8.00/3.57	YES
9.59/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.59/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.59/4.03	
9.59/4.03	
9.59/4.03	H-Termination with start terms of the given HASKELL could be proven:
9.59/4.03	
9.59/4.03	(0) HASKELL
9.59/4.03	(1) BR [EQUIVALENT, 0 ms]
9.59/4.03	(2) HASKELL
9.59/4.03	(3) COR [EQUIVALENT, 0 ms]
9.59/4.03	(4) HASKELL
9.59/4.03	(5) Narrow [SOUND, 0 ms]
9.59/4.03	(6) QDP
9.59/4.03	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.59/4.03	(8) YES
9.59/4.03	
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(0)
9.59/4.03	Obligation:
9.59/4.03	mainModule Main 
9.59/4.03	module Main where {
9.59/4.03	import qualified Prelude;
9.59/4.03	}
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(1) BR (EQUIVALENT)
9.59/4.03	Replaced joker patterns by fresh variables and removed binding patterns.
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(2)
9.59/4.03	Obligation:
9.59/4.03	mainModule Main 
9.59/4.03	module Main where {
9.59/4.03	import qualified Prelude;
9.59/4.03	}
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(3) COR (EQUIVALENT)
9.59/4.03	Cond Reductions:
9.59/4.03	The following Function with conditions
9.59/4.03	"undefined |Falseundefined;
9.59/4.03	"
9.59/4.03	is transformed to
9.59/4.03	"undefined  = undefined1;
9.59/4.03	"
9.59/4.03	"undefined0 True = undefined;
9.59/4.03	"
9.59/4.03	"undefined1  = undefined0 False;
9.59/4.03	"
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(4)
9.59/4.03	Obligation:
9.59/4.03	mainModule Main 
9.59/4.03	module Main where {
9.59/4.03	import qualified Prelude;
9.59/4.03	}
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(5) Narrow (SOUND)
9.59/4.03	Haskell To QDPs
9.59/4.03	
9.59/4.03	digraph dp_graph {
9.59/4.03	node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.59/4.03	3[label="notElem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.59/4.03	4[label="notElem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.59/4.03	5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.59/4.03	6[label="all ((/=) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.59/4.03	7[label="and . map ((/=) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.59/4.03	8[label="and (map ((/=) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.59/4.03	9[label="foldr (&&) True (map ((/=) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];45[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 45[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	45 -> 10[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	46[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 46[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	46 -> 11[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	10[label="foldr (&&) True (map ((/=) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.59/4.03	11[label="foldr (&&) True (map ((/=) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.59/4.03	12[label="foldr (&&) True (((/=) vx3 vx40) : map ((/=) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.59/4.03	13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.59/4.03	14 -> 16[label="",style="dashed", color="red", weight=0];
9.59/4.03	14[label="(&&) (/=) vx3 vx40 foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.59/4.03	15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.59/4.03	17[label="foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.59/4.03	16[label="(&&) (/=) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.59/4.03	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vx3 == vx40) vx5",fontsize=16,color="burlywood",shape="box"];47[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	47 -> 20[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	48[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 48[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	48 -> 21[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	49[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 49[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	49 -> 22[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	20[label="(&&) not (LT == 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];
9.59/4.03	50 -> 23[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	51[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 51[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	51 -> 24[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	52[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 52[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	52 -> 25[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	21[label="(&&) not (EQ == 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];
9.59/4.03	53 -> 26[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	54[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];21 -> 54[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	54 -> 27[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	55[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];21 -> 55[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	55 -> 28[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	22[label="(&&) not (GT == vx40) vx5",fontsize=16,color="burlywood",shape="box"];56[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];22 -> 56[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	56 -> 29[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	57[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];22 -> 57[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	57 -> 30[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	58[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];22 -> 58[label="",style="solid", color="burlywood", weight=9];
9.59/4.03	58 -> 31[label="",style="solid", color="burlywood", weight=3];
9.59/4.03	23[label="(&&) not (LT == LT) vx5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.59/4.03	24[label="(&&) not (LT == EQ) vx5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.59/4.03	25[label="(&&) not (LT == GT) vx5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.59/4.03	26[label="(&&) not (EQ == LT) vx5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.59/4.03	27[label="(&&) not (EQ == EQ) vx5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.59/4.03	28[label="(&&) not (EQ == GT) vx5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.59/4.03	29[label="(&&) not (GT == LT) vx5",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.59/4.03	30[label="(&&) not (GT == EQ) vx5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.59/4.03	31[label="(&&) not (GT == GT) vx5",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
9.59/4.03	32[label="(&&) not True vx5",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
9.59/4.03	33[label="(&&) not False vx5",fontsize=16,color="black",shape="triangle"];33 -> 42[label="",style="solid", color="black", weight=3];
9.59/4.03	34 -> 33[label="",style="dashed", color="red", weight=0];
9.59/4.03	34[label="(&&) not False vx5",fontsize=16,color="magenta"];35 -> 33[label="",style="dashed", color="red", weight=0];
9.59/4.03	35[label="(&&) not False vx5",fontsize=16,color="magenta"];36 -> 32[label="",style="dashed", color="red", weight=0];
9.59/4.03	36[label="(&&) not True vx5",fontsize=16,color="magenta"];37 -> 33[label="",style="dashed", color="red", weight=0];
9.59/4.03	37[label="(&&) not False vx5",fontsize=16,color="magenta"];38 -> 33[label="",style="dashed", color="red", weight=0];
9.59/4.03	38[label="(&&) not False vx5",fontsize=16,color="magenta"];39 -> 33[label="",style="dashed", color="red", weight=0];
9.59/4.03	39[label="(&&) not False vx5",fontsize=16,color="magenta"];40 -> 32[label="",style="dashed", color="red", weight=0];
9.59/4.03	40[label="(&&) not True vx5",fontsize=16,color="magenta"];41[label="(&&) False vx5",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
9.59/4.03	42[label="(&&) True vx5",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
9.59/4.03	43[label="False",fontsize=16,color="green",shape="box"];44[label="vx5",fontsize=16,color="green",shape="box"];}
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(6)
9.59/4.03	Obligation:
9.59/4.03	Q DP problem:
9.59/4.03	The TRS P consists of the following rules:
9.59/4.03	
9.59/4.03	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.59/4.03	
9.59/4.03	R is empty.
9.59/4.03	Q is empty.
9.59/4.03	We have to consider all minimal (P,Q,R)-chains.
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(7) QDPSizeChangeProof (EQUIVALENT)
9.59/4.03	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. 
9.59/4.03	
9.59/4.03	From the DPs we obtained the following set of size-change graphs:
9.59/4.03	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.59/4.03	The graph contains the following edges 1 >= 1, 2 > 2
9.59/4.03	
9.59/4.03	
9.59/4.03	----------------------------------------
9.59/4.03	
9.59/4.03	(8)
9.59/4.03	YES
9.77/4.08	EOF