9.87/4.34	YES
12.27/4.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.27/4.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.27/4.98	
12.27/4.98	
12.27/4.98	H-Termination with start terms of the given HASKELL could be proven:
12.27/4.98	
12.27/4.98	(0) HASKELL
12.27/4.98	(1) LR [EQUIVALENT, 0 ms]
12.27/4.98	(2) HASKELL
12.27/4.98	(3) CR [EQUIVALENT, 0 ms]
12.27/4.98	(4) HASKELL
12.27/4.98	(5) IFR [EQUIVALENT, 0 ms]
12.27/4.98	(6) HASKELL
12.27/4.98	(7) BR [EQUIVALENT, 0 ms]
12.27/4.98	(8) HASKELL
12.27/4.98	(9) COR [EQUIVALENT, 28 ms]
12.27/4.98	(10) HASKELL
12.27/4.98	(11) Narrow [SOUND, 0 ms]
12.27/4.98	(12) AND
12.27/4.98	    (13) QDP
12.27/4.98	        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
12.27/4.98	        (15) AND
12.27/4.98	            (16) QDP
12.27/4.98	                (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.27/4.98	                (18) YES
12.27/4.98	            (19) QDP
12.27/4.98	                (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.27/4.98	                (21) YES
12.27/4.98	    (22) QDP
12.27/4.98	        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.27/4.98	        (24) YES
12.27/4.98	
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(0)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (\vv2 ->case vv2 of { 
12.27/4.98	x->  if any (eq x) ys then x : [] else []; 
12.27/4.98	_-> []; 
12.27/4.98	 } ) xs;
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(1) LR (EQUIVALENT)
12.27/4.98	Lambda Reductions:
12.27/4.98	The following Lambda expression
12.27/4.98	"\vv2->case vv2 of {
12.27/4.98	x  -> if any (eq x) ys then x : [] else [];
12.27/4.98	_  -> []}
12.27/4.98	"
12.27/4.98	is transformed to
12.27/4.98	"intersectBy0 eq ys vv2 = case vv2 of {
12.27/4.98	x  -> if any (eq x) ys then x : [] else [];
12.27/4.98	_  -> []}
12.27/4.98	;
12.27/4.98	"
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(2)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
12.27/4.98	
12.27/4.98	intersectBy0 eq ys vv2 = case vv2 of { 
12.27/4.98	x->  if any (eq x) ys then x : [] else []; 
12.27/4.98	_-> []; 
12.27/4.98	 } ;
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(3) CR (EQUIVALENT)
12.27/4.98	Case Reductions:
12.27/4.98	The following Case expression
12.27/4.98	"case vv2 of {
12.27/4.98	x  -> if any (eq x) ys then x : [] else [];
12.27/4.98	_  -> []}
12.27/4.98	"
12.27/4.98	is transformed to
12.27/4.98	"intersectBy00 eq ys x = if any (eq x) ys then x : [] else [];
12.27/4.98	intersectBy00 eq ys _ = [];
12.27/4.98	"
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(4)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
12.27/4.98	
12.27/4.98	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
12.27/4.98	
12.27/4.98	intersectBy00 eq ys x =  if any (eq x) ys then x : [] else [];
12.27/4.98	intersectBy00 eq ys _ = [];
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(5) IFR (EQUIVALENT)
12.27/4.98	If Reductions:
12.27/4.98	The following If expression
12.27/4.98	"if any (eq x) ys then x : [] else []"
12.27/4.98	is transformed to
12.27/4.98	"intersectBy000 x True = x : [];
12.27/4.98	intersectBy000 x False = [];
12.27/4.98	"
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(6)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
12.27/4.98	
12.27/4.98	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
12.27/4.98	
12.27/4.98	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
12.27/4.98	intersectBy00 eq ys _ = [];
12.27/4.98	
12.27/4.98	intersectBy000 x True = x : [];
12.27/4.98	intersectBy000 x False = [];
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(7) BR (EQUIVALENT)
12.27/4.98	Replaced joker patterns by fresh variables and removed binding patterns.
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(8)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
12.27/4.98	
12.27/4.98	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
12.27/4.98	
12.27/4.98	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
12.27/4.98	intersectBy00 eq ys vy = [];
12.27/4.98	
12.27/4.98	intersectBy000 x True = x : [];
12.27/4.98	intersectBy000 x False = [];
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(9) COR (EQUIVALENT)
12.27/4.98	Cond Reductions:
12.27/4.98	The following Function with conditions
12.27/4.98	"undefined |Falseundefined;
12.27/4.98	"
12.27/4.98	is transformed to
12.27/4.98	"undefined  = undefined1;
12.27/4.98	"
12.27/4.98	"undefined0 True = undefined;
12.27/4.98	"
12.27/4.98	"undefined1  = undefined0 False;
12.27/4.98	"
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(10)
12.27/4.98	Obligation:
12.27/4.98	mainModule Main 
12.27/4.98	module Maybe where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	module List where {
12.27/4.98	import qualified Main;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	intersect :: Eq a => [a]  ->  [a]  ->  [a];
12.27/4.98	intersect = intersectBy (==);
12.27/4.98	
12.27/4.98	intersectBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.27/4.98	intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs;
12.27/4.98	
12.27/4.98	intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2;
12.27/4.98	
12.27/4.98	intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys);
12.27/4.98	intersectBy00 eq ys vy = [];
12.27/4.98	
12.27/4.98	intersectBy000 x True = x : [];
12.27/4.98	intersectBy000 x False = [];
12.27/4.98	
12.27/4.98	}
12.27/4.98	module Main where {
12.27/4.98	import qualified List;
12.27/4.98	import qualified Maybe;
12.27/4.98	import qualified Prelude;
12.27/4.98	}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(11) Narrow (SOUND)
12.27/4.98	Haskell To QDPs
12.27/4.98	
12.27/4.98	digraph dp_graph {
12.27/4.98	node [outthreshold=100, inthreshold=100];1[label="List.intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.27/4.98	3[label="List.intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.27/4.98	4[label="List.intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.27/4.98	5[label="List.intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
12.27/4.98	6[label="concatMap (List.intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
12.27/4.98	7[label="concat . map (List.intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
12.27/4.98	8[label="concat (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
12.27/4.98	9[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];271[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 271[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	271 -> 10[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	272[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 272[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	272 -> 11[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	10[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
12.27/4.98	11[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12.27/4.98	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];
12.27/4.98	13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
12.27/4.98	14 -> 16[label="",style="dashed", color="red", weight=0];
12.27/4.98	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];
12.27/4.98	15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
12.27/4.98	17[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	16[label="(++) List.intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
12.27/4.98	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];
12.27/4.98	20[label="(++) List.intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
12.27/4.98	21[label="(++) List.intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
12.27/4.98	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];
12.27/4.98	23[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="box"];273[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 273[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	273 -> 24[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	274[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 274[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	274 -> 25[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	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];
12.27/4.98	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];
12.27/4.98	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];
12.27/4.98	27[label="(++) List.intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="triangle"];27 -> 29[label="",style="solid", color="black", weight=3];
12.27/4.98	28[label="(++) List.intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
12.27/4.98	29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
12.27/4.98	30[label="(++) List.intersectBy000 vz30 ((||) primEqChar vz30 vz40 foldr (||) False (map (primEqChar vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="triangle"];275[label="vz30/Char vz300",fontsize=10,color="white",style="solid",shape="box"];30 -> 275[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	275 -> 32[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	31[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
12.27/4.98	32[label="(++) List.intersectBy000 (Char vz300) ((||) primEqChar (Char vz300) vz40 foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];276[label="vz40/Char vz400",fontsize=10,color="white",style="solid",shape="box"];32 -> 276[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	276 -> 34[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	33[label="vz5",fontsize=16,color="green",shape="box"];34[label="(++) List.intersectBy000 (Char vz300) ((||) primEqChar (Char vz300) (Char vz400) foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3];
12.27/4.98	35[label="(++) List.intersectBy000 (Char vz300) ((||) primEqNat vz300 vz400 foldr (||) False (map (primEqChar (Char vz300)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];277[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 277[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	277 -> 36[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	278[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 278[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	278 -> 37[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	36[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) vz400 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];279[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];36 -> 279[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	279 -> 38[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	280[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 280[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	280 -> 39[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	37[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero vz400 foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];281[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];37 -> 281[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	281 -> 40[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	282[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];37 -> 282[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	282 -> 41[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	38[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) (Succ vz4000) foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
12.27/4.98	39[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat (Succ vz3000) Zero foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3];
12.27/4.98	40[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero (Succ vz4000) foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
12.27/4.98	41[label="(++) List.intersectBy000 (Char Zero) ((||) primEqNat Zero Zero foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3];
12.27/4.98	42 -> 202[label="",style="dashed", color="red", weight=0];
12.27/4.98	42[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqNat vz3000 vz4000 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="magenta"];42 -> 203[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	42 -> 204[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	42 -> 205[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	42 -> 206[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	42 -> 207[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	43[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="black",shape="triangle"];43 -> 48[label="",style="solid", color="black", weight=3];
12.27/4.98	44[label="(++) List.intersectBy000 (Char Zero) ((||) False foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
12.27/4.98	45[label="(++) List.intersectBy000 (Char Zero) ((||) True foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
12.27/4.98	203[label="vz3000",fontsize=16,color="green",shape="box"];204[label="vz4000",fontsize=16,color="green",shape="box"];205[label="vz3000",fontsize=16,color="green",shape="box"];206[label="vz41",fontsize=16,color="green",shape="box"];207[label="vz5",fontsize=16,color="green",shape="box"];202[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat vz11 vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="triangle"];283[label="vz11/Succ vz110",fontsize=10,color="white",style="solid",shape="box"];202 -> 283[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	283 -> 248[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	284[label="vz11/Zero",fontsize=10,color="white",style="solid",shape="box"];202 -> 284[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	284 -> 249[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	48[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];285[label="vz41/vz410 : vz411",fontsize=10,color="white",style="solid",shape="box"];48 -> 285[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	285 -> 55[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	286[label="vz41/[]",fontsize=10,color="white",style="solid",shape="box"];48 -> 286[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	286 -> 56[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	49[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];287[label="vz41/vz410 : vz411",fontsize=10,color="white",style="solid",shape="box"];49 -> 287[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	287 -> 57[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	288[label="vz41/[]",fontsize=10,color="white",style="solid",shape="box"];49 -> 288[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	288 -> 58[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	50[label="(++) List.intersectBy000 (Char Zero) True vz5",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
12.27/4.98	248[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="box"];289[label="vz12/Succ vz120",fontsize=10,color="white",style="solid",shape="box"];248 -> 289[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	289 -> 250[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	290[label="vz12/Zero",fontsize=10,color="white",style="solid",shape="box"];248 -> 290[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	290 -> 251[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	249[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero vz12 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="burlywood",shape="box"];291[label="vz12/Succ vz120",fontsize=10,color="white",style="solid",shape="box"];249 -> 291[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	291 -> 252[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	292[label="vz12/Zero",fontsize=10,color="white",style="solid",shape="box"];249 -> 292[label="",style="solid", color="burlywood", weight=9];
12.27/4.98	292 -> 253[label="",style="solid", color="burlywood", weight=3];
12.27/4.98	55[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) (vz410 : vz411))) vz5",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
12.27/4.98	56[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (map (primEqChar (Char (Succ vz3000))) [])) vz5",fontsize=16,color="black",shape="box"];56 -> 65[label="",style="solid", color="black", weight=3];
12.27/4.98	57[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) (vz410 : vz411))) vz5",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
12.27/4.98	58[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (map (primEqChar (Char Zero)) [])) vz5",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
12.27/4.98	59[label="(++) (Char Zero : []) vz5",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3];
12.27/4.98	250[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) (Succ vz120) foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];250 -> 254[label="",style="solid", color="black", weight=3];
12.27/4.98	251[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat (Succ vz110) Zero foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];251 -> 255[label="",style="solid", color="black", weight=3];
12.27/4.98	252[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero (Succ vz120) foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];252 -> 256[label="",style="solid", color="black", weight=3];
12.27/4.98	253[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat Zero Zero foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];253 -> 257[label="",style="solid", color="black", weight=3];
12.27/4.98	64[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False (primEqChar (Char (Succ vz3000)) vz410 : map (primEqChar (Char (Succ vz3000))) vz411)) vz5",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
12.27/4.98	65 -> 27[label="",style="dashed", color="red", weight=0];
12.27/4.98	65[label="(++) List.intersectBy000 (Char (Succ vz3000)) (foldr (||) False []) vz5",fontsize=16,color="magenta"];65 -> 75[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	66[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False (primEqChar (Char Zero) vz410 : map (primEqChar (Char Zero)) vz411)) vz5",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
12.27/4.98	67 -> 27[label="",style="dashed", color="red", weight=0];
12.27/4.98	67[label="(++) List.intersectBy000 (Char Zero) (foldr (||) False []) vz5",fontsize=16,color="magenta"];67 -> 77[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	68[label="Char Zero : [] ++ vz5",fontsize=16,color="green",shape="box"];68 -> 78[label="",style="dashed", color="green", weight=3];
12.27/4.98	254 -> 202[label="",style="dashed", color="red", weight=0];
12.27/4.98	254[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) primEqNat vz110 vz120 foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];254 -> 258[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	254 -> 259[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	255 -> 43[label="",style="dashed", color="red", weight=0];
12.27/4.98	255[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];255 -> 260[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	255 -> 261[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	255 -> 262[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	256 -> 43[label="",style="dashed", color="red", weight=0];
12.27/4.98	256[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) False foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="magenta"];256 -> 263[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	256 -> 264[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	256 -> 265[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	257[label="(++) List.intersectBy000 (Char (Succ vz10)) ((||) True foldr (||) False (map (primEqChar (Char (Succ vz10))) vz13)) vz14",fontsize=16,color="black",shape="box"];257 -> 266[label="",style="solid", color="black", weight=3];
12.27/4.98	74 -> 30[label="",style="dashed", color="red", weight=0];
12.27/4.98	74[label="(++) List.intersectBy000 (Char (Succ vz3000)) ((||) primEqChar (Char (Succ vz3000)) vz410 foldr (||) False (map (primEqChar (Char (Succ vz3000))) vz411)) vz5",fontsize=16,color="magenta"];74 -> 84[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	74 -> 85[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	74 -> 86[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	75[label="Char (Succ vz3000)",fontsize=16,color="green",shape="box"];76 -> 30[label="",style="dashed", color="red", weight=0];
12.27/4.98	76[label="(++) List.intersectBy000 (Char Zero) ((||) primEqChar (Char Zero) vz410 foldr (||) False (map (primEqChar (Char Zero)) vz411)) vz5",fontsize=16,color="magenta"];76 -> 87[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	76 -> 88[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	76 -> 89[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	77[label="Char Zero",fontsize=16,color="green",shape="box"];78 -> 31[label="",style="dashed", color="red", weight=0];
12.27/4.98	78[label="[] ++ vz5",fontsize=16,color="magenta"];258[label="vz110",fontsize=16,color="green",shape="box"];259[label="vz120",fontsize=16,color="green",shape="box"];260[label="vz13",fontsize=16,color="green",shape="box"];261[label="vz10",fontsize=16,color="green",shape="box"];262[label="vz14",fontsize=16,color="green",shape="box"];263[label="vz13",fontsize=16,color="green",shape="box"];264[label="vz10",fontsize=16,color="green",shape="box"];265[label="vz14",fontsize=16,color="green",shape="box"];266[label="(++) List.intersectBy000 (Char (Succ vz10)) True vz14",fontsize=16,color="black",shape="box"];266 -> 267[label="",style="solid", color="black", weight=3];
12.27/4.98	84[label="vz411",fontsize=16,color="green",shape="box"];85[label="vz410",fontsize=16,color="green",shape="box"];86[label="Char (Succ vz3000)",fontsize=16,color="green",shape="box"];87[label="vz411",fontsize=16,color="green",shape="box"];88[label="vz410",fontsize=16,color="green",shape="box"];89[label="Char Zero",fontsize=16,color="green",shape="box"];267[label="(++) (Char (Succ vz10) : []) vz14",fontsize=16,color="black",shape="box"];267 -> 268[label="",style="solid", color="black", weight=3];
12.27/4.98	268[label="Char (Succ vz10) : [] ++ vz14",fontsize=16,color="green",shape="box"];268 -> 269[label="",style="dashed", color="green", weight=3];
12.27/4.98	269 -> 31[label="",style="dashed", color="red", weight=0];
12.27/4.98	269[label="[] ++ vz14",fontsize=16,color="magenta"];269 -> 270[label="",style="dashed", color="magenta", weight=3];
12.27/4.98	270[label="vz14",fontsize=16,color="green",shape="box"];}
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(12)
12.27/4.98	Complex Obligation (AND)
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(13)
12.27/4.98	Obligation:
12.27/4.98	Q DP problem:
12.27/4.98	The TRS P consists of the following rules:
12.27/4.98	
12.27/4.98	   new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
12.27/4.98	   new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.98	   new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.98	   new_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.98	   new_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)
12.27/4.98	   new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
12.27/4.98	   new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.98	
12.27/4.98	R is empty.
12.27/4.98	Q is empty.
12.27/4.98	We have to consider all minimal (P,Q,R)-chains.
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(14) DependencyGraphProof (EQUIVALENT)
12.27/4.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(15)
12.27/4.98	Complex Obligation (AND)
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(16)
12.27/4.98	Obligation:
12.27/4.98	Q DP problem:
12.27/4.98	The TRS P consists of the following rules:
12.27/4.98	
12.27/4.98	   new_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)
12.27/4.98	
12.27/4.98	R is empty.
12.27/4.98	Q is empty.
12.27/4.98	We have to consider all minimal (P,Q,R)-chains.
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(17) QDPSizeChangeProof (EQUIVALENT)
12.27/4.98	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
12.27/4.98	
12.27/4.98	From the DPs we obtained the following set of size-change graphs:
12.27/4.98	*new_psPs1(Char(Zero), Char(Succ(vz4000)), :(vz410, vz411), vz5) -> new_psPs1(Char(Zero), vz410, vz411, vz5)
12.27/4.98	The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4
12.27/4.98	
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(18)
12.27/4.98	YES
12.27/4.98	
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(19)
12.27/4.98	Obligation:
12.27/4.98	Q DP problem:
12.27/4.98	The TRS P consists of the following rules:
12.27/4.98	
12.27/4.98	   new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.98	   new_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.98	   new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
12.27/4.98	   new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
12.27/4.98	   new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.98	   new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.98	
12.27/4.98	R is empty.
12.27/4.98	Q is empty.
12.27/4.98	We have to consider all minimal (P,Q,R)-chains.
12.27/4.98	----------------------------------------
12.27/4.98	
12.27/4.98	(20) QDPSizeChangeProof (EQUIVALENT)
12.27/4.98	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
12.27/4.98	
12.27/4.98	From the DPs we obtained the following set of size-change graphs:
12.27/4.98	*new_psPs0(vz3000, :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.98	The graph contains the following edges 2 > 2, 2 > 3, 3 >= 4
12.27/4.98	
12.27/4.98	
12.27/4.98	*new_psPs1(Char(Succ(vz3000)), Char(Succ(vz4000)), vz41, vz5) -> new_psPs(vz3000, vz3000, vz4000, vz41, vz5)
12.27/4.99	The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 3 >= 4, 4 >= 5
12.27/4.99	
12.27/4.99	
12.27/4.99	*new_psPs(vz10, Succ(vz110), Succ(vz120), vz13, vz14) -> new_psPs(vz10, vz110, vz120, vz13, vz14)
12.27/4.99	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.27/4.99	
12.27/4.99	
12.27/4.99	*new_psPs1(Char(Succ(vz3000)), Char(Zero), :(vz410, vz411), vz5) -> new_psPs1(Char(Succ(vz3000)), vz410, vz411, vz5)
12.27/4.99	The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4
12.27/4.99	
12.27/4.99	
12.27/4.99	*new_psPs(vz10, Succ(vz110), Zero, vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.99	The graph contains the following edges 1 >= 1, 4 >= 2, 5 >= 3
12.27/4.99	
12.27/4.99	
12.27/4.99	*new_psPs(vz10, Zero, Succ(vz120), vz13, vz14) -> new_psPs0(vz10, vz13, vz14)
12.27/4.99	The graph contains the following edges 1 >= 1, 4 >= 2, 5 >= 3
12.27/4.99	
12.27/4.99	
12.27/4.99	----------------------------------------
12.27/4.99	
12.27/4.99	(21)
12.27/4.99	YES
12.27/4.99	
12.27/4.99	----------------------------------------
12.27/4.99	
12.27/4.99	(22)
12.27/4.99	Obligation:
12.27/4.99	Q DP problem:
12.27/4.99	The TRS P consists of the following rules:
12.27/4.99	
12.27/4.99	   new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)
12.27/4.99	
12.27/4.99	R is empty.
12.27/4.99	Q is empty.
12.27/4.99	We have to consider all minimal (P,Q,R)-chains.
12.27/4.99	----------------------------------------
12.27/4.99	
12.27/4.99	(23) QDPSizeChangeProof (EQUIVALENT)
12.27/4.99	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
12.27/4.99	
12.27/4.99	From the DPs we obtained the following set of size-change graphs:
12.27/4.99	*new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31)
12.27/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.27/4.99	
12.27/4.99	
12.27/4.99	----------------------------------------
12.27/4.99	
12.27/4.99	(24)
12.27/4.99	YES
12.39/9.42	EOF