7.73/3.52	YES
9.13/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.13/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.13/3.98	
9.13/3.98	
9.13/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.13/3.98	
9.13/3.98	(0) HASKELL
9.13/3.98	(1) BR [EQUIVALENT, 0 ms]
9.13/3.98	(2) HASKELL
9.13/3.98	(3) COR [EQUIVALENT, 0 ms]
9.13/3.98	(4) HASKELL
9.13/3.98	(5) Narrow [SOUND, 0 ms]
9.13/3.98	(6) AND
9.13/3.98	    (7) QDP
9.13/3.98	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.13/3.98	        (9) YES
9.13/3.98	    (10) QDP
9.13/3.98	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.13/3.98	        (12) YES
9.13/3.98	
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(0)
9.13/3.98	Obligation:
9.13/3.98	mainModule Main 
9.13/3.98	module Main where {
9.13/3.98	import qualified Prelude;
9.13/3.98	}
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(1) BR (EQUIVALENT)
9.13/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(2)
9.13/3.98	Obligation:
9.13/3.98	mainModule Main 
9.13/3.98	module Main where {
9.13/3.98	import qualified Prelude;
9.13/3.98	}
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(3) COR (EQUIVALENT)
9.13/3.98	Cond Reductions:
9.13/3.98	The following Function with conditions
9.13/3.98	"undefined |Falseundefined;
9.13/3.98	"
9.13/3.98	is transformed to
9.13/3.98	"undefined  = undefined1;
9.13/3.98	"
9.13/3.98	"undefined0 True = undefined;
9.13/3.98	"
9.13/3.98	"undefined1  = undefined0 False;
9.13/3.98	"
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(4)
9.13/3.98	Obligation:
9.13/3.98	mainModule Main 
9.13/3.98	module Main where {
9.13/3.98	import qualified Prelude;
9.13/3.98	}
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(5) Narrow (SOUND)
9.13/3.98	Haskell To QDPs
9.13/3.98	
9.13/3.98	digraph dp_graph {
9.13/3.98	node [outthreshold=100, inthreshold=100];1[label="(>>=)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.13/3.98	3[label="(>>=) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.13/3.98	4[label="(>>=) vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];25[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
9.13/3.98	25 -> 5[label="",style="solid", color="burlywood", weight=3];
9.13/3.98	26[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
9.13/3.98	26 -> 6[label="",style="solid", color="burlywood", weight=3];
9.13/3.98	5[label="(>>=) vx30 : vx31 vx4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.13/3.98	6[label="(>>=) [] vx4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.13/3.98	7 -> 12[label="",style="dashed", color="red", weight=0];
9.13/3.98	7[label="vx4 vx30 ++ (vx31 >>= vx4)",fontsize=16,color="magenta"];7 -> 13[label="",style="dashed", color="magenta", weight=3];
9.13/3.98	7 -> 14[label="",style="dashed", color="magenta", weight=3];
9.13/3.98	8[label="[]",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
9.13/3.98	13[label="vx31 >>= vx4",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
9.13/3.98	14[label="vx4 vx30",fontsize=16,color="green",shape="box"];14 -> 17[label="",style="dashed", color="green", weight=3];
9.13/3.98	12[label="vx6 ++ vx5",fontsize=16,color="burlywood",shape="triangle"];27[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];12 -> 27[label="",style="solid", color="burlywood", weight=9];
9.13/3.98	27 -> 18[label="",style="solid", color="burlywood", weight=3];
9.13/3.98	28[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 28[label="",style="solid", color="burlywood", weight=9];
9.13/3.98	28 -> 19[label="",style="solid", color="burlywood", weight=3];
9.13/3.98	16[label="vx31",fontsize=16,color="green",shape="box"];17[label="vx30",fontsize=16,color="green",shape="box"];18[label="(vx60 : vx61) ++ vx5",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.13/3.98	19[label="[] ++ vx5",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
9.13/3.98	21[label="vx60 : vx61 ++ vx5",fontsize=16,color="green",shape="box"];21 -> 23[label="",style="dashed", color="green", weight=3];
9.13/3.98	22[label="vx5",fontsize=16,color="green",shape="box"];23 -> 12[label="",style="dashed", color="red", weight=0];
9.13/3.98	23[label="vx61 ++ vx5",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
9.13/3.98	24[label="vx61",fontsize=16,color="green",shape="box"];}
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(6)
9.13/3.98	Complex Obligation (AND)
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(7)
9.13/3.98	Obligation:
9.13/3.98	Q DP problem:
9.13/3.98	The TRS P consists of the following rules:
9.13/3.98	
9.13/3.98	   new_gtGtEs(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)
9.13/3.98	
9.13/3.98	R is empty.
9.13/3.98	Q is empty.
9.13/3.98	We have to consider all minimal (P,Q,R)-chains.
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(8) QDPSizeChangeProof (EQUIVALENT)
9.13/3.98	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. 
9.13/3.98	
9.13/3.98	From the DPs we obtained the following set of size-change graphs:
9.13/3.98	*new_gtGtEs(:(vx30, vx31), vx4, h, ba) -> new_gtGtEs(vx31, vx4, h, ba)
9.13/3.98	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
9.13/3.98	
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(9)
9.13/3.98	YES
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(10)
9.13/3.98	Obligation:
9.13/3.98	Q DP problem:
9.13/3.98	The TRS P consists of the following rules:
9.13/3.98	
9.13/3.98	   new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.13/3.98	
9.13/3.98	R is empty.
9.13/3.98	Q is empty.
9.13/3.98	We have to consider all minimal (P,Q,R)-chains.
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(11) QDPSizeChangeProof (EQUIVALENT)
9.13/3.98	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. 
9.13/3.98	
9.13/3.98	From the DPs we obtained the following set of size-change graphs:
9.13/3.98	*new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.13/3.98	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.13/3.98	
9.13/3.98	
9.13/3.98	----------------------------------------
9.13/3.98	
9.13/3.98	(12)
9.13/3.98	YES
9.49/4.02	EOF