7.98/3.58	YES
9.90/4.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.90/4.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.90/4.09	
9.90/4.09	
9.90/4.09	H-Termination with start terms of the given HASKELL could be proven:
9.90/4.09	
9.90/4.09	(0) HASKELL
9.90/4.09	(1) LR [EQUIVALENT, 0 ms]
9.90/4.09	(2) HASKELL
9.90/4.09	(3) BR [EQUIVALENT, 0 ms]
9.90/4.09	(4) HASKELL
9.90/4.09	(5) COR [EQUIVALENT, 0 ms]
9.90/4.09	(6) HASKELL
9.90/4.09	(7) Narrow [SOUND, 0 ms]
9.90/4.09	(8) AND
9.90/4.09	    (9) QDP
9.90/4.09	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.90/4.09	        (11) YES
9.90/4.09	    (12) QDP
9.90/4.09	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.90/4.09	        (14) YES
9.90/4.09	
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(0)
9.90/4.09	Obligation:
9.90/4.09	mainModule Main 
9.90/4.09	module Main where {
9.90/4.09	import qualified Prelude;
9.90/4.09	}
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(1) LR (EQUIVALENT)
9.90/4.09	Lambda Reductions:
9.90/4.09	The following Lambda expression
9.90/4.09	"\_->q"
9.90/4.09	is transformed to
9.90/4.09	"gtGt0 q _ = q;
9.90/4.09	"
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(2)
9.90/4.09	Obligation:
9.90/4.09	mainModule Main 
9.90/4.09	module Main where {
9.90/4.09	import qualified Prelude;
9.90/4.09	}
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(3) BR (EQUIVALENT)
9.90/4.09	Replaced joker patterns by fresh variables and removed binding patterns.
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(4)
9.90/4.09	Obligation:
9.90/4.09	mainModule Main 
9.90/4.09	module Main where {
9.90/4.09	import qualified Prelude;
9.90/4.09	}
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(5) COR (EQUIVALENT)
9.90/4.09	Cond Reductions:
9.90/4.09	The following Function with conditions
9.90/4.09	"undefined |Falseundefined;
9.90/4.09	"
9.90/4.09	is transformed to
9.90/4.09	"undefined  = undefined1;
9.90/4.09	"
9.90/4.09	"undefined0 True = undefined;
9.90/4.09	"
9.90/4.09	"undefined1  = undefined0 False;
9.90/4.09	"
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(6)
9.90/4.09	Obligation:
9.90/4.09	mainModule Main 
9.90/4.09	module Main where {
9.90/4.09	import qualified Prelude;
9.90/4.09	}
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(7) Narrow (SOUND)
9.90/4.09	Haskell To QDPs
9.90/4.09	
9.90/4.09	digraph dp_graph {
9.90/4.09	node [outthreshold=100, inthreshold=100];1[label="(>>)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.90/4.09	3[label="(>>) vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.90/4.09	4[label="(>>) vy3 vy4",fontsize=16,color="blue",shape="box"];40[label=">> :: (Maybe a) -> (Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="blue", weight=9];
9.90/4.09	40 -> 5[label="",style="solid", color="blue", weight=3];
9.90/4.09	41[label=">> :: (IO a) -> (IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];4 -> 41[label="",style="solid", color="blue", weight=9];
9.90/4.09	41 -> 6[label="",style="solid", color="blue", weight=3];
9.90/4.09	42[label=">> :: ([] a) -> ([] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];4 -> 42[label="",style="solid", color="blue", weight=9];
9.90/4.09	42 -> 7[label="",style="solid", color="blue", weight=3];
9.90/4.09	5[label="(>>) vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
9.90/4.09	6[label="(>>) vy3 vy4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.90/4.09	7[label="(>>) vy3 vy4",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.90/4.09	8[label="vy3 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="box"];43[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 43[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	43 -> 11[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	44[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 44[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	44 -> 12[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	9[label="vy3 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3];
9.90/4.09	10[label="vy3 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="triangle"];45[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];10 -> 45[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	45 -> 14[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	46[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 46[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	46 -> 15[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	11[label="Nothing >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
9.90/4.09	12[label="Just vy30 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
9.90/4.09	13[label="primbindIO vy3 (gtGt0 vy4)",fontsize=16,color="burlywood",shape="box"];47[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];13 -> 47[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	47 -> 18[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	48[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];13 -> 48[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	48 -> 19[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	49[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];13 -> 49[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	49 -> 20[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	50[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];13 -> 50[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	50 -> 21[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	14[label="vy30 : vy31 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
9.90/4.09	15[label="[] >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
9.90/4.09	16[label="Nothing",fontsize=16,color="green",shape="box"];17[label="gtGt0 vy4 vy30",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3];
9.90/4.09	18[label="primbindIO (IO vy30) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
9.90/4.09	19[label="primbindIO (AProVE_IO vy30) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
9.90/4.09	20[label="primbindIO (AProVE_Exception vy30) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
9.90/4.09	21[label="primbindIO (AProVE_Error vy30) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3];
9.90/4.09	22 -> 29[label="",style="dashed", color="red", weight=0];
9.90/4.09	22[label="gtGt0 vy4 vy30 ++ (vy31 >>= gtGt0 vy4)",fontsize=16,color="magenta"];22 -> 30[label="",style="dashed", color="magenta", weight=3];
9.90/4.09	23[label="[]",fontsize=16,color="green",shape="box"];24[label="vy4",fontsize=16,color="green",shape="box"];25[label="error []",fontsize=16,color="red",shape="box"];26[label="gtGt0 vy4 vy30",fontsize=16,color="black",shape="box"];26 -> 31[label="",style="solid", color="black", weight=3];
9.90/4.09	27[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];28[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];30 -> 10[label="",style="dashed", color="red", weight=0];
9.90/4.09	30[label="vy31 >>= gtGt0 vy4",fontsize=16,color="magenta"];30 -> 32[label="",style="dashed", color="magenta", weight=3];
9.90/4.09	29[label="gtGt0 vy4 vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];29 -> 33[label="",style="solid", color="black", weight=3];
9.90/4.09	31[label="vy4",fontsize=16,color="green",shape="box"];32[label="vy31",fontsize=16,color="green",shape="box"];33[label="vy4 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];51[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];33 -> 51[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	51 -> 34[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	52[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];33 -> 52[label="",style="solid", color="burlywood", weight=9];
9.90/4.09	52 -> 35[label="",style="solid", color="burlywood", weight=3];
9.90/4.09	34[label="(vy40 : vy41) ++ vy5",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
9.90/4.09	35[label="[] ++ vy5",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
9.90/4.09	36[label="vy40 : vy41 ++ vy5",fontsize=16,color="green",shape="box"];36 -> 38[label="",style="dashed", color="green", weight=3];
9.90/4.09	37[label="vy5",fontsize=16,color="green",shape="box"];38 -> 33[label="",style="dashed", color="red", weight=0];
9.90/4.09	38[label="vy41 ++ vy5",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
9.90/4.09	39[label="vy41",fontsize=16,color="green",shape="box"];}
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(8)
9.90/4.09	Complex Obligation (AND)
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(9)
9.90/4.09	Obligation:
9.90/4.09	Q DP problem:
9.90/4.09	The TRS P consists of the following rules:
9.90/4.09	
9.90/4.09	   new_gtGtEs(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs(vy31, vy4, h, ba)
9.90/4.09	
9.90/4.09	R is empty.
9.90/4.09	Q is empty.
9.90/4.09	We have to consider all minimal (P,Q,R)-chains.
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(10) QDPSizeChangeProof (EQUIVALENT)
9.90/4.09	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. 
9.90/4.09	
9.90/4.09	From the DPs we obtained the following set of size-change graphs:
9.90/4.09	*new_gtGtEs(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs(vy31, vy4, h, ba)
9.90/4.09	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
9.90/4.09	
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(11)
9.90/4.09	YES
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(12)
9.90/4.09	Obligation:
9.90/4.09	Q DP problem:
9.90/4.09	The TRS P consists of the following rules:
9.90/4.09	
9.90/4.09	   new_psPs(:(vy40, vy41), vy5, h) -> new_psPs(vy41, vy5, h)
9.90/4.09	
9.90/4.09	R is empty.
9.90/4.09	Q is empty.
9.90/4.09	We have to consider all minimal (P,Q,R)-chains.
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(13) QDPSizeChangeProof (EQUIVALENT)
9.90/4.09	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. 
9.90/4.09	
9.90/4.09	From the DPs we obtained the following set of size-change graphs:
9.90/4.09	*new_psPs(:(vy40, vy41), vy5, h) -> new_psPs(vy41, vy5, h)
9.90/4.09	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.90/4.09	
9.90/4.09	
9.90/4.09	----------------------------------------
9.90/4.09	
9.90/4.09	(14)
9.90/4.09	YES
9.90/4.13	EOF