8.05/3.57	YES
9.73/4.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.73/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.73/4.02	
9.73/4.02	
9.73/4.02	H-Termination with start terms of the given HASKELL could be proven:
9.73/4.02	
9.73/4.02	(0) HASKELL
9.73/4.02	(1) BR [EQUIVALENT, 0 ms]
9.73/4.02	(2) HASKELL
9.73/4.02	(3) COR [EQUIVALENT, 0 ms]
9.73/4.02	(4) HASKELL
9.73/4.02	(5) Narrow [SOUND, 0 ms]
9.73/4.02	(6) AND
9.73/4.02	    (7) QDP
9.73/4.02	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.73/4.02	        (9) YES
9.73/4.02	    (10) QDP
9.73/4.02	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.73/4.02	        (12) YES
9.73/4.02	
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(0)
9.73/4.02	Obligation:
9.73/4.02	mainModule Main 
9.73/4.02	module Main where {
9.73/4.02	import qualified Prelude;
9.73/4.02	}
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(1) BR (EQUIVALENT)
9.73/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(2)
9.73/4.02	Obligation:
9.73/4.02	mainModule Main 
9.73/4.02	module Main where {
9.73/4.02	import qualified Prelude;
9.73/4.02	}
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(3) COR (EQUIVALENT)
9.73/4.02	Cond Reductions:
9.73/4.02	The following Function with conditions
9.73/4.02	"undefined |Falseundefined;
9.73/4.02	"
9.73/4.02	is transformed to
9.73/4.02	"undefined  = undefined1;
9.73/4.02	"
9.73/4.02	"undefined0 True = undefined;
9.73/4.02	"
9.73/4.02	"undefined1  = undefined0 False;
9.73/4.02	"
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(4)
9.73/4.02	Obligation:
9.73/4.02	mainModule Main 
9.73/4.02	module Main where {
9.73/4.02	import qualified Prelude;
9.73/4.02	}
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(5) Narrow (SOUND)
9.73/4.02	Haskell To QDPs
9.73/4.02	
9.73/4.02	digraph dp_graph {
9.73/4.02	node [outthreshold=100, inthreshold=100];1[label="(=<<)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.73/4.02	3[label="(=<<) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.73/4.02	4[label="(=<<) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.73/4.02	5[label="vx4 >>= vx3",fontsize=16,color="blue",shape="box"];44[label=">>= :: ([] a) -> (a -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="blue", weight=9];
9.73/4.02	44 -> 6[label="",style="solid", color="blue", weight=3];
9.73/4.02	45[label=">>= :: (Maybe a) -> (a -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="blue", weight=9];
9.73/4.02	45 -> 7[label="",style="solid", color="blue", weight=3];
9.73/4.02	46[label=">>= :: (IO a) -> (a -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];5 -> 46[label="",style="solid", color="blue", weight=9];
9.73/4.02	46 -> 8[label="",style="solid", color="blue", weight=3];
9.73/4.02	6[label="vx4 >>= vx3",fontsize=16,color="burlywood",shape="triangle"];47[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 47[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	47 -> 9[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	48[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 48[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	48 -> 10[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	7[label="vx4 >>= vx3",fontsize=16,color="burlywood",shape="box"];49[label="vx4/Nothing",fontsize=10,color="white",style="solid",shape="box"];7 -> 49[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	49 -> 11[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	50[label="vx4/Just vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 50[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	50 -> 12[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	8[label="vx4 >>= vx3",fontsize=16,color="black",shape="box"];8 -> 13[label="",style="solid", color="black", weight=3];
9.73/4.02	9[label="vx40 : vx41 >>= vx3",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
9.73/4.02	10[label="[] >>= vx3",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
9.73/4.02	11[label="Nothing >>= vx3",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
9.73/4.02	12[label="Just vx40 >>= vx3",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
9.73/4.02	13[label="primbindIO vx4 vx3",fontsize=16,color="burlywood",shape="box"];51[label="vx4/IO vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 51[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	51 -> 18[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	52[label="vx4/AProVE_IO vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 52[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	52 -> 19[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	53[label="vx4/AProVE_Exception vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 53[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	53 -> 20[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	54[label="vx4/AProVE_Error vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 54[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	54 -> 21[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	14 -> 30[label="",style="dashed", color="red", weight=0];
9.73/4.02	14[label="vx3 vx40 ++ (vx41 >>= vx3)",fontsize=16,color="magenta"];14 -> 31[label="",style="dashed", color="magenta", weight=3];
9.73/4.02	14 -> 32[label="",style="dashed", color="magenta", weight=3];
9.73/4.02	15[label="[]",fontsize=16,color="green",shape="box"];16[label="Nothing",fontsize=16,color="green",shape="box"];17[label="vx3 vx40",fontsize=16,color="green",shape="box"];17 -> 24[label="",style="dashed", color="green", weight=3];
9.73/4.02	18[label="primbindIO (IO vx40) vx3",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
9.73/4.02	19[label="primbindIO (AProVE_IO vx40) vx3",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
9.73/4.02	20[label="primbindIO (AProVE_Exception vx40) vx3",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
9.73/4.02	21[label="primbindIO (AProVE_Error vx40) vx3",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3];
9.73/4.02	31[label="vx3 vx40",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3];
9.73/4.02	32 -> 6[label="",style="dashed", color="red", weight=0];
9.73/4.02	32[label="vx41 >>= vx3",fontsize=16,color="magenta"];32 -> 35[label="",style="dashed", color="magenta", weight=3];
9.73/4.02	30[label="vx6 ++ vx5",fontsize=16,color="burlywood",shape="triangle"];55[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	55 -> 36[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	56[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 56[label="",style="solid", color="burlywood", weight=9];
9.73/4.02	56 -> 37[label="",style="solid", color="burlywood", weight=3];
9.73/4.02	24[label="vx40",fontsize=16,color="green",shape="box"];25[label="error []",fontsize=16,color="red",shape="box"];26[label="vx3 vx40",fontsize=16,color="green",shape="box"];26 -> 38[label="",style="dashed", color="green", weight=3];
9.73/4.02	27[label="AProVE_Exception vx40",fontsize=16,color="green",shape="box"];28[label="AProVE_Error vx40",fontsize=16,color="green",shape="box"];34[label="vx40",fontsize=16,color="green",shape="box"];35[label="vx41",fontsize=16,color="green",shape="box"];36[label="(vx60 : vx61) ++ vx5",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
9.73/4.02	37[label="[] ++ vx5",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
9.73/4.02	38[label="vx40",fontsize=16,color="green",shape="box"];40[label="vx60 : vx61 ++ vx5",fontsize=16,color="green",shape="box"];40 -> 42[label="",style="dashed", color="green", weight=3];
9.73/4.02	41[label="vx5",fontsize=16,color="green",shape="box"];42 -> 30[label="",style="dashed", color="red", weight=0];
9.73/4.02	42[label="vx61 ++ vx5",fontsize=16,color="magenta"];42 -> 43[label="",style="dashed", color="magenta", weight=3];
9.73/4.02	43[label="vx61",fontsize=16,color="green",shape="box"];}
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(6)
9.73/4.02	Complex Obligation (AND)
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(7)
9.73/4.02	Obligation:
9.73/4.02	Q DP problem:
9.73/4.02	The TRS P consists of the following rules:
9.73/4.02	
9.73/4.02	   new_gtGtEs(:(vx40, vx41), vx3, h, ba) -> new_gtGtEs(vx41, vx3, h, ba)
9.73/4.02	
9.73/4.02	R is empty.
9.73/4.02	Q is empty.
9.73/4.02	We have to consider all minimal (P,Q,R)-chains.
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(8) QDPSizeChangeProof (EQUIVALENT)
9.73/4.02	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. 
9.73/4.02	
9.73/4.02	From the DPs we obtained the following set of size-change graphs:
9.73/4.02	*new_gtGtEs(:(vx40, vx41), vx3, h, ba) -> new_gtGtEs(vx41, vx3, h, ba)
9.73/4.02	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
9.73/4.02	
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(9)
9.73/4.02	YES
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(10)
9.73/4.02	Obligation:
9.73/4.02	Q DP problem:
9.73/4.02	The TRS P consists of the following rules:
9.73/4.02	
9.73/4.02	   new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.73/4.02	
9.73/4.02	R is empty.
9.73/4.02	Q is empty.
9.73/4.02	We have to consider all minimal (P,Q,R)-chains.
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(11) QDPSizeChangeProof (EQUIVALENT)
9.73/4.02	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. 
9.73/4.02	
9.73/4.02	From the DPs we obtained the following set of size-change graphs:
9.73/4.02	*new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.73/4.02	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.73/4.02	
9.73/4.02	
9.73/4.02	----------------------------------------
9.73/4.02	
9.73/4.02	(12)
9.73/4.02	YES
9.73/4.06	EOF