/4.98	mainModule Main 
12.58/4.98	module Maybe where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	module List where {
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	infix 5 \\;
12.58/4.98	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.58/4.98	(\\) = foldl (flip delete);
12.58/4.98	
12.58/4.98	delete :: Eq a => a  ->  [a]  ->  [a];
12.58/4.98	delete = deleteBy (==);
12.58/4.98	
12.58/4.98	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.58/4.98	deleteBy vy vz [] = [];
12.58/4.98	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.58/4.98	
12.58/4.98	deleteBy0 ys y eq x True = ys;
12.58/4.98	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.58/4.98	
12.58/4.98	}
12.58/4.98	module Main where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(5) COR (EQUIVALENT)
12.58/4.98	Cond Reductions:
12.58/4.98	The following Function with conditions
12.58/4.98	"undefined |Falseundefined;
12.58/4.98	"
12.58/4.98	is transformed to
12.58/4.98	"undefined  = undefined1;
12.58/4.98	"
12.58/4.98	"undefined0 True = undefined;
12.58/4.98	"
12.58/4.98	"undefined1  = undefined0 False;
12.58/4.98	"
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(6)
12.58/4.98	Obligation:
12.58/4.98	mainModule Main 
12.58/4.98	module Maybe where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	module List where {
12.58/4.98	import qualified Main;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	infix 5 \\;
12.58/4.98	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.58/4.98	(\\) = foldl (flip delete);
12.58/4.98	
12.58/4.98	delete :: Eq a => a  ->  [a]  ->  [a];
12.58/4.98	delete = deleteBy (==);
12.58/4.98	
12.58/4.98	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.58/4.98	deleteBy vy vz [] = [];
12.58/4.98	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.58/4.98	
12.58/4.98	deleteBy0 ys y eq x True = ys;
12.58/4.98	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.58/4.98	
12.58/4.98	}
12.58/4.98	module Main where {
12.58/4.98	import qualified List;
12.58/4.98	import qualified Maybe;
12.58/4.98	import qualified Prelude;
12.58/4.98	}
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(7) Narrow (SOUND)
12.58/4.98	Haskell To QDPs
12.58/4.98	
12.58/4.98	digraph dp_graph {
12.58/4.98	node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.58/4.98	3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.58/4.98	4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.58/4.98	5[label="foldl (flip List.delete) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];66[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 66[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	66 -> 6[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	67[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 67[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	67 -> 7[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.58/4.98	7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.58/4.98	8 -> 5[label="",style="dashed", color="red", weight=0];
12.58/4.98	8[label="foldl (flip List.delete) (flip List.delete wu3 wu40) wu41",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	8 -> 11[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	9[label="wu3",fontsize=16,color="green",shape="box"];10[label="flip List.delete wu3 wu40",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
12.58/4.98	11[label="wu41",fontsize=16,color="green",shape="box"];12[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.58/4.98	13[label="List.deleteBy (==) wu40 wu3",fontsize=16,color="burlywood",shape="triangle"];68[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];13 -> 68[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	68 -> 14[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	69[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 69[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	69 -> 15[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.58/4.98	15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.58/4.98	16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="burlywood",shape="box"];70[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 70[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	70 -> 18[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	71[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 71[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	71 -> 19[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	72[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 72[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	72 -> 20[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	17[label="[]",fontsize=16,color="green",shape="box"];18[label="List.deleteBy0 wu31 wu30 (==) LT ((==) LT wu30)",fontsize=16,color="burlywood",shape="box"];73[label="wu30/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 73[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	73 -> 21[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	74[label="wu30/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 74[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	74 -> 22[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	75[label="wu30/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 75[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	75 -> 23[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	19[label="List.deleteBy0 wu31 wu30 (==) EQ ((==) EQ wu30)",fontsize=16,color="burlywood",shape="box"];76[label="wu30/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 76[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	76 -> 24[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	77[label="wu30/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 77[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	77 -> 25[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	78[label="wu30/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 78[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	78 -> 26[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	20[label="List.deleteBy0 wu31 wu30 (==) GT ((==) GT wu30)",fontsize=16,color="burlywood",shape="box"];79[label="wu30/LT",fontsize=10,color="white",style="solid",shape="box"];20 -> 79[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	79 -> 27[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	80[label="wu30/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 80[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	80 -> 28[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	81[label="wu30/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 81[label="",style="solid", color="burlywood", weight=9];
12.58/4.98	81 -> 29[label="",style="solid", color="burlywood", weight=3];
12.58/4.98	21[label="List.deleteBy0 wu31 LT (==) LT ((==) LT LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
12.58/4.98	22[label="List.deleteBy0 wu31 EQ (==) LT ((==) LT EQ)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
12.58/4.98	23[label="List.deleteBy0 wu31 GT (==) LT ((==) LT GT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
12.58/4.98	24[label="List.deleteBy0 wu31 LT (==) EQ ((==) EQ LT)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
12.58/4.98	25[label="List.deleteBy0 wu31 EQ (==) EQ ((==) EQ EQ)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
12.58/4.98	26[label="List.deleteBy0 wu31 GT (==) EQ ((==) EQ GT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
12.58/4.98	27[label="List.deleteBy0 wu31 LT (==) GT ((==) GT LT)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
12.58/4.98	28[label="List.deleteBy0 wu31 EQ (==) GT ((==) GT EQ)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
12.58/4.98	29[label="List.deleteBy0 wu31 GT (==) GT ((==) GT GT)",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
12.58/4.98	30[label="List.deleteBy0 wu31 LT (==) LT True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
12.58/4.98	31[label="List.deleteBy0 wu31 EQ (==) LT False",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
12.58/4.98	32[label="List.deleteBy0 wu31 GT (==) LT False",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
12.58/4.98	33[label="List.deleteBy0 wu31 LT (==) EQ False",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
12.58/4.98	34[label="List.deleteBy0 wu31 EQ (==) EQ True",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
12.58/4.98	35[label="List.deleteBy0 wu31 GT (==) EQ False",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
12.58/4.98	36[label="List.deleteBy0 wu31 LT (==) GT False",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
12.58/4.98	37[label="List.deleteBy0 wu31 EQ (==) GT False",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
12.58/4.98	38[label="List.deleteBy0 wu31 GT (==) GT True",fontsize=16,color="black",shape="box"];38 -> 47[label="",style="solid", color="black", weight=3];
12.58/4.98	39[label="wu31",fontsize=16,color="green",shape="box"];40[label="EQ : List.deleteBy (==) LT wu31",fontsize=16,color="green",shape="box"];40 -> 48[label="",style="dashed", color="green", weight=3];
12.58/4.98	41[label="GT : List.deleteBy (==) LT wu31",fontsize=16,color="green",shape="box"];41 -> 49[label="",style="dashed", color="green", weight=3];
12.58/4.98	42[label="LT : List.deleteBy (==) EQ wu31",fontsize=16,color="green",shape="box"];42 -> 50[label="",style="dashed", color="green", weight=3];
12.58/4.98	43[label="wu31",fontsize=16,color="green",shape="box"];44[label="GT : List.deleteBy (==) EQ wu31",fontsize=16,color="green",shape="box"];44 -> 51[label="",style="dashed", color="green", weight=3];
12.58/4.98	45[label="LT : List.deleteBy (==) GT wu31",fontsize=16,color="green",shape="box"];45 -> 52[label="",style="dashed", color="green", weight=3];
12.58/4.98	46[label="EQ : List.deleteBy (==) GT wu31",fontsize=16,color="green",shape="box"];46 -> 53[label="",style="dashed", color="green", weight=3];
12.58/4.98	47[label="wu31",fontsize=16,color="green",shape="box"];48 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	48[label="List.deleteBy (==) LT wu31",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	48 -> 55[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	49 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	49[label="List.deleteBy (==) LT wu31",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	49 -> 57[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	50 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	50[label="List.deleteBy (==) EQ wu31",fontsize=16,color="magenta"];50 -> 58[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	50 -> 59[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	51 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	51[label="List.deleteBy (==) EQ wu31",fontsize=16,color="magenta"];51 -> 60[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	51 -> 61[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	52 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	52[label="List.deleteBy (==) GT wu31",fontsize=16,color="magenta"];52 -> 62[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	52 -> 63[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	53 -> 13[label="",style="dashed", color="red", weight=0];
12.58/4.98	53[label="List.deleteBy (==) GT wu31",fontsize=16,color="magenta"];53 -> 64[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	53 -> 65[label="",style="dashed", color="magenta", weight=3];
12.58/4.98	54[label="wu31",fontsize=16,color="green",shape="box"];55[label="LT",fontsize=16,color="green",shape="box"];56[label="wu31",fontsize=16,color="green",shape="box"];57[label="LT",fontsize=16,color="green",shape="box"];58[label="wu31",fontsize=16,color="green",shape="box"];59[label="EQ",fontsize=16,color="green",shape="box"];60[label="wu31",fontsize=16,color="green",shape="box"];61[label="EQ",fontsize=16,color="green",shape="box"];62[label="wu31",fontsize=16,color="green",shape="box"];63[label="GT",fontsize=16,color="green",shape="box"];64[label="wu31",fontsize=16,color="green",shape="box"];65[label="GT",fontsize=16,color="green",shape="box"];}
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(8)
12.58/4.98	Complex Obligation (AND)
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(9)
12.58/4.98	Obligation:
12.58/4.98	Q DP problem:
12.58/4.98	The TRS P consists of the following rules:
12.58/4.98	
12.58/4.98	   new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)
12.58/4.98	
12.58/4.98	The TRS R consists of the following rules:
12.58/4.98	
12.58/4.98	   new_deleteBy0(wu40, []) -> []
12.58/4.98	   new_deleteBy0(LT, :(EQ, wu31)) -> :(EQ, new_deleteBy0(LT, wu31))
12.58/4.98	   new_deleteBy0(LT, :(LT, wu31)) -> wu31
12.58/4.98	   new_deleteBy0(EQ, :(EQ, wu31)) -> wu31
12.58/4.98	   new_deleteBy0(GT, :(LT, wu31)) -> :(LT, new_deleteBy0(GT, wu31))
12.58/4.98	   new_deleteBy0(LT, :(GT, wu31)) -> :(GT, new_deleteBy0(LT, wu31))
12.58/4.98	   new_deleteBy0(EQ, :(LT, wu31)) -> :(LT, new_deleteBy0(EQ, wu31))
12.58/4.98	   new_deleteBy0(GT, :(GT, wu31)) -> wu31
12.58/4.98	   new_deleteBy0(EQ, :(GT, wu31)) -> :(GT, new_deleteBy0(EQ, wu31))
12.58/4.98	   new_deleteBy0(GT, :(EQ, wu31)) -> :(EQ, new_deleteBy0(GT, wu31))
12.58/4.98	
12.58/4.98	The set Q consists of the following terms:
12.58/4.98	
12.58/4.98	   new_deleteBy0(GT, :(LT, x0))
12.58/4.98	   new_deleteBy0(EQ, :(GT, x0))
12.58/4.98	   new_deleteBy0(x0, [])
12.58/4.98	   new_deleteBy0(LT, :(GT, x0))
12.58/4.98	   new_deleteBy0(GT, :(EQ, x0))
12.58/4.98	   new_deleteBy0(GT, :(GT, x0))
12.58/4.98	   new_deleteBy0(LT, :(LT, x0))
12.58/4.98	   new_deleteBy0(EQ, :(EQ, x0))
12.58/4.98	   new_deleteBy0(EQ, :(LT, x0))
12.58/4.98	   new_deleteBy0(LT, :(EQ, x0))
12.58/4.98	
12.58/4.98	We have to consider all minimal (P,Q,R)-chains.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(10) QDPSizeChangeProof (EQUIVALENT)
12.58/4.98	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. 
12.58/4.98	
12.58/4.98	From the DPs we obtained the following set of size-change graphs:
12.58/4.98	*new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)
12.58/4.98	The graph contains the following edges 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(11)
12.58/4.98	YES
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(12)
12.58/4.98	Obligation:
12.58/4.98	Q DP problem:
12.58/4.98	The TRS P consists of the following rules:
12.58/4.98	
12.58/4.98	   new_deleteBy(GT, :(EQ, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	   new_deleteBy(EQ, :(GT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	   new_deleteBy(EQ, :(LT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	   new_deleteBy(LT, :(GT, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	   new_deleteBy(LT, :(EQ, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	   new_deleteBy(GT, :(LT, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	
12.58/4.98	R is empty.
12.58/4.98	Q is empty.
12.58/4.98	We have to consider all minimal (P,Q,R)-chains.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(13) DependencyGraphProof (EQUIVALENT)
12.58/4.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(14)
12.58/4.98	Complex Obligation (AND)
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(15)
12.58/4.98	Obligation:
12.58/4.98	Q DP problem:
12.58/4.98	The TRS P consists of the following rules:
12.58/4.98	
12.58/4.98	   new_deleteBy(LT, :(EQ, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	   new_deleteBy(LT, :(GT, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	
12.58/4.98	R is empty.
12.58/4.98	Q is empty.
12.58/4.98	We have to consider all minimal (P,Q,R)-chains.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(16) QDPSizeChangeProof (EQUIVALENT)
12.58/4.98	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. 
12.58/4.98	
12.58/4.98	From the DPs we obtained the following set of size-change graphs:
12.58/4.98	*new_deleteBy(LT, :(EQ, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	*new_deleteBy(LT, :(GT, wu31)) -> new_deleteBy(LT, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(17)
12.58/4.98	YES
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(18)
12.58/4.98	Obligation:
12.58/4.98	Q DP problem:
12.58/4.98	The TRS P consists of the following rules:
12.58/4.98	
12.58/4.98	   new_deleteBy(EQ, :(LT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	   new_deleteBy(EQ, :(GT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	
12.58/4.98	R is empty.
12.58/4.98	Q is empty.
12.58/4.98	We have to consider all minimal (P,Q,R)-chains.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(19) QDPSizeChangeProof (EQUIVALENT)
12.58/4.98	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. 
12.58/4.98	
12.58/4.98	From the DPs we obtained the following set of size-change graphs:
12.58/4.98	*new_deleteBy(EQ, :(LT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	*new_deleteBy(EQ, :(GT, wu31)) -> new_deleteBy(EQ, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(20)
12.58/4.98	YES
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(21)
12.58/4.98	Obligation:
12.58/4.98	Q DP problem:
12.58/4.98	The TRS P consists of the following rules:
12.58/4.98	
12.58/4.98	   new_deleteBy(GT, :(LT, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	   new_deleteBy(GT, :(EQ, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	
12.58/4.98	R is empty.
12.58/4.98	Q is empty.
12.58/4.98	We have to consider all minimal (P,Q,R)-chains.
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(22) QDPSizeChangeProof (EQUIVALENT)
12.58/4.98	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. 
12.58/4.98	
12.58/4.98	From the DPs we obtained the following set of size-change graphs:
12.58/4.98	*new_deleteBy(GT, :(LT, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	*new_deleteBy(GT, :(EQ, wu31)) -> new_deleteBy(GT, wu31)
12.58/4.98	The graph contains the following edges 1 >= 1, 2 > 2
12.58/4.98	
12.58/4.98	
12.58/4.98	----------------------------------------
12.58/4.98	
12.58/4.98	(23)
12.58/4.98	YES
12.58/5.01	EOF