7.89/3.50	YES
9.49/3.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.49/3.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.49/3.95	
9.49/3.95	
9.49/3.95	H-Termination with start terms of the given HASKELL could be proven:
9.49/3.95	
9.49/3.95	(0) HASKELL
9.49/3.95	(1) BR [EQUIVALENT, 0 ms]
9.49/3.95	(2) HASKELL
9.49/3.95	(3) COR [EQUIVALENT, 0 ms]
9.49/3.95	(4) HASKELL
9.49/3.95	(5) Narrow [SOUND, 0 ms]
9.49/3.95	(6) QDP
9.49/3.95	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.49/3.95	(8) YES
9.49/3.95	
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(0)
9.49/3.95	Obligation:
9.49/3.95	mainModule Main 
9.49/3.95	module Main where {
9.49/3.95	import qualified Prelude;
9.49/3.95	}
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(1) BR (EQUIVALENT)
9.49/3.95	Replaced joker patterns by fresh variables and removed binding patterns.
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(2)
9.49/3.95	Obligation:
9.49/3.95	mainModule Main 
9.49/3.95	module Main where {
9.49/3.95	import qualified Prelude;
9.49/3.95	}
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(3) COR (EQUIVALENT)
9.49/3.95	Cond Reductions:
9.49/3.95	The following Function with conditions
9.49/3.95	"undefined |Falseundefined;
9.49/3.95	"
9.49/3.95	is transformed to
9.49/3.95	"undefined  = undefined1;
9.49/3.95	"
9.49/3.95	"undefined0 True = undefined;
9.49/3.95	"
9.49/3.95	"undefined1  = undefined0 False;
9.49/3.95	"
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(4)
9.49/3.95	Obligation:
9.49/3.95	mainModule Main 
9.49/3.95	module Main where {
9.49/3.95	import qualified Prelude;
9.49/3.95	}
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(5) Narrow (SOUND)
9.49/3.95	Haskell To QDPs
9.49/3.95	
9.49/3.95	digraph dp_graph {
9.49/3.95	node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.49/3.95	3[label="elem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.49/3.95	4[label="elem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.49/3.95	5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.49/3.95	6[label="any ((==) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.49/3.95	7[label="or . map ((==) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.49/3.95	8[label="or (map ((==) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.49/3.95	9[label="foldr (||) False (map ((==) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];23[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
9.49/3.95	23 -> 10[label="",style="solid", color="burlywood", weight=3];
9.49/3.95	24[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
9.49/3.95	24 -> 11[label="",style="solid", color="burlywood", weight=3];
9.49/3.95	10[label="foldr (||) False (map ((==) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.49/3.95	11[label="foldr (||) False (map ((==) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.49/3.95	12[label="foldr (||) False (((==) vx3 vx40) : map ((==) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.49/3.95	13[label="foldr (||) False []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.49/3.95	14 -> 16[label="",style="dashed", color="red", weight=0];
9.49/3.95	14[label="(||) (==) vx3 vx40 foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.49/3.95	15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.49/3.95	17[label="foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.49/3.95	16[label="(||) (==) vx3 vx40 vx5",fontsize=16,color="burlywood",shape="triangle"];25[label="vx3/()",fontsize=10,color="white",style="solid",shape="box"];16 -> 25[label="",style="solid", color="burlywood", weight=9];
9.49/3.95	25 -> 19[label="",style="solid", color="burlywood", weight=3];
9.49/3.95	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(||) (==) () vx40 vx5",fontsize=16,color="burlywood",shape="box"];26[label="vx40/()",fontsize=10,color="white",style="solid",shape="box"];19 -> 26[label="",style="solid", color="burlywood", weight=9];
9.49/3.95	26 -> 20[label="",style="solid", color="burlywood", weight=3];
9.49/3.95	20[label="(||) (==) () () vx5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.49/3.95	21[label="(||) True vx5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.49/3.95	22[label="True",fontsize=16,color="green",shape="box"];}
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(6)
9.49/3.95	Obligation:
9.49/3.95	Q DP problem:
9.49/3.95	The TRS P consists of the following rules:
9.49/3.95	
9.49/3.95	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.49/3.95	
9.49/3.95	R is empty.
9.49/3.95	Q is empty.
9.49/3.95	We have to consider all minimal (P,Q,R)-chains.
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(7) QDPSizeChangeProof (EQUIVALENT)
9.49/3.95	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. 
9.49/3.95	
9.49/3.95	From the DPs we obtained the following set of size-change graphs:
9.49/3.95	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.49/3.95	The graph contains the following edges 1 >= 1, 2 > 2
9.49/3.95	
9.49/3.95	
9.49/3.95	----------------------------------------
9.49/3.95	
9.49/3.95	(8)
9.49/3.95	YES
9.76/4.03	EOF