7.97/3.62	YES
9.45/4.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.45/4.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.45/4.07	
9.45/4.07	
9.45/4.07	H-Termination with start terms of the given HASKELL could be proven:
9.45/4.07	
9.45/4.07	(0) HASKELL
9.45/4.07	(1) BR [EQUIVALENT, 0 ms]
9.45/4.07	(2) HASKELL
9.45/4.07	(3) COR [EQUIVALENT, 0 ms]
9.45/4.07	(4) HASKELL
9.45/4.07	(5) Narrow [SOUND, 0 ms]
9.45/4.07	(6) QDP
9.45/4.07	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.45/4.07	(8) YES
9.45/4.07	
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(0)
9.45/4.07	Obligation:
9.45/4.07	mainModule Main 
9.45/4.07	module Main where {
9.45/4.07	import qualified Prelude;
9.45/4.07	}
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(1) BR (EQUIVALENT)
9.45/4.07	Replaced joker patterns by fresh variables and removed binding patterns.
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(2)
9.45/4.07	Obligation:
9.45/4.07	mainModule Main 
9.45/4.07	module Main where {
9.45/4.07	import qualified Prelude;
9.45/4.07	}
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(3) COR (EQUIVALENT)
9.45/4.07	Cond Reductions:
9.45/4.07	The following Function with conditions
9.45/4.07	"undefined |Falseundefined;
9.45/4.07	"
9.45/4.07	is transformed to
9.45/4.07	"undefined  = undefined1;
9.45/4.07	"
9.45/4.07	"undefined0 True = undefined;
9.45/4.07	"
9.45/4.07	"undefined1  = undefined0 False;
9.45/4.07	"
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(4)
9.45/4.07	Obligation:
9.45/4.07	mainModule Main 
9.45/4.07	module Main where {
9.45/4.07	import qualified Prelude;
9.45/4.07	}
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(5) Narrow (SOUND)
9.45/4.07	Haskell To QDPs
9.45/4.07	
9.45/4.07	digraph dp_graph {
9.45/4.07	node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.45/4.07	3[label="elem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.45/4.07	4[label="elem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.45/4.07	5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.45/4.07	6[label="any ((==) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.45/4.07	7[label="or . map ((==) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.45/4.07	8[label="or (map ((==) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.45/4.07	9[label="foldr (||) False (map ((==) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];31[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 31[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	31 -> 10[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	32[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 32[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	32 -> 11[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	10[label="foldr (||) False (map ((==) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.45/4.07	11[label="foldr (||) False (map ((==) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.45/4.07	12[label="foldr (||) False (((==) vx3 vx40) : map ((==) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.45/4.07	13[label="foldr (||) False []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.45/4.07	14 -> 16[label="",style="dashed", color="red", weight=0];
9.45/4.07	14[label="(||) (==) vx3 vx40 foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.45/4.07	15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.45/4.07	17[label="foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.45/4.07	16[label="(||) (==) vx3 vx40 vx5",fontsize=16,color="burlywood",shape="triangle"];33[label="vx3/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 33[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	33 -> 19[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	34[label="vx3/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 34[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	34 -> 20[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(||) (==) False vx40 vx5",fontsize=16,color="burlywood",shape="box"];35[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];19 -> 35[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	35 -> 21[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	36[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];19 -> 36[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	36 -> 22[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	20[label="(||) (==) True vx40 vx5",fontsize=16,color="burlywood",shape="box"];37[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 37[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	37 -> 23[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	38[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 38[label="",style="solid", color="burlywood", weight=9];
9.45/4.07	38 -> 24[label="",style="solid", color="burlywood", weight=3];
9.45/4.07	21[label="(||) (==) False False vx5",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.45/4.07	22[label="(||) (==) False True vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.45/4.07	23[label="(||) (==) True False vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.45/4.07	24[label="(||) (==) True True vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.45/4.07	25[label="(||) True vx5",fontsize=16,color="black",shape="triangle"];25 -> 29[label="",style="solid", color="black", weight=3];
9.45/4.07	26[label="(||) False vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
9.45/4.07	27 -> 26[label="",style="dashed", color="red", weight=0];
9.45/4.07	27[label="(||) False vx5",fontsize=16,color="magenta"];28 -> 25[label="",style="dashed", color="red", weight=0];
9.45/4.07	28[label="(||) True vx5",fontsize=16,color="magenta"];29[label="True",fontsize=16,color="green",shape="box"];30[label="vx5",fontsize=16,color="green",shape="box"];}
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(6)
9.45/4.07	Obligation:
9.45/4.07	Q DP problem:
9.45/4.07	The TRS P consists of the following rules:
9.45/4.07	
9.45/4.07	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.45/4.07	
9.45/4.07	R is empty.
9.45/4.07	Q is empty.
9.45/4.07	We have to consider all minimal (P,Q,R)-chains.
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(7) QDPSizeChangeProof (EQUIVALENT)
9.45/4.07	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. 
9.45/4.07	
9.45/4.07	From the DPs we obtained the following set of size-change graphs:
9.45/4.07	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.45/4.07	The graph contains the following edges 1 >= 1, 2 > 2
9.45/4.07	
9.45/4.07	
9.45/4.07	----------------------------------------
9.45/4.07	
9.45/4.07	(8)
9.45/4.07	YES
9.45/4.12	EOF