13.50/5.59	YES
16.11/6.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
16.11/6.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.11/6.23	
16.11/6.23	
16.11/6.23	H-Termination with start terms of the given HASKELL could be proven:
16.11/6.23	
16.11/6.23	(0) HASKELL
16.11/6.23	(1) IFR [EQUIVALENT, 0 ms]
16.11/6.23	(2) HASKELL
16.11/6.23	(3) BR [EQUIVALENT, 0 ms]
16.11/6.23	(4) HASKELL
16.11/6.23	(5) COR [EQUIVALENT, 19 ms]
16.11/6.23	(6) HASKELL
16.11/6.23	(7) LetRed [EQUIVALENT, 0 ms]
16.11/6.23	(8) HASKELL
16.11/6.23	(9) Narrow [SOUND, 0 ms]
16.11/6.23	(10) AND
16.11/6.23	    (11) QDP
16.11/6.23	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.11/6.23	        (13) YES
16.11/6.23	    (14) QDP
16.11/6.23	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.11/6.23	        (16) YES
16.11/6.23	    (17) QDP
16.11/6.23	        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
16.11/6.23	        (19) QDP
16.11/6.23	        (20) TransformationProof [EQUIVALENT, 0 ms]
16.11/6.23	        (21) QDP
16.11/6.23	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.11/6.23	        (23) YES
16.11/6.23	    (24) QDP
16.11/6.23	        (25) TransformationProof [EQUIVALENT, 0 ms]
16.11/6.23	        (26) QDP
16.11/6.23	        (27) UsableRulesProof [EQUIVALENT, 0 ms]
16.11/6.23	        (28) QDP
16.11/6.23	        (29) QReductionProof [EQUIVALENT, 0 ms]
16.11/6.23	        (30) QDP
16.11/6.23	        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.11/6.23	        (32) YES
16.11/6.23	    (33) QDP
16.11/6.23	        (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.11/6.23	        (35) YES
16.11/6.23	
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(0)
16.11/6.23	Obligation:
16.11/6.23	mainModule Main 
16.11/6.23	module Maybe where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	module List where {
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
16.11/6.23	deleteBy _ _ [] = [];
16.11/6.23	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
16.11/6.23	
16.11/6.23	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
16.11/6.23	elem_by _ _ [] = False;
16.11/6.23	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
16.11/6.23	
16.11/6.23	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
16.11/6.23	nubBy eq l = nubBy' l [] where { 
16.11/6.23	nubBy' [] _ = [];
16.11/6.23	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
16.11/6.23	 | otherwise = y : nubBy' ys (y : xs);
16.11/6.23	 };
16.11/6.23	
16.11/6.23	union :: Eq a => [a]  ->  [a]  ->  [a];
16.11/6.23	union = unionBy (==);
16.11/6.23	
16.11/6.23	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
16.11/6.23	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
16.11/6.23	
16.11/6.23	}
16.11/6.23	module Main where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(1) IFR (EQUIVALENT)
16.11/6.23	If Reductions:
16.11/6.23	The following If expression
16.11/6.23	"if eq x y then ys else y : deleteBy eq x ys"
16.11/6.23	is transformed to
16.11/6.23	"deleteBy0 ys y eq x True = ys;
16.11/6.23	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
16.11/6.23	"
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(2)
16.11/6.23	Obligation:
16.11/6.23	mainModule Main 
16.11/6.23	module Maybe where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	module List where {
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
16.11/6.23	deleteBy _ _ [] = [];
16.11/6.23	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
16.11/6.23	
16.11/6.23	deleteBy0 ys y eq x True = ys;
16.11/6.23	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
16.11/6.23	
16.11/6.23	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
16.11/6.23	elem_by _ _ [] = False;
16.11/6.23	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
16.11/6.23	
16.11/6.23	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
16.11/6.23	nubBy eq l = nubBy' l [] where { 
16.11/6.23	nubBy' [] _ = [];
16.11/6.23	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
16.11/6.23	 | otherwise = y : nubBy' ys (y : xs);
16.11/6.23	 };
16.11/6.23	
16.11/6.23	union :: Eq a => [a]  ->  [a]  ->  [a];
16.11/6.23	union = unionBy (==);
16.11/6.23	
16.11/6.23	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
16.11/6.23	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
16.11/6.23	
16.11/6.23	}
16.11/6.23	module Main where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(3) BR (EQUIVALENT)
16.11/6.23	Replaced joker patterns by fresh variables and removed binding patterns.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(4)
16.11/6.23	Obligation:
16.11/6.23	mainModule Main 
16.11/6.23	module Maybe where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	module List where {
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
16.11/6.23	deleteBy wv ww [] = [];
16.11/6.23	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
16.11/6.23	
16.11/6.23	deleteBy0 ys y eq x True = ys;
16.11/6.23	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
16.11/6.23	
16.11/6.23	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
16.11/6.23	elem_by vy vz [] = False;
16.11/6.23	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
16.11/6.23	
16.11/6.23	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
16.11/6.23	nubBy eq l = nubBy' l [] where { 
16.11/6.23	nubBy' [] wu = [];
16.11/6.23	nubBy' (y : ys) xs  | elem_by eq y xs = nubBy' ys xs
16.11/6.23	 | otherwise = y : nubBy' ys (y : xs);
16.11/6.23	 };
16.11/6.23	
16.11/6.23	union :: Eq a => [a]  ->  [a]  ->  [a];
16.11/6.23	union = unionBy (==);
16.11/6.23	
16.11/6.23	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
16.11/6.23	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
16.11/6.23	
16.11/6.23	}
16.11/6.23	module Main where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(5) COR (EQUIVALENT)
16.11/6.23	Cond Reductions:
16.11/6.23	The following Function with conditions
16.11/6.23	"undefined |Falseundefined;
16.11/6.23	"
16.11/6.23	is transformed to
16.11/6.23	"undefined  = undefined1;
16.11/6.23	"
16.11/6.23	"undefined0 True = undefined;
16.11/6.23	"
16.11/6.23	"undefined1  = undefined0 False;
16.11/6.23	"
16.11/6.23	The following Function with conditions
16.11/6.23	"nubBy' [] wu = [];
16.11/6.23	nubBy' (y : ys) xs|elem_by eq y xsnubBy' ys xs|otherwisey : nubBy' ys (y : xs);
16.11/6.23	"
16.11/6.23	is transformed to
16.11/6.23	"nubBy' [] wu = nubBy'3 [] wu;
16.11/6.23	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
16.11/6.23	"
16.11/6.23	"nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
16.11/6.23	"
16.11/6.23	"nubBy'1 y ys xs True = nubBy' ys xs;
16.11/6.23	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
16.11/6.23	"
16.11/6.23	"nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
16.11/6.23	"
16.11/6.23	"nubBy'3 [] wu = [];
16.11/6.23	nubBy'3 wz xu = nubBy'2 wz xu;
16.11/6.23	"
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(6)
16.11/6.23	Obligation:
16.11/6.23	mainModule Main 
16.11/6.23	module Maybe where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	module List where {
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
16.11/6.23	deleteBy wv ww [] = [];
16.11/6.23	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
16.11/6.23	
16.11/6.23	deleteBy0 ys y eq x True = ys;
16.11/6.23	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
16.11/6.23	
16.11/6.23	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
16.11/6.23	elem_by vy vz [] = False;
16.11/6.23	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
16.11/6.23	
16.11/6.23	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
16.11/6.23	nubBy eq l = nubBy' l [] where { 
16.11/6.23	nubBy' [] wu = nubBy'3 [] wu;
16.11/6.23	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
16.11/6.23	nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
16.11/6.23	nubBy'1 y ys xs True = nubBy' ys xs;
16.11/6.23	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
16.11/6.23	nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
16.11/6.23	nubBy'3 [] wu = [];
16.11/6.23	nubBy'3 wz xu = nubBy'2 wz xu;
16.11/6.23	 };
16.11/6.23	
16.11/6.23	union :: Eq a => [a]  ->  [a]  ->  [a];
16.11/6.23	union = unionBy (==);
16.11/6.23	
16.11/6.23	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
16.11/6.23	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
16.11/6.23	
16.11/6.23	}
16.11/6.23	module Main where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(7) LetRed (EQUIVALENT)
16.11/6.23	Let/Where Reductions:
16.11/6.23	The bindings of the following Let/Where expression
16.11/6.23	"nubBy' l [] where {
16.11/6.23	nubBy' [] wu = nubBy'3 [] wu;
16.11/6.23	nubBy' (y : ys) xs = nubBy'2 (y : ys) xs;
16.11/6.23	;
16.11/6.23	nubBy'0 y ys xs True = y : nubBy' ys (y : xs);
16.11/6.23	;
16.11/6.23	nubBy'1 y ys xs True = nubBy' ys xs;
16.11/6.23	nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise;
16.11/6.23	;
16.11/6.23	nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs);
16.11/6.23	;
16.11/6.23	nubBy'3 [] wu = [];
16.11/6.23	nubBy'3 wz xu = nubBy'2 wz xu;
16.11/6.23	} 
16.11/6.23	"
16.11/6.23	are unpacked to the following functions on top level
16.11/6.23	"nubByNubBy'3 xv [] wu = [];
16.11/6.23	nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
16.11/6.23	"
16.11/6.23	"nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);
16.11/6.23	"
16.11/6.23	"nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);
16.11/6.23	"
16.11/6.23	"nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
16.11/6.23	nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;
16.11/6.23	"
16.11/6.23	"nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
16.11/6.23	nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;
16.11/6.23	"
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(8)
16.11/6.23	Obligation:
16.11/6.23	mainModule Main 
16.11/6.23	module Maybe where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	module List where {
16.11/6.23	import qualified Main;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
16.11/6.23	deleteBy wv ww [] = [];
16.11/6.23	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
16.11/6.23	
16.11/6.23	deleteBy0 ys y eq x True = ys;
16.11/6.23	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
16.11/6.23	
16.11/6.23	elem_by :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  Bool;
16.11/6.23	elem_by vy vz [] = False;
16.11/6.23	elem_by eq y (x : xs) = x `eq` y || elem_by eq y xs;
16.11/6.23	
16.11/6.23	nubBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a];
16.11/6.23	nubBy eq l = nubByNubBy' eq l [];
16.11/6.23	
16.11/6.23	nubByNubBy' xv [] wu = nubByNubBy'3 xv [] wu;
16.11/6.23	nubByNubBy' xv (y : ys) xs = nubByNubBy'2 xv (y : ys) xs;
16.11/6.23	
16.11/6.23	nubByNubBy'0 xv y ys xs True = y : nubByNubBy' xv ys (y : xs);
16.11/6.23	
16.11/6.23	nubByNubBy'1 xv y ys xs True = nubByNubBy' xv ys xs;
16.11/6.23	nubByNubBy'1 xv y ys xs False = nubByNubBy'0 xv y ys xs otherwise;
16.11/6.23	
16.11/6.23	nubByNubBy'2 xv (y : ys) xs = nubByNubBy'1 xv y ys xs (elem_by xv y xs);
16.11/6.23	
16.11/6.23	nubByNubBy'3 xv [] wu = [];
16.11/6.23	nubByNubBy'3 xv wz xu = nubByNubBy'2 xv wz xu;
16.11/6.23	
16.11/6.23	union :: Eq a => [a]  ->  [a]  ->  [a];
16.11/6.23	union = unionBy (==);
16.11/6.23	
16.11/6.23	unionBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
16.11/6.23	unionBy eq xs ys = xs ++ foldl (flip (deleteBy eq)) (nubBy eq ys) xs;
16.11/6.23	
16.11/6.23	}
16.11/6.23	module Main where {
16.11/6.23	import qualified List;
16.11/6.23	import qualified Maybe;
16.11/6.23	import qualified Prelude;
16.11/6.23	}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(9) Narrow (SOUND)
16.11/6.23	Haskell To QDPs
16.11/6.23	
16.11/6.23	digraph dp_graph {
16.11/6.23	node [outthreshold=100, inthreshold=100];1[label="List.union",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
16.11/6.23	3[label="List.union xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
16.11/6.23	4[label="List.union xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
16.11/6.23	5[label="List.unionBy (==) xw3 xw4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
16.11/6.23	6[label="xw3 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) xw3",fontsize=16,color="burlywood",shape="box"];2440[label="xw3/xw30 : xw31",fontsize=10,color="white",style="solid",shape="box"];6 -> 2440[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2440 -> 7[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2441[label="xw3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 2441[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2441 -> 8[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	7[label="(xw30 : xw31) ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
16.11/6.23	8[label="[] ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
16.11/6.23	9[label="xw30 : xw31 ++ foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) (xw30 : xw31)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
16.11/6.23	10[label="foldl (flip (List.deleteBy (==))) (List.nubBy (==) xw4) []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
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16.11/6.23	19[label="List.nubByNubBy' (==) [] []",fontsize=16,color="black",shape="box"];19 -> 24[label="",style="solid", color="black", weight=3];
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16.11/6.23	24[label="List.nubByNubBy'3 (==) [] []",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
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16.11/6.23	2446 -> 149[label="",style="solid", color="burlywood", weight=3];
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16.11/6.23	33[label="List.nubByNubBy'1 (==) xw40 xw41 [] False",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
16.11/6.23	155 -> 143[label="",style="dashed", color="red", weight=0];
16.11/6.23	155[label="foldl (flip (List.deleteBy (==))) (flip (List.deleteBy (==)) (flip (List.deleteBy (==)) xw9 xw10) xw110) xw111",fontsize=16,color="magenta"];155 -> 162[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	155 -> 163[label="",style="dashed", color="magenta", weight=3];
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16.11/6.23	162 -> 156[label="",style="dashed", color="red", weight=0];
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16.11/6.23	2448 -> 173[label="",style="solid", color="burlywood", weight=3];
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16.11/6.23	2449 -> 174[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	42[label="List.nubByNubBy'0 (==) xw40 xw41 [] True",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
16.11/6.23	173[label="List.deleteBy (==) xw10 (xw90 : xw91)",fontsize=16,color="black",shape="box"];173 -> 183[label="",style="solid", color="black", weight=3];
16.11/6.23	174[label="List.deleteBy (==) xw10 []",fontsize=16,color="black",shape="box"];174 -> 184[label="",style="solid", color="black", weight=3];
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16.11/6.23	183 -> 194[label="",style="dashed", color="red", weight=0];
16.11/6.23	183[label="List.deleteBy0 xw91 xw90 (==) xw10 ((==) xw10 xw90)",fontsize=16,color="magenta"];183 -> 195[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	183 -> 196[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	183 -> 197[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	183 -> 198[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	184[label="[]",fontsize=16,color="green",shape="box"];54[label="List.nubByNubBy' (==) xw41 (xw40 : [])",fontsize=16,color="burlywood",shape="box"];2450[label="xw41/xw410 : xw411",fontsize=10,color="white",style="solid",shape="box"];54 -> 2450[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2450 -> 58[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2451[label="xw41/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 2451[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2451 -> 59[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	195[label="xw91",fontsize=16,color="green",shape="box"];196[label="xw90",fontsize=16,color="green",shape="box"];197[label="xw10",fontsize=16,color="green",shape="box"];198[label="(==) xw10 xw90",fontsize=16,color="blue",shape="box"];2452[label="== :: Char -> Char -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2452[label="",style="solid", color="blue", weight=9];
16.11/6.23	2452 -> 199[label="",style="solid", color="blue", weight=3];
16.11/6.23	2453[label="== :: () -> () -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2453[label="",style="solid", color="blue", weight=9];
16.11/6.23	2453 -> 200[label="",style="solid", color="blue", weight=3];
16.11/6.23	2454[label="== :: ([] a) -> ([] a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2454[label="",style="solid", color="blue", weight=9];
16.11/6.23	2454 -> 201[label="",style="solid", color="blue", weight=3];
16.11/6.23	2455[label="== :: Int -> Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2455[label="",style="solid", color="blue", weight=9];
16.11/6.23	2455 -> 202[label="",style="solid", color="blue", weight=3];
16.11/6.23	2456[label="== :: Ordering -> Ordering -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2456[label="",style="solid", color="blue", weight=9];
16.11/6.23	2456 -> 203[label="",style="solid", color="blue", weight=3];
16.11/6.23	2457[label="== :: ((@2) a b) -> ((@2) a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2457[label="",style="solid", color="blue", weight=9];
16.11/6.23	2457 -> 204[label="",style="solid", color="blue", weight=3];
16.11/6.23	2458[label="== :: Float -> Float -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2458[label="",style="solid", color="blue", weight=9];
16.11/6.23	2458 -> 205[label="",style="solid", color="blue", weight=3];
16.11/6.23	2459[label="== :: Bool -> Bool -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2459[label="",style="solid", color="blue", weight=9];
16.11/6.23	2459 -> 206[label="",style="solid", color="blue", weight=3];
16.11/6.23	2460[label="== :: Integer -> Integer -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2460[label="",style="solid", color="blue", weight=9];
16.11/6.23	2460 -> 207[label="",style="solid", color="blue", weight=3];
16.11/6.23	2461[label="== :: (Ratio a) -> (Ratio a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2461[label="",style="solid", color="blue", weight=9];
16.11/6.23	2461 -> 208[label="",style="solid", color="blue", weight=3];
16.11/6.23	2462[label="== :: Double -> Double -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2462[label="",style="solid", color="blue", weight=9];
16.11/6.23	2462 -> 209[label="",style="solid", color="blue", weight=3];
16.11/6.23	2463[label="== :: (Either a b) -> (Either a b) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2463[label="",style="solid", color="blue", weight=9];
16.11/6.23	2463 -> 210[label="",style="solid", color="blue", weight=3];
16.11/6.23	2464[label="== :: (Maybe a) -> (Maybe a) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2464[label="",style="solid", color="blue", weight=9];
16.11/6.23	2464 -> 211[label="",style="solid", color="blue", weight=3];
16.11/6.23	2465[label="== :: ((@3) a b c) -> ((@3) a b c) -> Bool",fontsize=10,color="white",style="solid",shape="box"];198 -> 2465[label="",style="solid", color="blue", weight=9];
16.11/6.23	2465 -> 212[label="",style="solid", color="blue", weight=3];
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16.11/6.23	2466 -> 213[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2467[label="xw20/True",fontsize=10,color="white",style="solid",shape="box"];194 -> 2467[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2467 -> 214[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	58[label="List.nubByNubBy' (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3];
16.11/6.23	59[label="List.nubByNubBy' (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3];
16.11/6.23	199[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];199 -> 226[label="",style="solid", color="black", weight=3];
16.11/6.23	200[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];200 -> 227[label="",style="solid", color="black", weight=3];
16.11/6.23	201[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];201 -> 228[label="",style="solid", color="black", weight=3];
16.11/6.23	202[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];202 -> 229[label="",style="solid", color="black", weight=3];
16.11/6.23	203[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];203 -> 230[label="",style="solid", color="black", weight=3];
16.11/6.23	204[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];204 -> 231[label="",style="solid", color="black", weight=3];
16.11/6.23	205[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];205 -> 232[label="",style="solid", color="black", weight=3];
16.11/6.23	206[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];206 -> 233[label="",style="solid", color="black", weight=3];
16.11/6.23	207[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];207 -> 234[label="",style="solid", color="black", weight=3];
16.11/6.23	208[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];208 -> 235[label="",style="solid", color="black", weight=3];
16.11/6.23	209[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];209 -> 236[label="",style="solid", color="black", weight=3];
16.11/6.23	210[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];210 -> 237[label="",style="solid", color="black", weight=3];
16.11/6.23	211[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];211 -> 238[label="",style="solid", color="black", weight=3];
16.11/6.23	212[label="(==) xw10 xw90",fontsize=16,color="black",shape="box"];212 -> 239[label="",style="solid", color="black", weight=3];
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16.11/6.23	214[label="List.deleteBy0 xw17 xw18 (==) xw19 True",fontsize=16,color="black",shape="box"];214 -> 241[label="",style="solid", color="black", weight=3];
16.11/6.23	66[label="List.nubByNubBy'2 (==) (xw410 : xw411) (xw40 : [])",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3];
16.11/6.23	67[label="List.nubByNubBy'3 (==) [] (xw40 : [])",fontsize=16,color="black",shape="box"];67 -> 73[label="",style="solid", color="black", weight=3];
16.11/6.23	226[label="primEqChar xw10 xw90",fontsize=16,color="burlywood",shape="triangle"];2468[label="xw10/Char xw100",fontsize=10,color="white",style="solid",shape="box"];226 -> 2468[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2468 -> 253[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	227[label="error []",fontsize=16,color="red",shape="box"];228[label="error []",fontsize=16,color="red",shape="box"];229[label="error []",fontsize=16,color="red",shape="box"];230[label="error []",fontsize=16,color="red",shape="box"];231[label="error []",fontsize=16,color="red",shape="box"];232[label="error []",fontsize=16,color="red",shape="box"];233[label="error []",fontsize=16,color="red",shape="box"];234[label="error []",fontsize=16,color="red",shape="box"];235[label="error []",fontsize=16,color="red",shape="box"];236[label="error []",fontsize=16,color="red",shape="box"];237[label="error []",fontsize=16,color="red",shape="box"];238[label="error []",fontsize=16,color="red",shape="box"];239[label="error []",fontsize=16,color="red",shape="box"];240[label="xw18 : List.deleteBy (==) xw19 xw17",fontsize=16,color="green",shape="box"];240 -> 254[label="",style="dashed", color="green", weight=3];
16.11/6.23	241[label="xw17",fontsize=16,color="green",shape="box"];72[label="List.nubByNubBy'1 (==) xw410 xw411 (xw40 : []) (List.elem_by (==) xw410 (xw40 : []))",fontsize=16,color="black",shape="box"];72 -> 78[label="",style="solid", color="black", weight=3];
16.11/6.23	73[label="[]",fontsize=16,color="green",shape="box"];253[label="primEqChar (Char xw100) xw90",fontsize=16,color="burlywood",shape="box"];2469[label="xw90/Char xw900",fontsize=10,color="white",style="solid",shape="box"];253 -> 2469[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2469 -> 271[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	254 -> 165[label="",style="dashed", color="red", weight=0];
16.11/6.23	254[label="List.deleteBy (==) xw19 xw17",fontsize=16,color="magenta"];254 -> 272[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	254 -> 273[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	78[label="List.nubByNubBy'1 (==) xw410 xw411 (xw40 : []) ((==) xw40 xw410 || List.elem_by (==) xw410 [])",fontsize=16,color="black",shape="box"];78 -> 89[label="",style="solid", color="black", weight=3];
16.11/6.23	271[label="primEqChar (Char xw100) (Char xw900)",fontsize=16,color="black",shape="box"];271 -> 289[label="",style="solid", color="black", weight=3];
16.11/6.23	272[label="xw17",fontsize=16,color="green",shape="box"];273[label="xw19",fontsize=16,color="green",shape="box"];89 -> 2402[label="",style="dashed", color="red", weight=0];
16.11/6.23	89[label="List.nubByNubBy'1 primEqChar xw410 xw411 (xw40 : []) (primEqChar xw40 xw410 || List.elem_by primEqChar xw410 [])",fontsize=16,color="magenta"];89 -> 2403[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	89 -> 2404[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	89 -> 2405[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	89 -> 2406[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	89 -> 2407[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	89 -> 2408[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	289[label="primEqNat xw100 xw900",fontsize=16,color="burlywood",shape="triangle"];2470[label="xw100/Succ xw1000",fontsize=10,color="white",style="solid",shape="box"];289 -> 2470[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2470 -> 305[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2471[label="xw100/Zero",fontsize=10,color="white",style="solid",shape="box"];289 -> 2471[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2471 -> 306[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2403[label="[]",fontsize=16,color="green",shape="box"];2404[label="xw40",fontsize=16,color="green",shape="box"];2405[label="[]",fontsize=16,color="green",shape="box"];2406[label="xw411",fontsize=16,color="green",shape="box"];2407 -> 226[label="",style="dashed", color="red", weight=0];
16.11/6.23	2407[label="primEqChar xw40 xw410",fontsize=16,color="magenta"];2407 -> 2410[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2407 -> 2411[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2408[label="xw410",fontsize=16,color="green",shape="box"];2402[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (xw161 || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2472[label="xw161/False",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2472[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2472 -> 2412[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2473[label="xw161/True",fontsize=10,color="white",style="solid",shape="box"];2402 -> 2473[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2473 -> 2413[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	305[label="primEqNat (Succ xw1000) xw900",fontsize=16,color="burlywood",shape="box"];2474[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];305 -> 2474[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2474 -> 309[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2475[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];305 -> 2475[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2475 -> 310[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	306[label="primEqNat Zero xw900",fontsize=16,color="burlywood",shape="box"];2476[label="xw900/Succ xw9000",fontsize=10,color="white",style="solid",shape="box"];306 -> 2476[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2476 -> 311[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2477[label="xw900/Zero",fontsize=10,color="white",style="solid",shape="box"];306 -> 2477[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2477 -> 312[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2410[label="xw410",fontsize=16,color="green",shape="box"];2411[label="xw40",fontsize=16,color="green",shape="box"];2412[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (False || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2412 -> 2414[label="",style="solid", color="black", weight=3];
16.11/6.23	2413[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (True || List.elem_by primEqChar xw155 xw160)",fontsize=16,color="black",shape="box"];2413 -> 2415[label="",style="solid", color="black", weight=3];
16.11/6.23	309[label="primEqNat (Succ xw1000) (Succ xw9000)",fontsize=16,color="black",shape="box"];309 -> 333[label="",style="solid", color="black", weight=3];
16.11/6.23	310[label="primEqNat (Succ xw1000) Zero",fontsize=16,color="black",shape="box"];310 -> 334[label="",style="solid", color="black", weight=3];
16.11/6.23	311[label="primEqNat Zero (Succ xw9000)",fontsize=16,color="black",shape="box"];311 -> 335[label="",style="solid", color="black", weight=3];
16.11/6.23	312[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];312 -> 336[label="",style="solid", color="black", weight=3];
16.11/6.23	2414[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 xw160)",fontsize=16,color="burlywood",shape="triangle"];2478[label="xw160/xw1600 : xw1601",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2478[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2478 -> 2416[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2479[label="xw160/[]",fontsize=10,color="white",style="solid",shape="box"];2414 -> 2479[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2479 -> 2417[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2415[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2415 -> 2418[label="",style="solid", color="black", weight=3];
16.11/6.23	333 -> 289[label="",style="dashed", color="red", weight=0];
16.11/6.23	333[label="primEqNat xw1000 xw9000",fontsize=16,color="magenta"];333 -> 350[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	333 -> 351[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	334[label="False",fontsize=16,color="green",shape="box"];335[label="False",fontsize=16,color="green",shape="box"];336[label="True",fontsize=16,color="green",shape="box"];2416[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 (xw1600 : xw1601))",fontsize=16,color="black",shape="box"];2416 -> 2419[label="",style="solid", color="black", weight=3];
16.11/6.23	2417[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (List.elem_by primEqChar xw155 [])",fontsize=16,color="black",shape="box"];2417 -> 2420[label="",style="solid", color="black", weight=3];
16.11/6.23	2418[label="List.nubByNubBy' primEqChar xw156 (xw157 : xw158)",fontsize=16,color="burlywood",shape="triangle"];2480[label="xw156/xw1560 : xw1561",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2480[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2480 -> 2421[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	2481[label="xw156/[]",fontsize=10,color="white",style="solid",shape="box"];2418 -> 2481[label="",style="solid", color="burlywood", weight=9];
16.11/6.23	2481 -> 2422[label="",style="solid", color="burlywood", weight=3];
16.11/6.23	350[label="xw9000",fontsize=16,color="green",shape="box"];351[label="xw1000",fontsize=16,color="green",shape="box"];2419 -> 2402[label="",style="dashed", color="red", weight=0];
16.11/6.23	2419[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) (primEqChar xw1600 xw155 || List.elem_by primEqChar xw155 xw1601)",fontsize=16,color="magenta"];2419 -> 2423[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2419 -> 2424[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2420[label="List.nubByNubBy'1 primEqChar xw155 xw156 (xw157 : xw158) False",fontsize=16,color="black",shape="box"];2420 -> 2425[label="",style="solid", color="black", weight=3];
16.11/6.23	2421[label="List.nubByNubBy' primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2421 -> 2426[label="",style="solid", color="black", weight=3];
16.11/6.23	2422[label="List.nubByNubBy' primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2422 -> 2427[label="",style="solid", color="black", weight=3];
16.11/6.23	2423[label="xw1601",fontsize=16,color="green",shape="box"];2424 -> 226[label="",style="dashed", color="red", weight=0];
16.11/6.23	2424[label="primEqChar xw1600 xw155",fontsize=16,color="magenta"];2424 -> 2428[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2424 -> 2429[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2425[label="List.nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) otherwise",fontsize=16,color="black",shape="box"];2425 -> 2430[label="",style="solid", color="black", weight=3];
16.11/6.23	2426[label="List.nubByNubBy'2 primEqChar (xw1560 : xw1561) (xw157 : xw158)",fontsize=16,color="black",shape="box"];2426 -> 2431[label="",style="solid", color="black", weight=3];
16.11/6.23	2427[label="List.nubByNubBy'3 primEqChar [] (xw157 : xw158)",fontsize=16,color="black",shape="box"];2427 -> 2432[label="",style="solid", color="black", weight=3];
16.11/6.23	2428[label="xw155",fontsize=16,color="green",shape="box"];2429[label="xw1600",fontsize=16,color="green",shape="box"];2430[label="List.nubByNubBy'0 primEqChar xw155 xw156 (xw157 : xw158) True",fontsize=16,color="black",shape="box"];2430 -> 2433[label="",style="solid", color="black", weight=3];
16.11/6.23	2431 -> 2414[label="",style="dashed", color="red", weight=0];
16.11/6.23	2431[label="List.nubByNubBy'1 primEqChar xw1560 xw1561 (xw157 : xw158) (List.elem_by primEqChar xw1560 (xw157 : xw158))",fontsize=16,color="magenta"];2431 -> 2434[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2431 -> 2435[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2431 -> 2436[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2432[label="[]",fontsize=16,color="green",shape="box"];2433[label="xw155 : List.nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="green",shape="box"];2433 -> 2437[label="",style="dashed", color="green", weight=3];
16.11/6.23	2434[label="xw157 : xw158",fontsize=16,color="green",shape="box"];2435[label="xw1561",fontsize=16,color="green",shape="box"];2436[label="xw1560",fontsize=16,color="green",shape="box"];2437 -> 2418[label="",style="dashed", color="red", weight=0];
16.11/6.23	2437[label="List.nubByNubBy' primEqChar xw156 (xw155 : xw157 : xw158)",fontsize=16,color="magenta"];2437 -> 2438[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2437 -> 2439[label="",style="dashed", color="magenta", weight=3];
16.11/6.23	2438[label="xw155",fontsize=16,color="green",shape="box"];2439[label="xw157 : xw158",fontsize=16,color="green",shape="box"];}
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(10)
16.11/6.23	Complex Obligation (AND)
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(11)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) -> new_psPs(xw81, xw9, xw10, xw11, ba)
16.11/6.23	
16.11/6.23	R is empty.
16.11/6.23	Q is empty.
16.11/6.23	We have to consider all minimal (P,Q,R)-chains.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(12) QDPSizeChangeProof (EQUIVALENT)
16.11/6.23	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. 
16.11/6.23	
16.11/6.23	From the DPs we obtained the following set of size-change graphs:
16.11/6.23	*new_psPs(:(xw80, xw81), xw9, xw10, xw11, ba) -> new_psPs(xw81, xw9, xw10, xw11, ba)
16.11/6.23	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5
16.11/6.23	
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(13)
16.11/6.23	YES
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(14)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_deleteBy0(xw17, xw18, xw19, False, ba) -> new_deleteBy(xw19, xw17, ba)
16.11/6.23	   new_deleteBy(xw10, :(xw90, xw91), bb) -> new_deleteBy0(xw91, xw90, xw10, new_esEs(xw10, xw90, bb), bb)
16.11/6.23	
16.11/6.23	The TRS R consists of the following rules:
16.11/6.23	
16.11/6.23	   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
16.11/6.23	   new_esEs(xw10, xw90, app(ty_Ratio, bf)) -> error([])
16.11/6.23	   new_primEqNat0(Zero, Zero) -> True
16.11/6.23	   new_esEs(xw10, xw90, app(app(app(ty_@3, cb), cc), cd)) -> error([])
16.11/6.23	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.23	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.23	   new_esEs(xw10, xw90, ty_Integer) -> error([])
16.11/6.23	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.23	   new_esEs(xw10, xw90, app(ty_[], bc)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Ordering) -> error([])
16.11/6.23	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.23	   new_esEs(xw10, xw90, app(app(ty_@2, bd), be)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Float) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Bool) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_@0) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Int) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, app(app(ty_Either, bg), bh)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, app(ty_Maybe, ca)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Double) -> error([])
16.11/6.23	
16.11/6.23	The set Q consists of the following terms:
16.11/6.23	
16.11/6.23	   new_esEs(x0, x1, app(ty_[], x2))
16.11/6.23	   new_esEs(x0, x1, ty_Float)
16.11/6.23	   new_primEqNat0(Zero, Zero)
16.11/6.23	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Zero)
16.11/6.23	   new_primEqNat0(Zero, Succ(x0))
16.11/6.23	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
16.11/6.23	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
16.11/6.23	   new_esEs(x0, x1, ty_Char)
16.11/6.23	   new_esEs(x0, x1, ty_Int)
16.11/6.23	   new_esEs(x0, x1, ty_Ordering)
16.11/6.23	   new_esEs(x0, x1, ty_Integer)
16.11/6.23	   new_primEqChar(Char(x0), Char(x1))
16.11/6.23	   new_esEs(x0, x1, ty_@0)
16.11/6.23	   new_esEs(x0, x1, ty_Double)
16.11/6.23	   new_esEs(x0, x1, app(ty_Ratio, x2))
16.11/6.23	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
16.11/6.23	   new_esEs(x0, x1, ty_Bool)
16.11/6.23	   new_esEs(x0, x1, app(ty_Maybe, x2))
16.11/6.23	
16.11/6.23	We have to consider all minimal (P,Q,R)-chains.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(15) QDPSizeChangeProof (EQUIVALENT)
16.11/6.23	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. 
16.11/6.23	
16.11/6.23	From the DPs we obtained the following set of size-change graphs:
16.11/6.23	*new_deleteBy(xw10, :(xw90, xw91), bb) -> new_deleteBy0(xw91, xw90, xw10, new_esEs(xw10, xw90, bb), bb)
16.11/6.23	The graph contains the following edges 2 > 1, 2 > 2, 1 >= 3, 3 >= 5
16.11/6.23	
16.11/6.23	
16.11/6.23	*new_deleteBy0(xw17, xw18, xw19, False, ba) -> new_deleteBy(xw19, xw17, ba)
16.11/6.23	The graph contains the following edges 3 >= 1, 1 >= 2, 5 >= 3
16.11/6.23	
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(16)
16.11/6.23	YES
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(17)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	   new_nubByNubBy'10(xw155, xw156, xw157, xw158, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	
16.11/6.23	The TRS R consists of the following rules:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero) -> True
16.11/6.23	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.23	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.23	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.23	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.23	
16.11/6.23	The set Q consists of the following terms:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero)
16.11/6.23	   new_primEqChar(Char(x0), Char(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Zero)
16.11/6.23	   new_primEqNat0(Zero, Succ(x0))
16.11/6.23	
16.11/6.23	We have to consider all minimal (P,Q,R)-chains.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(18) DependencyGraphProof (EQUIVALENT)
16.11/6.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(19)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	
16.11/6.23	The TRS R consists of the following rules:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero) -> True
16.11/6.23	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.23	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.23	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.23	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.23	
16.11/6.23	The set Q consists of the following terms:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero)
16.11/6.23	   new_primEqChar(Char(x0), Char(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Zero)
16.11/6.23	   new_primEqNat0(Zero, Succ(x0))
16.11/6.23	
16.11/6.23	We have to consider all minimal (P,Q,R)-chains.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(20) TransformationProof (EQUIVALENT)
16.11/6.23	By instantiating [LPAR04] the rule new_nubByNubBy'10(xw155, xw156, xw157, xw158, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601) we obtained the following new rules [LPAR04]:
16.11/6.23	
16.11/6.23	   (new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3),new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3))
16.11/6.23	
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(21)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	   new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	   new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3)
16.11/6.23	
16.11/6.23	The TRS R consists of the following rules:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero) -> True
16.11/6.23	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.23	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.23	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.23	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.23	
16.11/6.23	The set Q consists of the following terms:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero)
16.11/6.23	   new_primEqChar(Char(x0), Char(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.23	   new_primEqNat0(Succ(x0), Zero)
16.11/6.23	   new_primEqNat0(Zero, Succ(x0))
16.11/6.23	
16.11/6.23	We have to consider all minimal (P,Q,R)-chains.
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(22) QDPSizeChangeProof (EQUIVALENT)
16.11/6.23	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. 
16.11/6.23	
16.11/6.23	From the DPs we obtained the following set of size-change graphs:
16.11/6.23	*new_nubByNubBy'(:(xw1560, xw1561), xw157, xw158) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 4
16.11/6.23	
16.11/6.23	
16.11/6.23	*new_nubByNubBy'10(z0, z1, z2, z3, :(z2, z3)) -> new_nubByNubBy'1(z0, z1, z2, z3, new_primEqChar(z2, z0), z3)
16.11/6.23	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 5 > 3, 4 >= 4, 5 > 4, 4 >= 6, 5 > 6
16.11/6.23	
16.11/6.23	
16.11/6.23	*new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, :(xw1600, xw1601)) -> new_nubByNubBy'1(xw155, xw156, xw157, xw158, new_primEqChar(xw1600, xw155), xw1601)
16.11/6.23	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 6 > 6
16.11/6.23	
16.11/6.23	
16.11/6.23	*new_nubByNubBy'1(xw155, xw156, xw157, xw158, False, []) -> new_nubByNubBy'(xw156, xw155, :(xw157, xw158))
16.11/6.23	The graph contains the following edges 2 >= 1, 1 >= 2
16.11/6.23	
16.11/6.23	
16.11/6.23	*new_nubByNubBy'1(xw155, :(xw1560, xw1561), xw157, xw158, True, xw160) -> new_nubByNubBy'10(xw1560, xw1561, xw157, xw158, :(xw157, xw158))
16.11/6.23	The graph contains the following edges 2 > 1, 2 > 2, 3 >= 3, 4 >= 4
16.11/6.23	
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(23)
16.11/6.23	YES
16.11/6.23	
16.11/6.23	----------------------------------------
16.11/6.23	
16.11/6.23	(24)
16.11/6.23	Obligation:
16.11/6.23	Q DP problem:
16.11/6.23	The TRS P consists of the following rules:
16.11/6.23	
16.11/6.23	   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba)
16.11/6.23	
16.11/6.23	The TRS R consists of the following rules:
16.11/6.23	
16.11/6.23	   new_primEqNat0(Zero, Zero) -> True
16.11/6.23	   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Integer) -> error([])
16.11/6.23	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.23	   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_@0) -> error([])
16.11/6.23	   new_esEs(xw10, xw90, ty_Double) -> error([])
16.11/6.23	   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
16.11/6.23	   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
16.11/6.23	   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
16.11/6.23	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.23	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.23	   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
16.11/6.23	   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
16.11/6.24	   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
16.11/6.24	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.24	   new_esEs(xw10, xw90, ty_Ordering) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Float) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Bool) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Int) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
16.11/6.24	   new_deleteBy1(xw10, [], ba) -> []
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
16.11/6.24	
16.11/6.24	The set Q consists of the following terms:
16.11/6.24	
16.11/6.24	   new_esEs(x0, x1, ty_Float)
16.11/6.24	   new_primEqNat0(Zero, Zero)
16.11/6.24	   new_deleteBy00(x0, x1, x2, False, x3)
16.11/6.24	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
16.11/6.24	   new_primEqNat0(Succ(x0), Zero)
16.11/6.24	   new_esEs(x0, x1, app(ty_Ratio, x2))
16.11/6.24	   new_primEqNat0(Zero, Succ(x0))
16.11/6.24	   new_esEs(x0, x1, ty_Char)
16.11/6.24	   new_esEs(x0, x1, ty_Int)
16.11/6.24	   new_deleteBy1(x0, :(x1, x2), x3)
16.11/6.24	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
16.11/6.24	   new_esEs(x0, x1, ty_Ordering)
16.11/6.24	   new_esEs(x0, x1, app(ty_[], x2))
16.11/6.24	   new_deleteBy1(x0, [], x1)
16.11/6.24	   new_esEs(x0, x1, ty_Integer)
16.11/6.24	   new_primEqChar(Char(x0), Char(x1))
16.11/6.24	   new_flip(x0, x1, x2)
16.11/6.24	   new_esEs(x0, x1, ty_@0)
16.11/6.24	   new_esEs(x0, x1, ty_Double)
16.11/6.24	   new_deleteBy00(x0, x1, x2, True, x3)
16.11/6.24	   new_esEs(x0, x1, ty_Bool)
16.11/6.24	   new_esEs(x0, x1, app(ty_Maybe, x2))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
16.11/6.24	
16.11/6.24	We have to consider all minimal (P,Q,R)-chains.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(25) TransformationProof (EQUIVALENT)
16.11/6.24	By rewriting [LPAR04] the rule new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_flip(xw9, xw10, ba), xw110, xw111, ba) at position [0] we obtained the following new rules [LPAR04]:
16.11/6.24	
16.11/6.24	   (new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba),new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba))
16.11/6.24	
16.11/6.24	
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(26)
16.11/6.24	Obligation:
16.11/6.24	Q DP problem:
16.11/6.24	The TRS P consists of the following rules:
16.11/6.24	
16.11/6.24	   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)
16.11/6.24	
16.11/6.24	The TRS R consists of the following rules:
16.11/6.24	
16.11/6.24	   new_primEqNat0(Zero, Zero) -> True
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Integer) -> error([])
16.11/6.24	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_@0) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Double) -> error([])
16.11/6.24	   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
16.11/6.24	   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
16.11/6.24	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.24	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.24	   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
16.11/6.24	   new_flip(xw9, xw10, ba) -> new_deleteBy1(xw10, xw9, ba)
16.11/6.24	   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
16.11/6.24	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.24	   new_esEs(xw10, xw90, ty_Ordering) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Float) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Bool) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Int) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
16.11/6.24	   new_deleteBy1(xw10, [], ba) -> []
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
16.11/6.24	
16.11/6.24	The set Q consists of the following terms:
16.11/6.24	
16.11/6.24	   new_esEs(x0, x1, ty_Float)
16.11/6.24	   new_primEqNat0(Zero, Zero)
16.11/6.24	   new_deleteBy00(x0, x1, x2, False, x3)
16.11/6.24	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
16.11/6.24	   new_primEqNat0(Succ(x0), Zero)
16.11/6.24	   new_esEs(x0, x1, app(ty_Ratio, x2))
16.11/6.24	   new_primEqNat0(Zero, Succ(x0))
16.11/6.24	   new_esEs(x0, x1, ty_Char)
16.11/6.24	   new_esEs(x0, x1, ty_Int)
16.11/6.24	   new_deleteBy1(x0, :(x1, x2), x3)
16.11/6.24	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
16.11/6.24	   new_esEs(x0, x1, ty_Ordering)
16.11/6.24	   new_esEs(x0, x1, app(ty_[], x2))
16.11/6.24	   new_deleteBy1(x0, [], x1)
16.11/6.24	   new_esEs(x0, x1, ty_Integer)
16.11/6.24	   new_primEqChar(Char(x0), Char(x1))
16.11/6.24	   new_flip(x0, x1, x2)
16.11/6.24	   new_esEs(x0, x1, ty_@0)
16.11/6.24	   new_esEs(x0, x1, ty_Double)
16.11/6.24	   new_deleteBy00(x0, x1, x2, True, x3)
16.11/6.24	   new_esEs(x0, x1, ty_Bool)
16.11/6.24	   new_esEs(x0, x1, app(ty_Maybe, x2))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
16.11/6.24	
16.11/6.24	We have to consider all minimal (P,Q,R)-chains.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(27) UsableRulesProof (EQUIVALENT)
16.11/6.24	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(28)
16.11/6.24	Obligation:
16.11/6.24	Q DP problem:
16.11/6.24	The TRS P consists of the following rules:
16.11/6.24	
16.11/6.24	   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)
16.11/6.24	
16.11/6.24	The TRS R consists of the following rules:
16.11/6.24	
16.11/6.24	   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
16.11/6.24	   new_deleteBy1(xw10, [], ba) -> []
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Integer) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_@0) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Double) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
16.11/6.24	   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Ordering) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Float) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Bool) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Int) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
16.11/6.24	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.24	   new_primEqNat0(Zero, Zero) -> True
16.11/6.24	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.24	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.24	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.24	
16.11/6.24	The set Q consists of the following terms:
16.11/6.24	
16.11/6.24	   new_esEs(x0, x1, ty_Float)
16.11/6.24	   new_primEqNat0(Zero, Zero)
16.11/6.24	   new_deleteBy00(x0, x1, x2, False, x3)
16.11/6.24	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
16.11/6.24	   new_primEqNat0(Succ(x0), Zero)
16.11/6.24	   new_esEs(x0, x1, app(ty_Ratio, x2))
16.11/6.24	   new_primEqNat0(Zero, Succ(x0))
16.11/6.24	   new_esEs(x0, x1, ty_Char)
16.11/6.24	   new_esEs(x0, x1, ty_Int)
16.11/6.24	   new_deleteBy1(x0, :(x1, x2), x3)
16.11/6.24	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
16.11/6.24	   new_esEs(x0, x1, ty_Ordering)
16.11/6.24	   new_esEs(x0, x1, app(ty_[], x2))
16.11/6.24	   new_deleteBy1(x0, [], x1)
16.11/6.24	   new_esEs(x0, x1, ty_Integer)
16.11/6.24	   new_primEqChar(Char(x0), Char(x1))
16.11/6.24	   new_flip(x0, x1, x2)
16.11/6.24	   new_esEs(x0, x1, ty_@0)
16.11/6.24	   new_esEs(x0, x1, ty_Double)
16.11/6.24	   new_deleteBy00(x0, x1, x2, True, x3)
16.11/6.24	   new_esEs(x0, x1, ty_Bool)
16.11/6.24	   new_esEs(x0, x1, app(ty_Maybe, x2))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
16.11/6.24	
16.11/6.24	We have to consider all minimal (P,Q,R)-chains.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(29) QReductionProof (EQUIVALENT)
16.11/6.24	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
16.11/6.24	
16.11/6.24	   new_flip(x0, x1, x2)
16.11/6.24	
16.11/6.24	
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(30)
16.11/6.24	Obligation:
16.11/6.24	Q DP problem:
16.11/6.24	The TRS P consists of the following rules:
16.11/6.24	
16.11/6.24	   new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)
16.11/6.24	
16.11/6.24	The TRS R consists of the following rules:
16.11/6.24	
16.11/6.24	   new_deleteBy1(xw10, :(xw90, xw91), ba) -> new_deleteBy00(xw91, xw90, xw10, new_esEs(xw10, xw90, ba), ba)
16.11/6.24	   new_deleteBy1(xw10, [], ba) -> []
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Ratio, be)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Integer) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_@2, bc), bd)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_@0) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Double) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Char) -> new_primEqChar(xw10, xw90)
16.11/6.24	   new_esEs(xw10, xw90, app(app(app(ty_@3, ca), cb), cc)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(ty_[], bb)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Ordering) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Float) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Bool) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, ty_Int) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(app(ty_Either, bf), bg)) -> error([])
16.11/6.24	   new_esEs(xw10, xw90, app(ty_Maybe, bh)) -> error([])
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, True, cd) -> xw17
16.11/6.24	   new_deleteBy00(xw17, xw18, xw19, False, cd) -> :(xw18, new_deleteBy1(xw19, xw17, cd))
16.11/6.24	   new_primEqChar(Char(xw100), Char(xw900)) -> new_primEqNat0(xw100, xw900)
16.11/6.24	   new_primEqNat0(Zero, Zero) -> True
16.11/6.24	   new_primEqNat0(Succ(xw1000), Zero) -> False
16.11/6.24	   new_primEqNat0(Zero, Succ(xw9000)) -> False
16.11/6.24	   new_primEqNat0(Succ(xw1000), Succ(xw9000)) -> new_primEqNat0(xw1000, xw9000)
16.11/6.24	
16.11/6.24	The set Q consists of the following terms:
16.11/6.24	
16.11/6.24	   new_esEs(x0, x1, ty_Float)
16.11/6.24	   new_primEqNat0(Zero, Zero)
16.11/6.24	   new_deleteBy00(x0, x1, x2, False, x3)
16.11/6.24	   new_primEqNat0(Succ(x0), Succ(x1))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_Either, x2), x3))
16.11/6.24	   new_primEqNat0(Succ(x0), Zero)
16.11/6.24	   new_esEs(x0, x1, app(ty_Ratio, x2))
16.11/6.24	   new_primEqNat0(Zero, Succ(x0))
16.11/6.24	   new_esEs(x0, x1, ty_Char)
16.11/6.24	   new_esEs(x0, x1, ty_Int)
16.11/6.24	   new_deleteBy1(x0, :(x1, x2), x3)
16.11/6.24	   new_esEs(x0, x1, app(app(app(ty_@3, x2), x3), x4))
16.11/6.24	   new_esEs(x0, x1, ty_Ordering)
16.11/6.24	   new_esEs(x0, x1, app(ty_[], x2))
16.11/6.24	   new_deleteBy1(x0, [], x1)
16.11/6.24	   new_esEs(x0, x1, ty_Integer)
16.11/6.24	   new_primEqChar(Char(x0), Char(x1))
16.11/6.24	   new_esEs(x0, x1, ty_@0)
16.11/6.24	   new_esEs(x0, x1, ty_Double)
16.11/6.24	   new_deleteBy00(x0, x1, x2, True, x3)
16.11/6.24	   new_esEs(x0, x1, ty_Bool)
16.11/6.24	   new_esEs(x0, x1, app(ty_Maybe, x2))
16.11/6.24	   new_esEs(x0, x1, app(app(ty_@2, x2), x3))
16.11/6.24	
16.11/6.24	We have to consider all minimal (P,Q,R)-chains.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(31) QDPSizeChangeProof (EQUIVALENT)
16.11/6.24	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. 
16.11/6.24	
16.11/6.24	From the DPs we obtained the following set of size-change graphs:
16.11/6.24	*new_foldl(xw9, xw10, :(xw110, xw111), ba) -> new_foldl(new_deleteBy1(xw10, xw9, ba), xw110, xw111, ba)
16.11/6.24	The graph contains the following edges 3 > 2, 3 > 3, 4 >= 4
16.11/6.24	
16.11/6.24	
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(32)
16.11/6.24	YES
16.11/6.24	
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(33)
16.11/6.24	Obligation:
16.11/6.24	Q DP problem:
16.11/6.24	The TRS P consists of the following rules:
16.11/6.24	
16.11/6.24	   new_primEqNat(Succ(xw1000), Succ(xw9000)) -> new_primEqNat(xw1000, xw9000)
16.11/6.24	
16.11/6.24	R is empty.
16.11/6.24	Q is empty.
16.11/6.24	We have to consider all minimal (P,Q,R)-chains.
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(34) QDPSizeChangeProof (EQUIVALENT)
16.11/6.24	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. 
16.11/6.24	
16.11/6.24	From the DPs we obtained the following set of size-change graphs:
16.11/6.24	*new_primEqNat(Succ(xw1000), Succ(xw9000)) -> new_primEqNat(xw1000, xw9000)
16.11/6.24	The graph contains the following edges 1 > 1, 2 > 2
16.11/6.24	
16.11/6.24	
16.11/6.24	----------------------------------------
16.11/6.24	
16.11/6.24	(35)
16.11/6.24	YES
16.11/6.28	EOF