8.22/3.62	YES
9.71/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.71/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.71/4.06	
9.71/4.06	
9.71/4.06	H-Termination with start terms of the given HASKELL could be proven:
9.71/4.06	
9.71/4.06	(0) HASKELL
9.71/4.06	(1) BR [EQUIVALENT, 0 ms]
9.71/4.06	(2) HASKELL
9.71/4.06	(3) COR [EQUIVALENT, 0 ms]
9.71/4.06	(4) HASKELL
9.71/4.06	(5) Narrow [SOUND, 0 ms]
9.71/4.06	(6) AND
9.71/4.06	    (7) QDP
9.71/4.06	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.71/4.06	        (9) YES
9.71/4.06	    (10) QDP
9.71/4.06	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.71/4.06	        (12) YES
9.71/4.06	
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(0)
9.71/4.06	Obligation:
9.71/4.06	mainModule Main 
9.71/4.06	module Main where {
9.71/4.06	import qualified Prelude;
9.71/4.06	}
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(1) BR (EQUIVALENT)
9.71/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(2)
9.71/4.06	Obligation:
9.71/4.06	mainModule Main 
9.71/4.06	module Main where {
9.71/4.06	import qualified Prelude;
9.71/4.06	}
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(3) COR (EQUIVALENT)
9.71/4.06	Cond Reductions:
9.71/4.06	The following Function with conditions
9.71/4.06	"undefined |Falseundefined;
9.71/4.06	"
9.71/4.06	is transformed to
9.71/4.06	"undefined  = undefined1;
9.71/4.06	"
9.71/4.06	"undefined0 True = undefined;
9.71/4.06	"
9.71/4.06	"undefined1  = undefined0 False;
9.71/4.06	"
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(4)
9.71/4.06	Obligation:
9.71/4.06	mainModule Main 
9.71/4.06	module Main where {
9.71/4.06	import qualified Prelude;
9.71/4.06	}
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(5) Narrow (SOUND)
9.71/4.06	Haskell To QDPs
9.71/4.06	
9.71/4.06	digraph dp_graph {
9.71/4.06	node [outthreshold=100, inthreshold=100];1[label="(>>=)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.71/4.06	3[label="(>>=) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.71/4.06	4[label="(>>=) vx3 vx4",fontsize=16,color="blue",shape="box"];43[label=">>= :: (IO a) -> (a -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];4 -> 43[label="",style="solid", color="blue", weight=9];
9.71/4.06	43 -> 5[label="",style="solid", color="blue", weight=3];
9.71/4.06	44[label=">>= :: ([] a) -> (a -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];4 -> 44[label="",style="solid", color="blue", weight=9];
9.71/4.06	44 -> 6[label="",style="solid", color="blue", weight=3];
9.71/4.06	45[label=">>= :: (Maybe a) -> (a -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];4 -> 45[label="",style="solid", color="blue", weight=9];
9.71/4.06	45 -> 7[label="",style="solid", color="blue", weight=3];
9.71/4.06	5[label="(>>=) vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
9.71/4.06	6[label="(>>=) vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];46[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];6 -> 46[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	46 -> 9[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	47[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 47[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	47 -> 10[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	7[label="(>>=) vx3 vx4",fontsize=16,color="burlywood",shape="box"];48[label="vx3/Nothing",fontsize=10,color="white",style="solid",shape="box"];7 -> 48[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	48 -> 11[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	49[label="vx3/Just vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 49[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	49 -> 12[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	8[label="primbindIO vx3 vx4",fontsize=16,color="burlywood",shape="box"];50[label="vx3/IO vx30",fontsize=10,color="white",style="solid",shape="box"];8 -> 50[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	50 -> 13[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	51[label="vx3/AProVE_IO vx30",fontsize=10,color="white",style="solid",shape="box"];8 -> 51[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	51 -> 14[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	52[label="vx3/AProVE_Exception vx30",fontsize=10,color="white",style="solid",shape="box"];8 -> 52[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	52 -> 15[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	53[label="vx3/AProVE_Error vx30",fontsize=10,color="white",style="solid",shape="box"];8 -> 53[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	53 -> 16[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	9[label="(>>=) vx30 : vx31 vx4",fontsize=16,color="black",shape="box"];9 -> 17[label="",style="solid", color="black", weight=3];
9.71/4.06	10[label="(>>=) [] vx4",fontsize=16,color="black",shape="box"];10 -> 18[label="",style="solid", color="black", weight=3];
9.71/4.06	11[label="(>>=) Nothing vx4",fontsize=16,color="black",shape="box"];11 -> 19[label="",style="solid", color="black", weight=3];
9.71/4.06	12[label="(>>=) Just vx30 vx4",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
9.71/4.06	13[label="primbindIO (IO vx30) vx4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
9.71/4.06	14[label="primbindIO (AProVE_IO vx30) vx4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
9.71/4.06	15[label="primbindIO (AProVE_Exception vx30) vx4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
9.71/4.06	16[label="primbindIO (AProVE_Error vx30) vx4",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
9.71/4.06	17 -> 30[label="",style="dashed", color="red", weight=0];
9.71/4.06	17[label="vx4 vx30 ++ (vx31 >>= vx4)",fontsize=16,color="magenta"];17 -> 31[label="",style="dashed", color="magenta", weight=3];
9.71/4.06	17 -> 32[label="",style="dashed", color="magenta", weight=3];
9.71/4.06	18[label="[]",fontsize=16,color="green",shape="box"];19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="vx4 vx30",fontsize=16,color="green",shape="box"];20 -> 27[label="",style="dashed", color="green", weight=3];
9.71/4.06	21[label="error []",fontsize=16,color="red",shape="box"];22[label="vx4 vx30",fontsize=16,color="green",shape="box"];22 -> 28[label="",style="dashed", color="green", weight=3];
9.71/4.06	23[label="AProVE_Exception vx30",fontsize=16,color="green",shape="box"];24[label="AProVE_Error vx30",fontsize=16,color="green",shape="box"];31[label="vx4 vx30",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3];
9.71/4.06	32 -> 6[label="",style="dashed", color="red", weight=0];
9.71/4.06	32[label="vx31 >>= vx4",fontsize=16,color="magenta"];32 -> 35[label="",style="dashed", color="magenta", weight=3];
9.71/4.06	30[label="vx6 ++ vx5",fontsize=16,color="burlywood",shape="triangle"];54[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];30 -> 54[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	54 -> 36[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	55[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 55[label="",style="solid", color="burlywood", weight=9];
9.71/4.06	55 -> 37[label="",style="solid", color="burlywood", weight=3];
9.71/4.06	27[label="vx30",fontsize=16,color="green",shape="box"];28[label="vx30",fontsize=16,color="green",shape="box"];34[label="vx30",fontsize=16,color="green",shape="box"];35[label="vx31",fontsize=16,color="green",shape="box"];36[label="(vx60 : vx61) ++ vx5",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
9.71/4.06	37[label="[] ++ vx5",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
9.71/4.06	39[label="vx60 : vx61 ++ vx5",fontsize=16,color="green",shape="box"];39 -> 41[label="",style="dashed", color="green", weight=3];
9.71/4.06	40[label="vx5",fontsize=16,color="green",shape="box"];41 -> 30[label="",style="dashed", color="red", weight=0];
9.71/4.06	41[label="vx61 ++ vx5",fontsize=16,color="magenta"];41 -> 42[label="",style="dashed", color="magenta", weight=3];
9.71/4.06	42[label="vx61",fontsize=16,color="green",shape="box"];}
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(6)
9.71/4.06	Complex Obligation (AND)
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(7)
9.71/4.06	Obligation:
9.71/4.06	Q DP problem:
9.71/4.06	The TRS P consists of the following rules:
9.71/4.06	
9.71/4.06	   new_gtGtEs(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)
9.71/4.06	
9.71/4.06	R is empty.
9.71/4.06	Q is empty.
9.71/4.06	We have to consider all minimal (P,Q,R)-chains.
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(8) QDPSizeChangeProof (EQUIVALENT)
9.71/4.06	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. 
9.71/4.06	
9.71/4.06	From the DPs we obtained the following set of size-change graphs:
9.71/4.06	*new_gtGtEs(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)
9.71/4.06	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
9.71/4.06	
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(9)
9.71/4.06	YES
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(10)
9.71/4.06	Obligation:
9.71/4.06	Q DP problem:
9.71/4.06	The TRS P consists of the following rules:
9.71/4.06	
9.71/4.06	   new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.71/4.06	
9.71/4.06	R is empty.
9.71/4.06	Q is empty.
9.71/4.06	We have to consider all minimal (P,Q,R)-chains.
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(11) QDPSizeChangeProof (EQUIVALENT)
9.71/4.06	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. 
9.71/4.06	
9.71/4.06	From the DPs we obtained the following set of size-change graphs:
9.71/4.06	*new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.71/4.06	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.71/4.06	
9.71/4.06	
9.71/4.06	----------------------------------------
9.71/4.06	
9.71/4.06	(12)
9.71/4.06	YES
9.87/5.64	EOF