11.37/4.58	YES
13.82/5.25	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
13.82/5.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.82/5.25	
13.82/5.25	
13.82/5.25	H-Termination with start terms of the given HASKELL could be proven:
13.82/5.25	
13.82/5.25	(0) HASKELL
13.82/5.25	(1) IFR [EQUIVALENT, 0 ms]
13.82/5.25	(2) HASKELL
13.82/5.25	(3) BR [EQUIVALENT, 0 ms]
13.82/5.25	(4) HASKELL
13.82/5.25	(5) COR [EQUIVALENT, 0 ms]
13.82/5.25	(6) HASKELL
13.82/5.25	(7) LetRed [EQUIVALENT, 0 ms]
13.82/5.25	(8) HASKELL
13.82/5.25	(9) Narrow [SOUND, 0 ms]
13.82/5.25	(10) AND
13.82/5.25	    (11) QDP
13.82/5.25	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.82/5.25	        (13) YES
13.82/5.25	    (14) QDP
13.82/5.25	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.82/5.25	        (16) YES
13.82/5.25	    (17) QDP
13.82/5.25	        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
13.82/5.25	        (19) QDP
13.82/5.25	        (20) TransformationProof [EQUIVALENT, 0 ms]
13.82/5.25	        (21) QDP
13.82/5.25	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.82/5.25	        (23) YES
13.82/5.25	    (24) QDP
13.82/5.25	        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.82/5.25	        (26) YES
13.82/5.25	
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(0)
13.82/5.25	Obligation:
13.82/5.25	mainModule Main 
13.82/5.25	module Maybe where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	module List where {
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
13.82/5.25	deleteBy _ _ [] = [];
13.82/5.25	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
13.82/5.25	
13.82/5.25	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
13.82/5.25	elem_by _ _ [] = False;
13.82/5.25	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
13.82/5.25	
13.82/5.25	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
13.82/5.25	nubBy eq l = nubBy' l [] where { 
13.82/5.25	nubBy' [] _ = [];
13.82/5.25	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
13.82/5.25	 | otherwise = y : nubBy' ys (y : xs);
13.82/5.25	 };
13.82/5.25	
13.82/5.25	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.82/5.25	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
13.82/5.25	
13.82/5.25	}
13.82/5.25	module Main where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(1) IFR (EQUIVALENT)
13.82/5.25	If Reductions:
13.82/5.25	The following If expression
13.82/5.25	"if eq x y then ys else y : deleteBy eq x ys"
13.82/5.25	is transformed to
13.82/5.25	"deleteBy0 ys y eq x True = ys;
13.82/5.25	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
13.82/5.25	"
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(2)
13.82/5.25	Obligation:
13.82/5.25	mainModule Main 
13.82/5.25	module Maybe where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	module List where {
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
13.82/5.25	deleteBy _ _ [] = [];
13.82/5.25	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
13.82/5.25	
13.82/5.25	deleteBy0 ys y eq x True = ys;
13.82/5.25	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
13.82/5.25	
13.82/5.25	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
13.82/5.25	elem_by _ _ [] = False;
13.82/5.25	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
13.82/5.25	
13.82/5.25	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
13.82/5.25	nubBy eq l = nubBy' l [] where { 
13.82/5.25	nubBy' [] _ = [];
13.82/5.25	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
13.82/5.25	 | otherwise = y : nubBy' ys (y : xs);
13.82/5.25	 };
13.82/5.25	
13.82/5.25	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.82/5.25	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
13.82/5.25	
13.82/5.25	}
13.82/5.25	module Main where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(3) BR (EQUIVALENT)
13.82/5.25	Replaced joker patterns by fresh variables and removed binding patterns.
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(4)
13.82/5.25	Obligation:
13.82/5.25	mainModule Main 
13.82/5.25	module Maybe where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	module List where {
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
13.82/5.25	deleteBy wv ww [] = [];
13.82/5.25	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
13.82/5.25	
13.82/5.25	deleteBy0 ys y eq x True = ys;
13.82/5.25	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
13.82/5.25	
13.82/5.25	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
13.82/5.25	elem_by vy vz [] = False;
13.82/5.25	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
13.82/5.25	
13.82/5.25	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
13.82/5.25	nubBy eq l = nubBy' l [] where { 
13.82/5.25	nubBy' [] wu = [];
13.82/5.25	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
13.82/5.25	 | otherwise = y : nubBy' ys (y : xs);
13.82/5.25	 };
13.82/5.25	
13.82/5.25	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.82/5.25	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
13.82/5.25	
13.82/5.25	}
13.82/5.25	module Main where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(5) COR (EQUIVALENT)
13.82/5.25	Cond Reductions:
13.82/5.25	The following Function with conditions
13.82/5.25	"undefined |Falseundefined;
13.82/5.25	"
13.82/5.25	is transformed to
13.82/5.25	"undefined  = undefined1;
13.82/5.25	"
13.82/5.25	"undefined0 True = undefined;
13.82/5.25	"
13.82/5.25	"undefined1  = undefined0 False;
13.82/5.25	"
13.82/5.25	The following Function with conditions
13.82/5.25	"nubBy' [] wu = [];
13.82/5.25	nubBy' (y : ys) xs|elem_by eq y xsnubBy' ys xs|otherwisey : nubBy' ys (y : xs);
13.82/5.25	"
13.82/5.25	is transformed to
13.82/5.25	"nubBy' [] wu = nubBy'3 [] wu;
13.82/5.25	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
13.82/5.25	"
13.82/5.25	"nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
13.82/5.25	"
13.82/5.25	"nubBy'1 y ys xs True = nubBy' ys xs;
13.82/5.25	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
13.82/5.25	"
13.82/5.25	"nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
13.82/5.25	"
13.82/5.25	"nubBy'3 [] wu = [];
13.82/5.25	nubBy'3 wz xu = nubBy'2 wz xu;
13.82/5.25	"
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(6)
13.82/5.25	Obligation:
13.82/5.25	mainModule Main 
13.82/5.25	module Maybe where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	module List where {
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
13.82/5.25	deleteBy wv ww [] = [];
13.82/5.25	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
13.82/5.25	
13.82/5.25	deleteBy0 ys y eq x True = ys;
13.82/5.25	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
13.82/5.25	
13.82/5.25	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
13.82/5.25	elem_by vy vz [] = False;
13.82/5.25	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
13.82/5.25	
13.82/5.25	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
13.82/5.25	nubBy eq l = nubBy' l [] where { 
13.82/5.25	nubBy' [] wu = nubBy'3 [] wu;
13.82/5.25	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
13.82/5.25	nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
13.82/5.25	nubBy'1 y ys xs True = nubBy' ys xs;
13.82/5.25	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
13.82/5.25	nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
13.82/5.25	nubBy'3 [] wu = [];
13.82/5.25	nubBy'3 wz xu = nubBy'2 wz xu;
13.82/5.25	 };
13.82/5.25	
13.82/5.25	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.82/5.25	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
13.82/5.25	
13.82/5.25	}
13.82/5.25	module Main where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(7) LetRed (EQUIVALENT)
13.82/5.25	Let/Where Reductions:
13.82/5.25	The bindings of the following Let/Where expression
13.82/5.25	"nubBy' l [] where {
13.82/5.25	nubBy' [] wu = nubBy'3 [] wu;
13.82/5.25	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
13.82/5.25	;
13.82/5.25	nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
13.82/5.25	;
13.82/5.25	nubBy'1 y ys xs True = nubBy' ys xs;
13.82/5.25	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
13.82/5.25	;
13.82/5.25	nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
13.82/5.25	;
13.82/5.25	nubBy'3 [] wu = [];
13.82/5.25	nubBy'3 wz xu = nubBy'2 wz xu;
13.82/5.25	} 
13.82/5.25	"
13.82/5.25	are unpacked to the following functions on top level
13.82/5.25	"nubByNubBy'3 xv [] wu = [];
13.82/5.25	nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
13.82/5.25	"
13.82/5.25	"nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
13.82/5.25	nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;
13.82/5.25	"
13.82/5.25	"nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);
13.82/5.25	"
13.82/5.25	"nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
13.82/5.25	nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;
13.82/5.25	"
13.82/5.25	"nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);
13.82/5.25	"
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(8)
13.82/5.25	Obligation:
13.82/5.25	mainModule Main 
13.82/5.25	module Maybe where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	module List where {
13.82/5.25	import qualified Main;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
13.82/5.25	deleteBy wv ww [] = [];
13.82/5.25	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
13.82/5.25	
13.82/5.25	deleteBy0 ys y eq x True = ys;
13.82/5.25	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
13.82/5.25	
13.82/5.25	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
13.82/5.25	elem_by vy vz [] = False;
13.82/5.25	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
13.82/5.25	
13.82/5.25	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
13.82/5.25	nubBy eq l = nubByNubBy' eq l [];
13.82/5.25	
13.82/5.25	nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
13.82/5.25	nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;
13.82/5.25	
13.82/5.25	nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);
13.82/5.25	
13.82/5.25	nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
13.82/5.25	nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;
13.82/5.25	
13.82/5.25	nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);
13.82/5.25	
13.82/5.25	nubByNubBy'3 xv [] wu = [];
13.82/5.25	nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
13.82/5.25	
13.82/5.25	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.82/5.25	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
13.82/5.25	
13.82/5.25	}
13.82/5.25	module Main where {
13.82/5.25	import qualified List;
13.82/5.25	import qualified Maybe;
13.82/5.25	import qualified Prelude;
13.82/5.25	}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(9) Narrow (SOUND)
13.82/5.25	Haskell To QDPs
13.82/5.25	
13.82/5.25	digraph dp_graph {
13.82/5.25	node [outthreshold=100, inthreshold=100];1[label="List.unionBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
13.82/5.25	3[label="List.unionBy xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
13.82/5.25	4[label="List.unionBy xw3 xw4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
13.82/5.25	5[label="List.unionBy xw3 xw4 xw5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
13.82/5.25	6 -> 168[label="",style="dashed", color="red", weight=0];
13.82/5.25	6[label="xw4 ++ foldl (flip (List.deleteBy xw3)) (List.nubBy xw3 xw5) xw4",fontsize=16,color="magenta"];6 -> 169[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	6 -> 170[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	169 -> 209[label="",style="dashed", color="red", weight=0];
13.82/5.25	169[label="foldl (flip (List.deleteBy xw3)) (List.nubBy xw3 xw5) xw4",fontsize=16,color="magenta"];169 -> 210[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	169 -> 211[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	170[label="xw4",fontsize=16,color="green",shape="box"];168[label="xw41111111 ++ xw15",fontsize=16,color="burlywood",shape="triangle"];727[label="xw41111111/xw411111110 : xw411111111",fontsize=10,color="white",style="solid",shape="box"];168 -> 727[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	727 -> 188[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	728[label="xw41111111/[]",fontsize=10,color="white",style="solid",shape="box"];168 -> 728[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	728 -> 189[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	210[label="xw4",fontsize=16,color="green",shape="box"];211[label="List.nubBy xw3 xw5",fontsize=16,color="black",shape="box"];211 -> 216[label="",style="solid", color="black", weight=3];
13.82/5.25	209[label="foldl (flip (List.deleteBy xw3)) xw18 xw411",fontsize=16,color="burlywood",shape="triangle"];729[label="xw411/xw4110 : xw4111",fontsize=10,color="white",style="solid",shape="box"];209 -> 729[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	729 -> 217[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	730[label="xw411/[]",fontsize=10,color="white",style="solid",shape="box"];209 -> 730[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	730 -> 218[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	188[label="(xw411111110 : xw411111111) ++ xw15",fontsize=16,color="black",shape="box"];188 -> 192[label="",style="solid", color="black", weight=3];
13.82/5.25	189[label="[] ++ xw15",fontsize=16,color="black",shape="box"];189 -> 193[label="",style="solid", color="black", weight=3];
13.82/5.25	216[label="List.nubByNubBy' xw3 xw5 []",fontsize=16,color="burlywood",shape="box"];731[label="xw5/xw50 : xw51",fontsize=10,color="white",style="solid",shape="box"];216 -> 731[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	731 -> 219[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	732[label="xw5/[]",fontsize=10,color="white",style="solid",shape="box"];216 -> 732[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	732 -> 220[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	217[label="foldl (flip (List.deleteBy xw3)) xw18 (xw4110 : xw4111)",fontsize=16,color="black",shape="box"];217 -> 221[label="",style="solid", color="black", weight=3];
13.82/5.25	218[label="foldl (flip (List.deleteBy xw3)) xw18 []",fontsize=16,color="black",shape="box"];218 -> 222[label="",style="solid", color="black", weight=3];
13.82/5.25	192[label="xw411111110 : xw411111111 ++ xw15",fontsize=16,color="green",shape="box"];192 -> 197[label="",style="dashed", color="green", weight=3];
13.82/5.25	193[label="xw15",fontsize=16,color="green",shape="box"];219[label="List.nubByNubBy' xw3 (xw50 : xw51) []",fontsize=16,color="black",shape="box"];219 -> 223[label="",style="solid", color="black", weight=3];
13.82/5.25	220[label="List.nubByNubBy' xw3 [] []",fontsize=16,color="black",shape="box"];220 -> 224[label="",style="solid", color="black", weight=3];
13.82/5.25	221 -> 209[label="",style="dashed", color="red", weight=0];
13.82/5.25	221[label="foldl (flip (List.deleteBy xw3)) (flip (List.deleteBy xw3) xw18 xw4110) xw4111",fontsize=16,color="magenta"];221 -> 225[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	221 -> 226[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	222[label="xw18",fontsize=16,color="green",shape="box"];197 -> 168[label="",style="dashed", color="red", weight=0];
13.82/5.25	197[label="xw411111111 ++ xw15",fontsize=16,color="magenta"];197 -> 202[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	223[label="List.nubByNubBy'2 xw3 (xw50 : xw51) []",fontsize=16,color="black",shape="box"];223 -> 227[label="",style="solid", color="black", weight=3];
13.82/5.25	224[label="List.nubByNubBy'3 xw3 [] []",fontsize=16,color="black",shape="box"];224 -> 228[label="",style="solid", color="black", weight=3];
13.82/5.25	225[label="xw4111",fontsize=16,color="green",shape="box"];226[label="flip (List.deleteBy xw3) xw18 xw4110",fontsize=16,color="black",shape="box"];226 -> 229[label="",style="solid", color="black", weight=3];
13.82/5.25	202[label="xw411111111",fontsize=16,color="green",shape="box"];227[label="List.nubByNubBy'1 xw3 xw50 xw51 [] (List.elem_by xw3 xw50 [])",fontsize=16,color="black",shape="box"];227 -> 230[label="",style="solid", color="black", weight=3];
13.82/5.25	228[label="[]",fontsize=16,color="green",shape="box"];229[label="List.deleteBy xw3 xw4110 xw18",fontsize=16,color="burlywood",shape="triangle"];733[label="xw18/xw180 : xw181",fontsize=10,color="white",style="solid",shape="box"];229 -> 733[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	733 -> 231[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	734[label="xw18/[]",fontsize=10,color="white",style="solid",shape="box"];229 -> 734[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	734 -> 232[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	230[label="List.nubByNubBy'1 xw3 xw50 xw51 [] False",fontsize=16,color="black",shape="box"];230 -> 233[label="",style="solid", color="black", weight=3];
13.82/5.25	231[label="List.deleteBy xw3 xw4110 (xw180 : xw181)",fontsize=16,color="black",shape="box"];231 -> 234[label="",style="solid", color="black", weight=3];
13.82/5.25	232[label="List.deleteBy xw3 xw4110 []",fontsize=16,color="black",shape="box"];232 -> 235[label="",style="solid", color="black", weight=3];
13.82/5.25	233[label="List.nubByNubBy'0 xw3 xw50 xw51 [] otherwise",fontsize=16,color="black",shape="box"];233 -> 236[label="",style="solid", color="black", weight=3];
13.82/5.25	234 -> 237[label="",style="dashed", color="red", weight=0];
13.82/5.25	234[label="List.deleteBy0 xw181 xw180 xw3 xw4110 (xw3 xw4110 xw180)",fontsize=16,color="magenta"];234 -> 238[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	235[label="[]",fontsize=16,color="green",shape="box"];236[label="List.nubByNubBy'0 xw3 xw50 xw51 [] True",fontsize=16,color="black",shape="box"];236 -> 239[label="",style="solid", color="black", weight=3];
13.82/5.25	238[label="xw3 xw4110 xw180",fontsize=16,color="green",shape="box"];238 -> 244[label="",style="dashed", color="green", weight=3];
13.82/5.25	238 -> 245[label="",style="dashed", color="green", weight=3];
13.82/5.25	237[label="List.deleteBy0 xw181 xw180 xw3 xw4110 xw20",fontsize=16,color="burlywood",shape="triangle"];735[label="xw20/False",fontsize=10,color="white",style="solid",shape="box"];237 -> 735[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	735 -> 242[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	736[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];237 -> 736[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	736 -> 243[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	239[label="xw50 : List.nubByNubBy' xw3 xw51 (xw50 : [])",fontsize=16,color="green",shape="box"];239 -> 246[label="",style="dashed", color="green", weight=3];
13.82/5.25	244[label="xw4110",fontsize=16,color="green",shape="box"];245[label="xw180",fontsize=16,color="green",shape="box"];242[label="List.deleteBy0 xw181 xw180 xw3 xw4110 False",fontsize=16,color="black",shape="box"];242 -> 247[label="",style="solid", color="black", weight=3];
13.82/5.25	243[label="List.deleteBy0 xw181 xw180 xw3 xw4110 True",fontsize=16,color="black",shape="box"];243 -> 248[label="",style="solid", color="black", weight=3];
13.82/5.25	246[label="List.nubByNubBy' xw3 xw51 (xw50 : [])",fontsize=16,color="burlywood",shape="triangle"];737[label="xw51/xw510 : xw511",fontsize=10,color="white",style="solid",shape="box"];246 -> 737[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	737 -> 249[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	738[label="xw51/[]",fontsize=10,color="white",style="solid",shape="box"];246 -> 738[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	738 -> 250[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	247[label="xw180 : List.deleteBy xw3 xw4110 xw181",fontsize=16,color="green",shape="box"];247 -> 251[label="",style="dashed", color="green", weight=3];
13.82/5.25	248[label="xw181",fontsize=16,color="green",shape="box"];249[label="List.nubByNubBy' xw3 (xw510 : xw511) (xw50 : [])",fontsize=16,color="black",shape="box"];249 -> 252[label="",style="solid", color="black", weight=3];
13.82/5.25	250[label="List.nubByNubBy' xw3 [] (xw50 : [])",fontsize=16,color="black",shape="box"];250 -> 253[label="",style="solid", color="black", weight=3];
13.82/5.25	251 -> 229[label="",style="dashed", color="red", weight=0];
13.82/5.25	251[label="List.deleteBy xw3 xw4110 xw181",fontsize=16,color="magenta"];251 -> 254[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	252[label="List.nubByNubBy'2 xw3 (xw510 : xw511) (xw50 : [])",fontsize=16,color="black",shape="box"];252 -> 255[label="",style="solid", color="black", weight=3];
13.82/5.25	253[label="List.nubByNubBy'3 xw3 [] (xw50 : [])",fontsize=16,color="black",shape="box"];253 -> 256[label="",style="solid", color="black", weight=3];
13.82/5.25	254[label="xw181",fontsize=16,color="green",shape="box"];255[label="List.nubByNubBy'1 xw3 xw510 xw511 (xw50 : []) (List.elem_by xw3 xw510 (xw50 : []))",fontsize=16,color="black",shape="box"];255 -> 257[label="",style="solid", color="black", weight=3];
13.82/5.25	256[label="[]",fontsize=16,color="green",shape="box"];257 -> 686[label="",style="dashed", color="red", weight=0];
13.82/5.25	257[label="List.nubByNubBy'1 xw3 xw510 xw511 (xw50 : []) (xw3 xw50 xw510 || List.elem_by xw3 xw510 [])",fontsize=16,color="magenta"];257 -> 687[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 688[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 689[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 690[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 691[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 692[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	257 -> 693[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	687[label="xw3 xw50 xw510",fontsize=16,color="green",shape="box"];687 -> 695[label="",style="dashed", color="green", weight=3];
13.82/5.25	687 -> 696[label="",style="dashed", color="green", weight=3];
13.82/5.25	688[label="[]",fontsize=16,color="green",shape="box"];689[label="xw3",fontsize=16,color="green",shape="box"];690[label="xw510",fontsize=16,color="green",shape="box"];691[label="xw50",fontsize=16,color="green",shape="box"];692[label="xw511",fontsize=16,color="green",shape="box"];693[label="[]",fontsize=16,color="green",shape="box"];686[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (xw55 || List.elem_by xw48 xw49 xw54)",fontsize=16,color="burlywood",shape="triangle"];739[label="xw55/False",fontsize=10,color="white",style="solid",shape="box"];686 -> 739[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	739 -> 697[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	740[label="xw55/True",fontsize=10,color="white",style="solid",shape="box"];686 -> 740[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	740 -> 698[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	695[label="xw50",fontsize=16,color="green",shape="box"];696[label="xw510",fontsize=16,color="green",shape="box"];697[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (False || List.elem_by xw48 xw49 xw54)",fontsize=16,color="black",shape="box"];697 -> 701[label="",style="solid", color="black", weight=3];
13.82/5.25	698[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (True || List.elem_by xw48 xw49 xw54)",fontsize=16,color="black",shape="box"];698 -> 702[label="",style="solid", color="black", weight=3];
13.82/5.25	701[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (List.elem_by xw48 xw49 xw54)",fontsize=16,color="burlywood",shape="triangle"];741[label="xw54/xw540 : xw541",fontsize=10,color="white",style="solid",shape="box"];701 -> 741[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	741 -> 703[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	742[label="xw54/[]",fontsize=10,color="white",style="solid",shape="box"];701 -> 742[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	742 -> 704[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	702[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) True",fontsize=16,color="black",shape="box"];702 -> 705[label="",style="solid", color="black", weight=3];
13.82/5.25	703[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (List.elem_by xw48 xw49 (xw540 : xw541))",fontsize=16,color="black",shape="box"];703 -> 706[label="",style="solid", color="black", weight=3];
13.82/5.25	704[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (List.elem_by xw48 xw49 [])",fontsize=16,color="black",shape="box"];704 -> 707[label="",style="solid", color="black", weight=3];
13.82/5.25	705[label="List.nubByNubBy' xw48 xw50 (xw51 : xw52)",fontsize=16,color="burlywood",shape="triangle"];743[label="xw50/xw500 : xw501",fontsize=10,color="white",style="solid",shape="box"];705 -> 743[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	743 -> 708[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	744[label="xw50/[]",fontsize=10,color="white",style="solid",shape="box"];705 -> 744[label="",style="solid", color="burlywood", weight=9];
13.82/5.25	744 -> 709[label="",style="solid", color="burlywood", weight=3];
13.82/5.25	706 -> 686[label="",style="dashed", color="red", weight=0];
13.82/5.25	706[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) (xw48 xw540 xw49 || List.elem_by xw48 xw49 xw541)",fontsize=16,color="magenta"];706 -> 710[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	706 -> 711[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	707[label="List.nubByNubBy'1 xw48 xw49 xw50 (xw51 : xw52) False",fontsize=16,color="black",shape="box"];707 -> 712[label="",style="solid", color="black", weight=3];
13.82/5.25	708[label="List.nubByNubBy' xw48 (xw500 : xw501) (xw51 : xw52)",fontsize=16,color="black",shape="box"];708 -> 713[label="",style="solid", color="black", weight=3];
13.82/5.25	709[label="List.nubByNubBy' xw48 [] (xw51 : xw52)",fontsize=16,color="black",shape="box"];709 -> 714[label="",style="solid", color="black", weight=3];
13.82/5.25	710[label="xw48 xw540 xw49",fontsize=16,color="green",shape="box"];710 -> 715[label="",style="dashed", color="green", weight=3];
13.82/5.25	710 -> 716[label="",style="dashed", color="green", weight=3];
13.82/5.25	711[label="xw541",fontsize=16,color="green",shape="box"];712[label="List.nubByNubBy'0 xw48 xw49 xw50 (xw51 : xw52) otherwise",fontsize=16,color="black",shape="box"];712 -> 717[label="",style="solid", color="black", weight=3];
13.82/5.25	713[label="List.nubByNubBy'2 xw48 (xw500 : xw501) (xw51 : xw52)",fontsize=16,color="black",shape="box"];713 -> 718[label="",style="solid", color="black", weight=3];
13.82/5.25	714[label="List.nubByNubBy'3 xw48 [] (xw51 : xw52)",fontsize=16,color="black",shape="box"];714 -> 719[label="",style="solid", color="black", weight=3];
13.82/5.25	715[label="xw540",fontsize=16,color="green",shape="box"];716[label="xw49",fontsize=16,color="green",shape="box"];717[label="List.nubByNubBy'0 xw48 xw49 xw50 (xw51 : xw52) True",fontsize=16,color="black",shape="box"];717 -> 720[label="",style="solid", color="black", weight=3];
13.82/5.25	718 -> 701[label="",style="dashed", color="red", weight=0];
13.82/5.25	718[label="List.nubByNubBy'1 xw48 xw500 xw501 (xw51 : xw52) (List.elem_by xw48 xw500 (xw51 : xw52))",fontsize=16,color="magenta"];718 -> 721[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	718 -> 722[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	718 -> 723[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	719[label="[]",fontsize=16,color="green",shape="box"];720[label="xw49 : List.nubByNubBy' xw48 xw50 (xw49 : xw51 : xw52)",fontsize=16,color="green",shape="box"];720 -> 724[label="",style="dashed", color="green", weight=3];
13.82/5.25	721[label="xw500",fontsize=16,color="green",shape="box"];722[label="xw501",fontsize=16,color="green",shape="box"];723[label="xw51 : xw52",fontsize=16,color="green",shape="box"];724 -> 705[label="",style="dashed", color="red", weight=0];
13.82/5.25	724[label="List.nubByNubBy' xw48 xw50 (xw49 : xw51 : xw52)",fontsize=16,color="magenta"];724 -> 725[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	724 -> 726[label="",style="dashed", color="magenta", weight=3];
13.82/5.25	725[label="xw51 : xw52",fontsize=16,color="green",shape="box"];726[label="xw49",fontsize=16,color="green",shape="box"];}
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(10)
13.82/5.25	Complex Obligation (AND)
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(11)
13.82/5.25	Obligation:
13.82/5.25	Q DP problem:
13.82/5.25	The TRS P consists of the following rules:
13.82/5.25	
13.82/5.25	   new_foldl(xw3, :(xw4110, xw4111), ba) -> new_foldl(xw3, xw4111, ba)
13.82/5.25	
13.82/5.25	R is empty.
13.82/5.25	Q is empty.
13.82/5.25	We have to consider all minimal (P,Q,R)-chains.
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(12) QDPSizeChangeProof (EQUIVALENT)
13.82/5.25	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. 
13.82/5.25	
13.82/5.25	From the DPs we obtained the following set of size-change graphs:
13.82/5.25	*new_foldl(xw3, :(xw4110, xw4111), ba) -> new_foldl(xw3, xw4111, ba)
13.82/5.25	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
13.82/5.25	
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(13)
13.82/5.25	YES
13.82/5.25	
13.82/5.25	----------------------------------------
13.82/5.25	
13.82/5.25	(14)
13.82/5.25	Obligation:
13.82/5.25	Q DP problem:
13.82/5.25	The TRS P consists of the following rules:
13.82/5.26	
13.82/5.26	   new_psPs(:(xw411111110, xw411111111), xw15, ba) -> new_psPs(xw411111111, xw15, ba)
13.82/5.26	
13.82/5.26	R is empty.
13.82/5.26	Q is empty.
13.82/5.26	We have to consider all minimal (P,Q,R)-chains.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(15) QDPSizeChangeProof (EQUIVALENT)
13.82/5.26	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. 
13.82/5.26	
13.82/5.26	From the DPs we obtained the following set of size-change graphs:
13.82/5.26	*new_psPs(:(xw411111110, xw411111111), xw15, ba) -> new_psPs(xw411111111, xw15, ba)
13.82/5.26	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
13.82/5.26	
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(16)
13.82/5.26	YES
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(17)
13.82/5.26	Obligation:
13.82/5.26	Q DP problem:
13.82/5.26	The TRS P consists of the following rules:
13.82/5.26	
13.82/5.26	   new_nubByNubBy'(xw48, :(xw500, xw501), xw51, xw52, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'1(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, [], ba) -> new_nubByNubBy'(xw48, xw50, xw49, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, :(xw500, xw501), xw51, xw52, xw54, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'1(xw48, xw49, xw50, xw51, xw52, [], ba) -> new_nubByNubBy'(xw48, xw50, xw49, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	
13.82/5.26	R is empty.
13.82/5.26	Q is empty.
13.82/5.26	We have to consider all minimal (P,Q,R)-chains.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(18) DependencyGraphProof (EQUIVALENT)
13.82/5.26	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(19)
13.82/5.26	Obligation:
13.82/5.26	Q DP problem:
13.82/5.26	The TRS P consists of the following rules:
13.82/5.26	
13.82/5.26	   new_nubByNubBy'1(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, [], ba) -> new_nubByNubBy'(xw48, xw50, xw49, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'(xw48, :(xw500, xw501), xw51, xw52, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, :(xw500, xw501), xw51, xw52, xw54, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	
13.82/5.26	R is empty.
13.82/5.26	Q is empty.
13.82/5.26	We have to consider all minimal (P,Q,R)-chains.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(20) TransformationProof (EQUIVALENT)
13.82/5.26	By instantiating [LPAR04] the rule new_nubByNubBy'1(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba) we obtained the following new rules [LPAR04]:
13.82/5.26	
13.82/5.26	   (new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5),new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5))
13.82/5.26	
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(21)
13.82/5.26	Obligation:
13.82/5.26	Q DP problem:
13.82/5.26	The TRS P consists of the following rules:
13.82/5.26	
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, [], ba) -> new_nubByNubBy'(xw48, xw50, xw49, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'(xw48, :(xw500, xw501), xw51, xw52, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, :(xw500, xw501), xw51, xw52, xw54, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	   new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	   new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5)
13.82/5.26	
13.82/5.26	R is empty.
13.82/5.26	Q is empty.
13.82/5.26	We have to consider all minimal (P,Q,R)-chains.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(22) QDPSizeChangeProof (EQUIVALENT)
13.82/5.26	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. 
13.82/5.26	
13.82/5.26	From the DPs we obtained the following set of size-change graphs:
13.82/5.26	*new_nubByNubBy'(xw48, :(xw500, xw501), xw51, xw52, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4, 4 >= 5, 5 >= 7
13.82/5.26	
13.82/5.26	
13.82/5.26	*new_nubByNubBy'1(z0, z1, z2, z3, z4, :(z3, z4), z5) -> new_nubByNubBy'10(z0, z1, z2, z3, z4, z4, z5)
13.82/5.26	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 4, 5 >= 5, 6 > 5, 5 >= 6, 6 > 6, 7 >= 7
13.82/5.26	
13.82/5.26	
13.82/5.26	*new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, :(xw540, xw541), ba) -> new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, xw541, ba)
13.82/5.26	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 7 >= 7
13.82/5.26	
13.82/5.26	
13.82/5.26	*new_nubByNubBy'10(xw48, xw49, xw50, xw51, xw52, [], ba) -> new_nubByNubBy'(xw48, xw50, xw49, :(xw51, xw52), ba)
13.82/5.26	The graph contains the following edges 1 >= 1, 3 >= 2, 2 >= 3, 7 >= 5
13.82/5.26	
13.82/5.26	
13.82/5.26	*new_nubByNubBy'10(xw48, xw49, :(xw500, xw501), xw51, xw52, xw54, ba) -> new_nubByNubBy'1(xw48, xw500, xw501, xw51, xw52, :(xw51, xw52), ba)
13.82/5.26	The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5, 7 >= 7
13.82/5.26	
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(23)
13.82/5.26	YES
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(24)
13.82/5.26	Obligation:
13.82/5.26	Q DP problem:
13.82/5.26	The TRS P consists of the following rules:
13.82/5.26	
13.82/5.26	   new_deleteBy(xw3, xw4110, :(xw180, xw181), ba) -> new_deleteBy0(xw181, xw180, xw3, xw4110, ba)
13.82/5.26	   new_deleteBy0(xw181, xw180, xw3, xw4110, ba) -> new_deleteBy(xw3, xw4110, xw181, ba)
13.82/5.26	
13.82/5.26	R is empty.
13.82/5.26	Q is empty.
13.82/5.26	We have to consider all minimal (P,Q,R)-chains.
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(25) QDPSizeChangeProof (EQUIVALENT)
13.82/5.26	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. 
13.82/5.26	
13.82/5.26	From the DPs we obtained the following set of size-change graphs:
13.82/5.26	*new_deleteBy0(xw181, xw180, xw3, xw4110, ba) -> new_deleteBy(xw3, xw4110, xw181, ba)
13.82/5.26	The graph contains the following edges 3 >= 1, 4 >= 2, 1 >= 3, 5 >= 4
13.82/5.26	
13.82/5.26	
13.82/5.26	*new_deleteBy(xw3, xw4110, :(xw180, xw181), ba) -> new_deleteBy0(xw181, xw180, xw3, xw4110, ba)
13.82/5.26	The graph contains the following edges 3 > 1, 3 > 2, 1 >= 3, 2 >= 4, 4 >= 5
13.82/5.26	
13.82/5.26	
13.82/5.26	----------------------------------------
13.82/5.26	
13.82/5.26	(26)
13.82/5.26	YES
14.01/5.32	EOF