7.84/3.65	YES
9.75/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.75/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.75/4.16	
9.75/4.16	
9.75/4.16	H-Termination with start terms of the given HASKELL could be proven:
9.75/4.16	
9.75/4.16	(0) HASKELL
9.75/4.16	(1) LR [EQUIVALENT, 0 ms]
9.75/4.16	(2) HASKELL
9.75/4.16	(3) BR [EQUIVALENT, 0 ms]
9.75/4.16	(4) HASKELL
9.75/4.16	(5) COR [EQUIVALENT, 0 ms]
9.75/4.16	(6) HASKELL
9.75/4.16	(7) Narrow [SOUND, 0 ms]
9.75/4.16	(8) QDP
9.75/4.16	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.75/4.16	(10) YES
9.75/4.16	
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(0)
9.75/4.16	Obligation:
9.75/4.16	mainModule Main 
9.75/4.16	module Main where {
9.75/4.16	import qualified Prelude;
9.75/4.16	}
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(1) LR (EQUIVALENT)
9.75/4.16	Lambda Reductions:
9.75/4.16	The following Lambda expression
9.75/4.16	"\xs->return (x : xs)"
9.75/4.16	is transformed to
9.75/4.16	"sequence0 x xs = return (x : xs);
9.75/4.16	"
9.75/4.16	The following Lambda expression
9.75/4.16	"\x->sequence cs >>= sequence0 x"
9.75/4.16	is transformed to
9.75/4.16	"sequence1 cs x = sequence cs >>= sequence0 x;
9.75/4.16	"
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(2)
9.75/4.16	Obligation:
9.75/4.16	mainModule Main 
9.75/4.16	module Main where {
9.75/4.16	import qualified Prelude;
9.75/4.16	}
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(3) BR (EQUIVALENT)
9.75/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(4)
9.75/4.16	Obligation:
9.75/4.16	mainModule Main 
9.75/4.16	module Main where {
9.75/4.16	import qualified Prelude;
9.75/4.16	}
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(5) COR (EQUIVALENT)
9.75/4.16	Cond Reductions:
9.75/4.16	The following Function with conditions
9.75/4.16	"undefined |Falseundefined;
9.75/4.16	"
9.75/4.16	is transformed to
9.75/4.16	"undefined  = undefined1;
9.75/4.16	"
9.75/4.16	"undefined0 True = undefined;
9.75/4.16	"
9.75/4.16	"undefined1  = undefined0 False;
9.75/4.16	"
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(6)
9.75/4.16	Obligation:
9.75/4.16	mainModule Main 
9.75/4.16	module Main where {
9.75/4.16	import qualified Prelude;
9.75/4.16	}
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(7) Narrow (SOUND)
9.75/4.16	Haskell To QDPs
9.75/4.16	
9.75/4.16	digraph dp_graph {
9.75/4.16	node [outthreshold=100, inthreshold=100];1[label="sequence",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.75/4.16	3[label="sequence vx3",fontsize=16,color="burlywood",shape="triangle"];23[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 23[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	23 -> 4[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	24[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 24[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	24 -> 5[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	4[label="sequence (vx30 : vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
9.75/4.16	5[label="sequence []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.75/4.16	6[label="vx30 >>= sequence1 vx31",fontsize=16,color="burlywood",shape="box"];25[label="vx30/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 25[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	25 -> 8[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	26[label="vx30/Just vx300",fontsize=10,color="white",style="solid",shape="box"];6 -> 26[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	26 -> 9[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	7[label="return []",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.75/4.16	8[label="Nothing >>= sequence1 vx31",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.75/4.16	9[label="Just vx300 >>= sequence1 vx31",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
9.75/4.16	10[label="Just []",fontsize=16,color="green",shape="box"];11[label="Nothing",fontsize=16,color="green",shape="box"];12[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
9.75/4.16	13 -> 14[label="",style="dashed", color="red", weight=0];
9.75/4.16	13[label="sequence vx31 >>= sequence0 vx300",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
9.75/4.16	15 -> 3[label="",style="dashed", color="red", weight=0];
9.75/4.16	15[label="sequence vx31",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
9.75/4.16	14[label="vx4 >>= sequence0 vx300",fontsize=16,color="burlywood",shape="triangle"];27[label="vx4/Nothing",fontsize=10,color="white",style="solid",shape="box"];14 -> 27[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	27 -> 17[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	28[label="vx4/Just vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 28[label="",style="solid", color="burlywood", weight=9];
9.75/4.16	28 -> 18[label="",style="solid", color="burlywood", weight=3];
9.75/4.16	16[label="vx31",fontsize=16,color="green",shape="box"];17[label="Nothing >>= sequence0 vx300",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
9.75/4.16	18[label="Just vx40 >>= sequence0 vx300",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
9.75/4.16	19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="sequence0 vx300 vx40",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.75/4.16	21[label="return (vx300 : vx40)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.75/4.16	22[label="Just (vx300 : vx40)",fontsize=16,color="green",shape="box"];}
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(8)
9.75/4.16	Obligation:
9.75/4.16	Q DP problem:
9.75/4.16	The TRS P consists of the following rules:
9.75/4.16	
9.75/4.16	   new_sequence(:(Just(vx300), vx31), h) -> new_sequence(vx31, h)
9.75/4.16	
9.75/4.16	R is empty.
9.75/4.16	Q is empty.
9.75/4.16	We have to consider all minimal (P,Q,R)-chains.
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(9) QDPSizeChangeProof (EQUIVALENT)
9.75/4.16	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. 
9.75/4.16	
9.75/4.16	From the DPs we obtained the following set of size-change graphs:
9.75/4.16	*new_sequence(:(Just(vx300), vx31), h) -> new_sequence(vx31, h)
9.75/4.16	The graph contains the following edges 1 > 1, 2 >= 2
9.75/4.16	
9.75/4.16	
9.75/4.16	----------------------------------------
9.75/4.16	
9.75/4.16	(10)
9.75/4.16	YES
9.85/4.22	EOF