7.54/3.49	YES
9.25/3.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.25/3.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.25/3.94	
9.25/3.94	
9.25/3.94	H-Termination with start terms of the given HASKELL could be proven:
9.25/3.94	
9.25/3.94	(0) HASKELL
9.25/3.94	(1) BR [EQUIVALENT, 0 ms]
9.25/3.94	(2) HASKELL
9.25/3.94	(3) COR [EQUIVALENT, 0 ms]
9.25/3.94	(4) HASKELL
9.25/3.94	(5) Narrow [SOUND, 0 ms]
9.25/3.94	(6) QDP
9.25/3.94	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.25/3.94	(8) YES
9.25/3.94	
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(0)
9.25/3.94	Obligation:
9.25/3.94	mainModule Main 
9.25/3.94	module Main where {
9.25/3.94	import qualified Prelude;
9.25/3.94	}
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(1) BR (EQUIVALENT)
9.25/3.94	Replaced joker patterns by fresh variables and removed binding patterns.
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(2)
9.25/3.94	Obligation:
9.25/3.94	mainModule Main 
9.25/3.94	module Main where {
9.25/3.94	import qualified Prelude;
9.25/3.94	}
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(3) COR (EQUIVALENT)
9.25/3.94	Cond Reductions:
9.25/3.94	The following Function with conditions
9.25/3.94	"undefined |Falseundefined;
9.25/3.94	"
9.25/3.94	is transformed to
9.25/3.94	"undefined  = undefined1;
9.25/3.94	"
9.25/3.94	"undefined0 True = undefined;
9.25/3.94	"
9.25/3.94	"undefined1  = undefined0 False;
9.25/3.94	"
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(4)
9.25/3.94	Obligation:
9.25/3.94	mainModule Main 
9.25/3.94	module Main where {
9.25/3.94	import qualified Prelude;
9.25/3.94	}
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(5) Narrow (SOUND)
9.25/3.94	Haskell To QDPs
9.25/3.94	
9.25/3.94	digraph dp_graph {
9.25/3.94	node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.25/3.94	3[label="elem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.25/3.94	4[label="elem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.25/3.94	5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.25/3.94	6[label="any ((==) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.25/3.94	7[label="or . map ((==) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.25/3.94	8[label="or (map ((==) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.25/3.94	9[label="foldr (||) False (map ((==) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];42[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 42[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	42 -> 10[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	43[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 43[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	43 -> 11[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	10[label="foldr (||) False (map ((==) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.25/3.94	11[label="foldr (||) False (map ((==) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.25/3.94	12[label="foldr (||) False (((==) vx3 vx40) : map ((==) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.25/3.94	13[label="foldr (||) False []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.25/3.94	14 -> 16[label="",style="dashed", color="red", weight=0];
9.25/3.94	14[label="(||) (==) vx3 vx40 foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.25/3.94	15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.25/3.94	17[label="foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.25/3.94	16[label="(||) (==) vx3 vx40 vx5",fontsize=16,color="burlywood",shape="triangle"];44[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 44[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	44 -> 19[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	45[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 45[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	45 -> 20[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	46[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 46[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	46 -> 21[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(||) (==) LT vx40 vx5",fontsize=16,color="burlywood",shape="box"];47[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	47 -> 22[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	48[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 48[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	48 -> 23[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	49[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 49[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	49 -> 24[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	20[label="(||) (==) EQ vx40 vx5",fontsize=16,color="burlywood",shape="box"];50[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];20 -> 50[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	50 -> 25[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	51[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 51[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	51 -> 26[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	52[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 52[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	52 -> 27[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	21[label="(||) (==) GT vx40 vx5",fontsize=16,color="burlywood",shape="box"];53[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];21 -> 53[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	53 -> 28[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	54[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];21 -> 54[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	54 -> 29[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	55[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];21 -> 55[label="",style="solid", color="burlywood", weight=9];
9.25/3.94	55 -> 30[label="",style="solid", color="burlywood", weight=3];
9.25/3.94	22[label="(||) (==) LT LT vx5",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
9.25/3.94	23[label="(||) (==) LT EQ vx5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.25/3.94	24[label="(||) (==) LT GT vx5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.25/3.94	25[label="(||) (==) EQ LT vx5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.25/3.94	26[label="(||) (==) EQ EQ vx5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.25/3.94	27[label="(||) (==) EQ GT vx5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.25/3.94	28[label="(||) (==) GT LT vx5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.25/3.94	29[label="(||) (==) GT EQ vx5",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.25/3.94	30[label="(||) (==) GT GT vx5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.25/3.94	31[label="(||) True vx5",fontsize=16,color="black",shape="triangle"];31 -> 40[label="",style="solid", color="black", weight=3];
9.25/3.94	32[label="(||) False vx5",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
9.25/3.94	33 -> 32[label="",style="dashed", color="red", weight=0];
9.25/3.94	33[label="(||) False vx5",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0];
9.25/3.94	34[label="(||) False vx5",fontsize=16,color="magenta"];35 -> 31[label="",style="dashed", color="red", weight=0];
9.25/3.94	35[label="(||) True vx5",fontsize=16,color="magenta"];36 -> 32[label="",style="dashed", color="red", weight=0];
9.25/3.94	36[label="(||) False vx5",fontsize=16,color="magenta"];37 -> 32[label="",style="dashed", color="red", weight=0];
9.25/3.94	37[label="(||) False vx5",fontsize=16,color="magenta"];38 -> 32[label="",style="dashed", color="red", weight=0];
9.25/3.94	38[label="(||) False vx5",fontsize=16,color="magenta"];39 -> 31[label="",style="dashed", color="red", weight=0];
9.25/3.94	39[label="(||) True vx5",fontsize=16,color="magenta"];40[label="True",fontsize=16,color="green",shape="box"];41[label="vx5",fontsize=16,color="green",shape="box"];}
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(6)
9.25/3.94	Obligation:
9.25/3.94	Q DP problem:
9.25/3.94	The TRS P consists of the following rules:
9.25/3.94	
9.25/3.94	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.25/3.94	
9.25/3.94	R is empty.
9.25/3.94	Q is empty.
9.25/3.94	We have to consider all minimal (P,Q,R)-chains.
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(7) QDPSizeChangeProof (EQUIVALENT)
9.25/3.94	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. 
9.25/3.94	
9.25/3.94	From the DPs we obtained the following set of size-change graphs:
9.25/3.94	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.25/3.94	The graph contains the following edges 1 >= 1, 2 > 2
9.25/3.94	
9.25/3.94	
9.25/3.94	----------------------------------------
9.25/3.94	
9.25/3.94	(8)
9.25/3.94	YES
9.25/3.98	EOF