10.75/4.42	YES
13.12/5.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
13.12/5.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.12/5.09	
13.12/5.09	
13.12/5.09	H-Termination with start terms of the given HASKELL could be proven:
13.12/5.09	
13.12/5.09	(0) HASKELL
13.12/5.09	(1) LR [EQUIVALENT, 0 ms]
13.12/5.09	(2) HASKELL
13.12/5.09	(3) CR [EQUIVALENT, 0 ms]
13.12/5.09	(4) HASKELL
13.12/5.09	(5) IFR [EQUIVALENT, 0 ms]
13.12/5.09	(6) HASKELL
13.12/5.09	(7) BR [EQUIVALENT, 0 ms]
13.12/5.09	(8) HASKELL
13.12/5.09	(9) COR [EQUIVALENT, 0 ms]
13.12/5.09	(10) HASKELL
13.12/5.09	(11) Narrow [SOUND, 0 ms]
13.12/5.09	(12) AND
13.12/5.09	    (13) QDP
13.12/5.09	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.12/5.09	        (15) YES
13.12/5.09	    (16) QDP
13.12/5.09	        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
13.12/5.09	        (18) AND
13.12/5.09	            (19) QDP
13.12/5.09	                (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.12/5.09	                (21) YES
13.12/5.09	            (22) QDP
13.12/5.09	                (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
13.12/5.09	                (24) YES
13.12/5.09	
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(0)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (\vv2 ->case vv2 of { 
13.12/5.09	x->  if any (eq x) ys then x : [] else []; 
13.12/5.09	_-> []; 
13.12/5.09	 } ) xs;
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(1) LR (EQUIVALENT)
13.12/5.09	Lambda Reductions:
13.12/5.09	The following Lambda expression
13.12/5.09	"\vv2->case vv2 of {
13.12/5.09	x  -> if any (eq x) ys then x : [] else [];
13.12/5.09	_  -> []}
13.12/5.09	"
13.12/5.09	is transformed to
13.12/5.09	"intersectBy0 eq ys vv2 = case vv2 of {
13.12/5.09	x  -> if any (eq x) ys then x : [] else [];
13.12/5.09	_  -> []}
13.12/5.09	;
13.12/5.09	"
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(2)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
13.12/5.09	
13.12/5.09	intersectBy0 eq ys vv2 = case vv2 of { 
13.12/5.09	x->  if any (eq x) ys then x : [] else []; 
13.12/5.09	_-> []; 
13.12/5.09	 } ;
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(3) CR (EQUIVALENT)
13.12/5.09	Case Reductions:
13.12/5.09	The following Case expression
13.12/5.09	"case vv2 of {
13.12/5.09	x  -> if any (eq x) ys then x : [] else [];
13.12/5.09	_  -> []}
13.12/5.09	"
13.12/5.09	is transformed to
13.12/5.09	"intersectBy00 eq ys x = if any (eq x) ys then x : [] else [];
13.12/5.09	intersectBy00 eq ys _ = [];
13.12/5.09	"
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(4)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
13.12/5.09	
13.12/5.09	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
13.12/5.09	
13.12/5.09	intersectBy00 eq ys x =  if any (eq x) ys then x : [] else [];
13.12/5.09	intersectBy00 eq ys _ = [];
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(5) IFR (EQUIVALENT)
13.12/5.09	If Reductions:
13.12/5.09	The following If expression
13.12/5.09	"if any (eq x) ys then x : [] else []"
13.12/5.09	is transformed to
13.12/5.09	"intersectBy000 x True = x : [];
13.12/5.09	intersectBy000 x False = [];
13.12/5.09	"
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(6)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
13.12/5.09	
13.12/5.09	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
13.12/5.09	
13.12/5.09	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
13.12/5.09	intersectBy00 eq ys _ = [];
13.12/5.09	
13.12/5.09	intersectBy000 x True = x : [];
13.12/5.09	intersectBy000 x False = [];
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(7) BR (EQUIVALENT)
13.12/5.09	Replaced joker patterns by fresh variables and removed binding patterns.
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(8)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
13.12/5.09	
13.12/5.09	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
13.12/5.09	
13.12/5.09	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
13.12/5.09	intersectBy00 eq ys vy = [];
13.12/5.09	
13.12/5.09	intersectBy000 x True = x : [];
13.12/5.09	intersectBy000 x False = [];
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(9) COR (EQUIVALENT)
13.12/5.09	Cond Reductions:
13.12/5.09	The following Function with conditions
13.12/5.09	"undefined |Falseundefined;
13.12/5.09	"
13.12/5.09	is transformed to
13.12/5.09	"undefined  = undefined1;
13.12/5.09	"
13.12/5.09	"undefined0 True = undefined;
13.12/5.09	"
13.12/5.09	"undefined1  = undefined0 False;
13.12/5.09	"
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(10)
13.12/5.09	Obligation:
13.12/5.09	mainModule Main 
13.12/5.09	module Maybe where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	module List where {
13.12/5.09	import qualified Main;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	intersect :: Eq a => [a]  ->  [a]  ->  [a];
13.12/5.09	intersect = intersectBy (==);
13.12/5.09	
13.12/5.09	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
13.12/5.09	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
13.12/5.09	
13.12/5.09	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
13.12/5.09	
13.12/5.09	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
13.12/5.09	intersectBy00 eq ys vy = [];
13.12/5.09	
13.12/5.09	intersectBy000 x True = x : [];
13.12/5.09	intersectBy000 x False = [];
13.12/5.09	
13.12/5.09	}
13.12/5.09	module Main where {
13.12/5.09	import qualified List;
13.12/5.09	import qualified Maybe;
13.12/5.09	import qualified Prelude;
13.12/5.09	}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(11) Narrow (SOUND)
13.12/5.09	Haskell To QDPs
13.12/5.09	
13.12/5.09	digraph dp_graph {
13.12/5.09	node [outthreshold=100, inthreshold=100];1[label="List.intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
13.12/5.09	3[label="List.intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
13.12/5.09	4[label="List.intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
13.12/5.09	5[label="List.intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
13.12/5.09	6[label="concatMap (List.intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
13.12/5.09	7[label="concat . map (List.intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
13.12/5.09	8[label="concat (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
13.12/5.09	9[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];56[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 56[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	56 -> 10[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	57[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 57[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	57 -> 11[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	10[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
13.12/5.09	11[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
13.12/5.09	12[label="foldr (++) [] (List.intersectBy0 (==) vz4 vz30 : map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13.12/5.09	13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
13.12/5.09	14 -> 16[label="",style="dashed", color="red", weight=0];
13.12/5.09	14[label="(++) List.intersectBy0 (==) vz4 vz30 foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
13.12/5.09	17[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	16[label="(++) List.intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
13.12/5.09	18[label="vz31",fontsize=16,color="green",shape="box"];19[label="(++) List.intersectBy00 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
13.12/5.09	20[label="(++) List.intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
13.12/5.09	21[label="(++) List.intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
13.12/5.09	22[label="(++) List.intersectBy000 vz30 (or (map ((==) vz30) vz4)) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
13.12/5.09	23[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="triangle"];58[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 58[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	58 -> 24[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	59[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 59[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	59 -> 25[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	24[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) (vz40 : vz41))) vz5",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
13.12/5.09	25[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) [])) vz5",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
13.12/5.09	26[label="(++) List.intersectBy000 vz30 (foldr (||) False (((==) vz30 vz40) : map ((==) vz30) vz41)) vz5",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
13.12/5.09	27[label="(++) List.intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
13.12/5.09	28[label="(++) List.intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];60[label="vz30/False",fontsize=10,color="white",style="solid",shape="box"];28 -> 60[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	60 -> 30[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	61[label="vz30/True",fontsize=10,color="white",style="solid",shape="box"];28 -> 61[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	61 -> 31[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
13.12/5.09	30[label="(++) List.intersectBy000 False ((||) (==) False vz40 foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];62[label="vz40/False",fontsize=10,color="white",style="solid",shape="box"];30 -> 62[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	62 -> 33[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	63[label="vz40/True",fontsize=10,color="white",style="solid",shape="box"];30 -> 63[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	63 -> 34[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	31[label="(++) List.intersectBy000 True ((||) (==) True vz40 foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];64[label="vz40/False",fontsize=10,color="white",style="solid",shape="box"];31 -> 64[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	64 -> 35[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	65[label="vz40/True",fontsize=10,color="white",style="solid",shape="box"];31 -> 65[label="",style="solid", color="burlywood", weight=9];
13.12/5.09	65 -> 36[label="",style="solid", color="burlywood", weight=3];
13.12/5.09	32[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3];
13.12/5.09	33[label="(++) List.intersectBy000 False ((||) (==) False False foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3];
13.12/5.09	34[label="(++) List.intersectBy000 False ((||) (==) False True foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
13.12/5.09	35[label="(++) List.intersectBy000 True ((||) (==) True False foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
13.12/5.09	36[label="(++) List.intersectBy000 True ((||) (==) True True foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
13.12/5.09	37[label="vz5",fontsize=16,color="green",shape="box"];38[label="(++) List.intersectBy000 False ((||) True foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
13.12/5.09	39[label="(++) List.intersectBy000 False ((||) False foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
13.12/5.09	40[label="(++) List.intersectBy000 True ((||) False foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
13.12/5.09	41[label="(++) List.intersectBy000 True ((||) True foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
13.12/5.09	42[label="(++) List.intersectBy000 False True vz5",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3];
13.12/5.09	43 -> 23[label="",style="dashed", color="red", weight=0];
13.12/5.09	43[label="(++) List.intersectBy000 False (foldr (||) False (map ((==) False) vz41)) vz5",fontsize=16,color="magenta"];43 -> 47[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	43 -> 48[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	44 -> 23[label="",style="dashed", color="red", weight=0];
13.12/5.09	44[label="(++) List.intersectBy000 True (foldr (||) False (map ((==) True) vz41)) vz5",fontsize=16,color="magenta"];44 -> 49[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	44 -> 50[label="",style="dashed", color="magenta", weight=3];
13.12/5.09	45[label="(++) List.intersectBy000 True True vz5",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
13.12/5.09	46[label="(++) (False : []) vz5",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
13.12/5.09	47[label="vz41",fontsize=16,color="green",shape="box"];48[label="False",fontsize=16,color="green",shape="box"];49[label="vz41",fontsize=16,color="green",shape="box"];50[label="True",fontsize=16,color="green",shape="box"];51[label="(++) (True : []) vz5",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3];
13.12/5.09	52[label="False : [] ++ vz5",fontsize=16,color="green",shape="box"];52 -> 54[label="",style="dashed", color="green", weight=3];
13.12/5.09	53[label="True : [] ++ vz5",fontsize=16,color="green",shape="box"];53 -> 55[label="",style="dashed", color="green", weight=3];
13.12/5.09	54 -> 32[label="",style="dashed", color="red", weight=0];
13.12/5.09	54[label="[] ++ vz5",fontsize=16,color="magenta"];55 -> 32[label="",style="dashed", color="red", weight=0];
13.12/5.09	55[label="[] ++ vz5",fontsize=16,color="magenta"];}
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(12)
13.12/5.09	Complex Obligation (AND)
13.12/5.09	
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(13)
13.12/5.09	Obligation:
13.12/5.09	Q DP problem:
13.12/5.09	The TRS P consists of the following rules:
13.12/5.09	
13.12/5.09	   new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)
13.12/5.09	
13.12/5.09	R is empty.
13.12/5.09	Q is empty.
13.12/5.09	We have to consider all minimal (P,Q,R)-chains.
13.12/5.09	----------------------------------------
13.12/5.09	
13.12/5.09	(14) QDPSizeChangeProof (EQUIVALENT)
13.12/5.09	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. 
13.12/5.09	
13.12/5.09	From the DPs we obtained the following set of size-change graphs:
13.12/5.09	*new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)
13.12/5.09	The graph contains the following edges 1 >= 1, 2 > 2
13.12/5.10	
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(15)
13.12/5.10	YES
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(16)
13.12/5.10	Obligation:
13.12/5.10	Q DP problem:
13.12/5.10	The TRS P consists of the following rules:
13.12/5.10	
13.12/5.10	   new_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)
13.12/5.10	   new_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)
13.12/5.10	
13.12/5.10	R is empty.
13.12/5.10	Q is empty.
13.12/5.10	We have to consider all minimal (P,Q,R)-chains.
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(17) DependencyGraphProof (EQUIVALENT)
13.12/5.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(18)
13.12/5.10	Complex Obligation (AND)
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(19)
13.12/5.10	Obligation:
13.12/5.10	Q DP problem:
13.12/5.10	The TRS P consists of the following rules:
13.12/5.10	
13.12/5.10	   new_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)
13.12/5.10	
13.12/5.10	R is empty.
13.12/5.10	Q is empty.
13.12/5.10	We have to consider all minimal (P,Q,R)-chains.
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(20) QDPSizeChangeProof (EQUIVALENT)
13.12/5.10	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. 
13.12/5.10	
13.12/5.10	From the DPs we obtained the following set of size-change graphs:
13.12/5.10	*new_psPs(True, :(False, vz41), vz5) -> new_psPs(True, vz41, vz5)
13.12/5.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
13.12/5.10	
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(21)
13.12/5.10	YES
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(22)
13.12/5.10	Obligation:
13.12/5.10	Q DP problem:
13.12/5.10	The TRS P consists of the following rules:
13.12/5.10	
13.12/5.10	   new_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)
13.12/5.10	
13.12/5.10	R is empty.
13.12/5.10	Q is empty.
13.12/5.10	We have to consider all minimal (P,Q,R)-chains.
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(23) QDPSizeChangeProof (EQUIVALENT)
13.12/5.10	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. 
13.12/5.10	
13.12/5.10	From the DPs we obtained the following set of size-change graphs:
13.12/5.10	*new_psPs(False, :(True, vz41), vz5) -> new_psPs(False, vz41, vz5)
13.12/5.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
13.12/5.10	
13.12/5.10	
13.12/5.10	----------------------------------------
13.12/5.10	
13.12/5.10	(24)
13.12/5.10	YES
13.12/5.13	EOF