8.57/3.66	YES
9.83/4.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.83/4.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.83/4.09	
9.83/4.09	
9.83/4.09	H-Termination with start terms of the given HASKELL could be proven:
9.83/4.09	
9.83/4.09	(0) HASKELL
9.83/4.09	(1) BR [EQUIVALENT, 0 ms]
9.83/4.09	(2) HASKELL
9.83/4.09	(3) COR [EQUIVALENT, 0 ms]
9.83/4.09	(4) HASKELL
9.83/4.09	(5) Narrow [SOUND, 0 ms]
9.83/4.09	(6) QDP
9.83/4.09	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.83/4.09	(8) YES
9.83/4.09	
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(0)
9.83/4.09	Obligation:
9.83/4.09	mainModule Main 
9.83/4.09	module Main where {
9.83/4.09	import qualified Prelude;
9.83/4.09	}
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(1) BR (EQUIVALENT)
9.83/4.09	Replaced joker patterns by fresh variables and removed binding patterns.
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(2)
9.83/4.09	Obligation:
9.83/4.09	mainModule Main 
9.83/4.09	module Main where {
9.83/4.09	import qualified Prelude;
9.83/4.09	}
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(3) COR (EQUIVALENT)
9.83/4.09	Cond Reductions:
9.83/4.09	The following Function with conditions
9.83/4.09	"undefined |Falseundefined;
9.83/4.09	"
9.83/4.09	is transformed to
9.83/4.09	"undefined  = undefined1;
9.83/4.09	"
9.83/4.09	"undefined0 True = undefined;
9.83/4.09	"
9.83/4.09	"undefined1  = undefined0 False;
9.83/4.09	"
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(4)
9.83/4.09	Obligation:
9.83/4.09	mainModule Main 
9.83/4.09	module Main where {
9.83/4.09	import qualified Prelude;
9.83/4.09	}
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(5) Narrow (SOUND)
9.83/4.09	Haskell To QDPs
9.83/4.09	
9.83/4.09	digraph dp_graph {
9.83/4.09	node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.83/4.09	3[label="notElem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.83/4.09	4[label="notElem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.83/4.09	5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.83/4.09	6[label="all ((/=) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.83/4.09	7[label="and . map ((/=) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.83/4.09	8[label="and (map ((/=) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.83/4.09	9[label="foldr (&&) True (map ((/=) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 34[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	34 -> 10[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	35[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 35[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	35 -> 11[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	10[label="foldr (&&) True (map ((/=) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.83/4.09	11[label="foldr (&&) True (map ((/=) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.83/4.09	12[label="foldr (&&) True (((/=) vx3 vx40) : map ((/=) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.83/4.09	13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.83/4.09	14 -> 16[label="",style="dashed", color="red", weight=0];
9.83/4.09	14[label="(&&) (/=) vx3 vx40 foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.83/4.09	15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.83/4.09	17[label="foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.83/4.09	16[label="(&&) (/=) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.83/4.09	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vx3 == vx40) vx5",fontsize=16,color="burlywood",shape="box"];36[label="vx3/False",fontsize=10,color="white",style="solid",shape="box"];19 -> 36[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	36 -> 20[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	37[label="vx3/True",fontsize=10,color="white",style="solid",shape="box"];19 -> 37[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	37 -> 21[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	20[label="(&&) not (False == vx40) vx5",fontsize=16,color="burlywood",shape="box"];38[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 38[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	38 -> 22[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	39[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 39[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	39 -> 23[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	21[label="(&&) not (True == vx40) vx5",fontsize=16,color="burlywood",shape="box"];40[label="vx40/False",fontsize=10,color="white",style="solid",shape="box"];21 -> 40[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	40 -> 24[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	41[label="vx40/True",fontsize=10,color="white",style="solid",shape="box"];21 -> 41[label="",style="solid", color="burlywood", weight=9];
9.83/4.09	41 -> 25[label="",style="solid", color="burlywood", weight=3];
9.83/4.09	22[label="(&&) not (False == False) vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.83/4.09	23[label="(&&) not (False == True) vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.83/4.09	24[label="(&&) not (True == False) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.83/4.09	25[label="(&&) not (True == True) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
9.83/4.09	26[label="(&&) not True vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
9.83/4.09	27[label="(&&) not False vx5",fontsize=16,color="black",shape="triangle"];27 -> 31[label="",style="solid", color="black", weight=3];
9.83/4.09	28 -> 27[label="",style="dashed", color="red", weight=0];
9.83/4.09	28[label="(&&) not False vx5",fontsize=16,color="magenta"];29 -> 26[label="",style="dashed", color="red", weight=0];
9.83/4.09	29[label="(&&) not True vx5",fontsize=16,color="magenta"];30[label="(&&) False vx5",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
9.83/4.09	31[label="(&&) True vx5",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
9.83/4.09	32[label="False",fontsize=16,color="green",shape="box"];33[label="vx5",fontsize=16,color="green",shape="box"];}
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(6)
9.83/4.09	Obligation:
9.83/4.09	Q DP problem:
9.83/4.09	The TRS P consists of the following rules:
9.83/4.09	
9.83/4.09	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.83/4.09	
9.83/4.09	R is empty.
9.83/4.09	Q is empty.
9.83/4.09	We have to consider all minimal (P,Q,R)-chains.
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(7) QDPSizeChangeProof (EQUIVALENT)
9.83/4.09	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. 
9.83/4.09	
9.83/4.09	From the DPs we obtained the following set of size-change graphs:
9.83/4.09	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.83/4.09	The graph contains the following edges 1 >= 1, 2 > 2
9.83/4.09	
9.83/4.09	
9.83/4.09	----------------------------------------
9.83/4.09	
9.83/4.09	(8)
9.83/4.09	YES
10.03/4.16	EOF