9.43/3.96	MAYBE
11.43/4.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.43/4.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.43/4.51	
11.43/4.51	
11.43/4.51	H-Termination with start terms of the given HASKELL could not be shown:
11.43/4.51	
11.43/4.51	(0) HASKELL
11.43/4.51	(1) LR [EQUIVALENT, 0 ms]
11.43/4.51	(2) HASKELL
11.43/4.51	(3) BR [EQUIVALENT, 0 ms]
11.43/4.51	(4) HASKELL
11.43/4.51	(5) COR [EQUIVALENT, 0 ms]
11.43/4.51	(6) HASKELL
11.43/4.51	(7) Narrow [SOUND, 0 ms]
11.43/4.51	(8) AND
11.43/4.51	    (9) QDP
11.43/4.51	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.43/4.51	        (11) YES
11.43/4.51	    (12) QDP
11.43/4.51	        (13) QDPOrderProof [EQUIVALENT, 41 ms]
11.43/4.51	        (14) QDP
11.43/4.51	        (15) DependencyGraphProof [EQUIVALENT, 0 ms]
11.43/4.51	        (16) QDP
11.43/4.51	        (17) MNOCProof [EQUIVALENT, 0 ms]
11.43/4.51	        (18) QDP
11.43/4.51	        (19) NonTerminationLoopProof [COMPLETE, 0 ms]
11.43/4.51	        (20) NO
11.43/4.51	    (21) QDP
11.43/4.51	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.43/4.51	        (23) YES
11.43/4.51	(24) Narrow [COMPLETE, 0 ms]
11.43/4.51	(25) TRUE
11.43/4.51	
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(0)
11.43/4.51	Obligation:
11.43/4.51	mainModule Main 
11.43/4.51	module Main where {
11.43/4.51	import qualified Prelude;
11.43/4.51	}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(1) LR (EQUIVALENT)
11.43/4.51	Lambda Reductions:
11.43/4.51	The following Lambda expression
11.43/4.51	"\xs->return (x : xs)"
11.43/4.51	is transformed to
11.43/4.51	"sequence0 x xs = return (x : xs);
11.43/4.51	"
11.43/4.51	The following Lambda expression
11.43/4.51	"\x->sequence cs >>= sequence0 x"
11.43/4.51	is transformed to
11.43/4.51	"sequence1 cs x = sequence cs >>= sequence0 x;
11.43/4.51	"
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(2)
11.43/4.51	Obligation:
11.43/4.51	mainModule Main 
11.43/4.51	module Main where {
11.43/4.51	import qualified Prelude;
11.43/4.51	}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(3) BR (EQUIVALENT)
11.43/4.51	Replaced joker patterns by fresh variables and removed binding patterns.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(4)
11.43/4.51	Obligation:
11.43/4.51	mainModule Main 
11.43/4.51	module Main where {
11.43/4.51	import qualified Prelude;
11.43/4.51	}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(5) COR (EQUIVALENT)
11.43/4.51	Cond Reductions:
11.43/4.51	The following Function with conditions
11.43/4.51	"undefined |Falseundefined;
11.43/4.51	"
11.43/4.51	is transformed to
11.43/4.51	"undefined  = undefined1;
11.43/4.51	"
11.43/4.51	"undefined0 True = undefined;
11.43/4.51	"
11.43/4.51	"undefined1  = undefined0 False;
11.43/4.51	"
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(6)
11.43/4.51	Obligation:
11.43/4.51	mainModule Main 
11.43/4.51	module Main where {
11.43/4.51	import qualified Prelude;
11.43/4.51	}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(7) Narrow (SOUND)
11.43/4.51	Haskell To QDPs
11.43/4.51	
11.43/4.51	digraph dp_graph {
11.43/4.51	node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.43/4.51	3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.43/4.51	4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.43/4.51	5[label="sequence . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.43/4.51	6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];57[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 57[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	57 -> 7[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	58[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 58[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	58 -> 8[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	7[label="sequence (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.43/4.51	8[label="sequence (map vx3 [])",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.43/4.51	9[label="sequence (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
11.43/4.51	10[label="sequence []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11.43/4.51	11 -> 13[label="",style="dashed", color="red", weight=0];
11.43/4.51	11[label="vx3 vx40 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	12[label="return []",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
11.43/4.51	14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
11.43/4.51	13[label="vx5 >>= sequence1 (map vx3 vx41)",fontsize=16,color="burlywood",shape="triangle"];59[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];13 -> 59[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	59 -> 17[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	60[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 60[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	60 -> 18[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	15[label="[] : []",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="vx50 : vx51 >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
11.43/4.51	18[label="[] >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
11.43/4.51	20 -> 22[label="",style="dashed", color="red", weight=0];
11.43/4.51	20[label="sequence1 (map vx3 vx41) vx50 ++ (vx51 >>= sequence1 (map vx3 vx41))",fontsize=16,color="magenta"];20 -> 23[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	21[label="[]",fontsize=16,color="green",shape="box"];23 -> 13[label="",style="dashed", color="red", weight=0];
11.43/4.51	23[label="vx51 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	22[label="sequence1 (map vx3 vx41) vx50 ++ vx6",fontsize=16,color="black",shape="triangle"];22 -> 25[label="",style="solid", color="black", weight=3];
11.43/4.51	24[label="vx51",fontsize=16,color="green",shape="box"];25 -> 26[label="",style="dashed", color="red", weight=0];
11.43/4.51	25[label="(sequence (map vx3 vx41) >>= sequence0 vx50) ++ vx6",fontsize=16,color="magenta"];25 -> 27[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	27 -> 6[label="",style="dashed", color="red", weight=0];
11.43/4.51	27[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];27 -> 28[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	26[label="(vx7 >>= sequence0 vx50) ++ vx6",fontsize=16,color="burlywood",shape="triangle"];61[label="vx7/vx70 : vx71",fontsize=10,color="white",style="solid",shape="box"];26 -> 61[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	61 -> 29[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	62[label="vx7/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 62[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	62 -> 30[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	28[label="vx41",fontsize=16,color="green",shape="box"];29[label="(vx70 : vx71 >>= sequence0 vx50) ++ vx6",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
11.43/4.51	30[label="([] >>= sequence0 vx50) ++ vx6",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
11.43/4.51	31[label="(sequence0 vx50 vx70 ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
11.43/4.51	32[label="[] ++ vx6",fontsize=16,color="black",shape="triangle"];32 -> 34[label="",style="solid", color="black", weight=3];
11.43/4.51	33[label="(return (vx50 : vx70) ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
11.43/4.51	34[label="vx6",fontsize=16,color="green",shape="box"];35[label="(((vx50 : vx70) : []) ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
11.43/4.51	36 -> 37[label="",style="dashed", color="red", weight=0];
11.43/4.51	36[label="((vx50 : vx70) : [] ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="magenta"];36 -> 38[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	38 -> 32[label="",style="dashed", color="red", weight=0];
11.43/4.51	38[label="[] ++ (vx71 >>= sequence0 vx50)",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	37[label="((vx50 : vx70) : vx8) ++ vx6",fontsize=16,color="black",shape="triangle"];37 -> 40[label="",style="solid", color="black", weight=3];
11.43/4.51	39[label="vx71 >>= sequence0 vx50",fontsize=16,color="burlywood",shape="triangle"];63[label="vx71/vx710 : vx711",fontsize=10,color="white",style="solid",shape="box"];39 -> 63[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	63 -> 41[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	64[label="vx71/[]",fontsize=10,color="white",style="solid",shape="box"];39 -> 64[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	64 -> 42[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	40[label="(vx50 : vx70) : vx8 ++ vx6",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3];
11.43/4.51	41[label="vx710 : vx711 >>= sequence0 vx50",fontsize=16,color="black",shape="box"];41 -> 44[label="",style="solid", color="black", weight=3];
11.43/4.51	42[label="[] >>= sequence0 vx50",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3];
11.43/4.51	43[label="vx8 ++ vx6",fontsize=16,color="burlywood",shape="triangle"];65[label="vx8/vx80 : vx81",fontsize=10,color="white",style="solid",shape="box"];43 -> 65[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	65 -> 46[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	66[label="vx8/[]",fontsize=10,color="white",style="solid",shape="box"];43 -> 66[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	66 -> 47[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	44 -> 43[label="",style="dashed", color="red", weight=0];
11.43/4.51	44[label="sequence0 vx50 vx710 ++ (vx711 >>= sequence0 vx50)",fontsize=16,color="magenta"];44 -> 48[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	44 -> 49[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	45[label="[]",fontsize=16,color="green",shape="box"];46[label="(vx80 : vx81) ++ vx6",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
11.43/4.51	47[label="[] ++ vx6",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
11.43/4.51	48[label="sequence0 vx50 vx710",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
11.43/4.51	49 -> 39[label="",style="dashed", color="red", weight=0];
11.43/4.51	49[label="vx711 >>= sequence0 vx50",fontsize=16,color="magenta"];49 -> 53[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	50[label="vx80 : vx81 ++ vx6",fontsize=16,color="green",shape="box"];50 -> 54[label="",style="dashed", color="green", weight=3];
11.43/4.51	51[label="vx6",fontsize=16,color="green",shape="box"];52[label="return (vx50 : vx710)",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
11.43/4.51	53[label="vx711",fontsize=16,color="green",shape="box"];54 -> 43[label="",style="dashed", color="red", weight=0];
11.43/4.51	54[label="vx81 ++ vx6",fontsize=16,color="magenta"];54 -> 56[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	55[label="(vx50 : vx710) : []",fontsize=16,color="green",shape="box"];56[label="vx81",fontsize=16,color="green",shape="box"];}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(8)
11.43/4.51	Complex Obligation (AND)
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(9)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_gtGtEs(:(vx710, vx711), vx50, h) -> new_gtGtEs(vx711, vx50, h)
11.43/4.51	
11.43/4.51	R is empty.
11.43/4.51	Q is empty.
11.43/4.51	We have to consider all minimal (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(10) QDPSizeChangeProof (EQUIVALENT)
11.43/4.51	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. 
11.43/4.51	
11.43/4.51	From the DPs we obtained the following set of size-change graphs:
11.43/4.51	*new_gtGtEs(:(vx710, vx711), vx50, h) -> new_gtGtEs(vx711, vx50, h)
11.43/4.51	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.43/4.51	
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(11)
11.43/4.51	YES
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(12)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_sequence(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_psPs0(vx3, vx41, h, ba)
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	   new_psPs0(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
11.43/4.51	
11.43/4.51	The TRS R consists of the following rules:
11.43/4.51	
11.43/4.51	   new_psPs1([], vx50, vx6, h) -> new_psPs3(vx6, h)
11.43/4.51	   new_psPs5(:(vx80, vx81), vx6, h) -> :(vx80, new_psPs5(vx81, vx6, h))
11.43/4.51	   new_gtGtEs1(:(vx50, vx51), vx3, vx41, h, ba) -> new_psPs4(vx3, vx41, vx50, new_gtGtEs1(vx51, vx3, vx41, h, ba), h, ba)
11.43/4.51	   new_psPs4(vx3, vx41, vx50, vx6, h, ba) -> new_psPs1(new_sequence0(vx3, vx41, h, ba), vx50, vx6, h)
11.43/4.51	   new_gtGtEs1([], vx3, vx41, h, ba) -> []
11.43/4.51	   new_psPs1(:(vx70, vx71), vx50, vx6, h) -> new_psPs2(vx50, vx70, new_psPs3(new_gtGtEs2(vx71, vx50, h), h), vx6, h)
11.43/4.51	   new_gtGtEs2([], vx50, h) -> []
11.43/4.51	   new_psPs5([], vx6, h) -> vx6
11.43/4.51	   new_psPs2(vx50, vx70, vx8, vx6, h) -> :(:(vx50, vx70), new_psPs5(vx8, vx6, h))
11.43/4.51	   new_psPs3(vx6, h) -> vx6
11.43/4.51	   new_gtGtEs2(:(vx710, vx711), vx50, h) -> new_psPs5(:(:(vx50, vx710), []), new_gtGtEs2(vx711, vx50, h), h)
11.43/4.51	
11.43/4.51	The set Q consists of the following terms:
11.43/4.51	
11.43/4.51	   new_psPs4(x0, x1, x2, x3, x4, x5)
11.43/4.51	   new_psPs1([], x0, x1, x2)
11.43/4.51	   new_gtGtEs2([], x0, x1)
11.43/4.51	   new_psPs1(:(x0, x1), x2, x3, x4)
11.43/4.51	   new_gtGtEs2(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1([], x0, x1, x2, x3)
11.43/4.51	   new_psPs5([], x0, x1)
11.43/4.51	   new_psPs3(x0, x1)
11.43/4.51	   new_psPs2(x0, x1, x2, x3, x4)
11.43/4.51	   new_psPs5(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
11.43/4.51	
11.43/4.51	We have to consider all minimal (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(13) QDPOrderProof (EQUIVALENT)
11.43/4.51	We use the reduction pair processor [LPAR04,JAR06].
11.43/4.51	
11.43/4.51	
11.43/4.51	The following pairs can be oriented strictly and are deleted.
11.43/4.51	
11.43/4.51	   new_psPs0(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
11.43/4.51	The remaining pairs can at least be oriented weakly.
11.43/4.51	Used ordering:  Polynomial interpretation [POLO]:
11.43/4.51	
11.43/4.51	   POL(:(x_1, x_2)) = 1 + x_2
11.43/4.51	   POL(new_gtGtEs0(x_1, x_2, x_3, x_4)) = 1 + x_2
11.43/4.51	   POL(new_psPs0(x_1, x_2, x_3, x_4)) = 1 + x_2
11.43/4.51	   POL(new_sequence(x_1, x_2, x_3, x_4)) = x_2
11.43/4.51	
11.43/4.51	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.43/4.51	none
11.43/4.51	
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(14)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_sequence(vx3, :(vx40, vx41), h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_psPs0(vx3, vx41, h, ba)
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	
11.43/4.51	The TRS R consists of the following rules:
11.43/4.51	
11.43/4.51	   new_psPs1([], vx50, vx6, h) -> new_psPs3(vx6, h)
11.43/4.51	   new_psPs5(:(vx80, vx81), vx6, h) -> :(vx80, new_psPs5(vx81, vx6, h))
11.43/4.51	   new_gtGtEs1(:(vx50, vx51), vx3, vx41, h, ba) -> new_psPs4(vx3, vx41, vx50, new_gtGtEs1(vx51, vx3, vx41, h, ba), h, ba)
11.43/4.51	   new_psPs4(vx3, vx41, vx50, vx6, h, ba) -> new_psPs1(new_sequence0(vx3, vx41, h, ba), vx50, vx6, h)
11.43/4.51	   new_gtGtEs1([], vx3, vx41, h, ba) -> []
11.43/4.51	   new_psPs1(:(vx70, vx71), vx50, vx6, h) -> new_psPs2(vx50, vx70, new_psPs3(new_gtGtEs2(vx71, vx50, h), h), vx6, h)
11.43/4.51	   new_gtGtEs2([], vx50, h) -> []
11.43/4.51	   new_psPs5([], vx6, h) -> vx6
11.43/4.51	   new_psPs2(vx50, vx70, vx8, vx6, h) -> :(:(vx50, vx70), new_psPs5(vx8, vx6, h))
11.43/4.51	   new_psPs3(vx6, h) -> vx6
11.43/4.51	   new_gtGtEs2(:(vx710, vx711), vx50, h) -> new_psPs5(:(:(vx50, vx710), []), new_gtGtEs2(vx711, vx50, h), h)
11.43/4.51	
11.43/4.51	The set Q consists of the following terms:
11.43/4.51	
11.43/4.51	   new_psPs4(x0, x1, x2, x3, x4, x5)
11.43/4.51	   new_psPs1([], x0, x1, x2)
11.43/4.51	   new_gtGtEs2([], x0, x1)
11.43/4.51	   new_psPs1(:(x0, x1), x2, x3, x4)
11.43/4.51	   new_gtGtEs2(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1([], x0, x1, x2, x3)
11.43/4.51	   new_psPs5([], x0, x1)
11.43/4.51	   new_psPs3(x0, x1)
11.43/4.51	   new_psPs2(x0, x1, x2, x3, x4)
11.43/4.51	   new_psPs5(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
11.43/4.51	
11.43/4.51	We have to consider all minimal (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(15) DependencyGraphProof (EQUIVALENT)
11.43/4.51	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(16)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	
11.43/4.51	The TRS R consists of the following rules:
11.43/4.51	
11.43/4.51	   new_psPs1([], vx50, vx6, h) -> new_psPs3(vx6, h)
11.43/4.51	   new_psPs5(:(vx80, vx81), vx6, h) -> :(vx80, new_psPs5(vx81, vx6, h))
11.43/4.51	   new_gtGtEs1(:(vx50, vx51), vx3, vx41, h, ba) -> new_psPs4(vx3, vx41, vx50, new_gtGtEs1(vx51, vx3, vx41, h, ba), h, ba)
11.43/4.51	   new_psPs4(vx3, vx41, vx50, vx6, h, ba) -> new_psPs1(new_sequence0(vx3, vx41, h, ba), vx50, vx6, h)
11.43/4.51	   new_gtGtEs1([], vx3, vx41, h, ba) -> []
11.43/4.51	   new_psPs1(:(vx70, vx71), vx50, vx6, h) -> new_psPs2(vx50, vx70, new_psPs3(new_gtGtEs2(vx71, vx50, h), h), vx6, h)
11.43/4.51	   new_gtGtEs2([], vx50, h) -> []
11.43/4.51	   new_psPs5([], vx6, h) -> vx6
11.43/4.51	   new_psPs2(vx50, vx70, vx8, vx6, h) -> :(:(vx50, vx70), new_psPs5(vx8, vx6, h))
11.43/4.51	   new_psPs3(vx6, h) -> vx6
11.43/4.51	   new_gtGtEs2(:(vx710, vx711), vx50, h) -> new_psPs5(:(:(vx50, vx710), []), new_gtGtEs2(vx711, vx50, h), h)
11.43/4.51	
11.43/4.51	The set Q consists of the following terms:
11.43/4.51	
11.43/4.51	   new_psPs4(x0, x1, x2, x3, x4, x5)
11.43/4.51	   new_psPs1([], x0, x1, x2)
11.43/4.51	   new_gtGtEs2([], x0, x1)
11.43/4.51	   new_psPs1(:(x0, x1), x2, x3, x4)
11.43/4.51	   new_gtGtEs2(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1([], x0, x1, x2, x3)
11.43/4.51	   new_psPs5([], x0, x1)
11.43/4.51	   new_psPs3(x0, x1)
11.43/4.51	   new_psPs2(x0, x1, x2, x3, x4)
11.43/4.51	   new_psPs5(:(x0, x1), x2, x3)
11.43/4.51	   new_gtGtEs1(:(x0, x1), x2, x3, x4, x5)
11.43/4.51	
11.43/4.51	We have to consider all minimal (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(17) MNOCProof (EQUIVALENT)
11.43/4.51	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(18)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_gtGtEs0(vx3, vx41, h, ba) -> new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	
11.43/4.51	The TRS R consists of the following rules:
11.43/4.51	
11.43/4.51	   new_psPs1([], vx50, vx6, h) -> new_psPs3(vx6, h)
11.43/4.51	   new_psPs5(:(vx80, vx81), vx6, h) -> :(vx80, new_psPs5(vx81, vx6, h))
11.43/4.51	   new_gtGtEs1(:(vx50, vx51), vx3, vx41, h, ba) -> new_psPs4(vx3, vx41, vx50, new_gtGtEs1(vx51, vx3, vx41, h, ba), h, ba)
11.43/4.51	   new_psPs4(vx3, vx41, vx50, vx6, h, ba) -> new_psPs1(new_sequence0(vx3, vx41, h, ba), vx50, vx6, h)
11.43/4.51	   new_gtGtEs1([], vx3, vx41, h, ba) -> []
11.43/4.51	   new_psPs1(:(vx70, vx71), vx50, vx6, h) -> new_psPs2(vx50, vx70, new_psPs3(new_gtGtEs2(vx71, vx50, h), h), vx6, h)
11.43/4.51	   new_gtGtEs2([], vx50, h) -> []
11.43/4.51	   new_psPs5([], vx6, h) -> vx6
11.43/4.51	   new_psPs2(vx50, vx70, vx8, vx6, h) -> :(:(vx50, vx70), new_psPs5(vx8, vx6, h))
11.43/4.51	   new_psPs3(vx6, h) -> vx6
11.43/4.51	   new_gtGtEs2(:(vx710, vx711), vx50, h) -> new_psPs5(:(:(vx50, vx710), []), new_gtGtEs2(vx711, vx50, h), h)
11.43/4.51	
11.43/4.51	Q is empty.
11.43/4.51	We have to consider all (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(19) NonTerminationLoopProof (COMPLETE)
11.43/4.51	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.43/4.51	Found a loop by semiunifying a rule from P directly.
11.43/4.51	
11.43/4.51	s = new_gtGtEs0(vx3, vx41, h, ba) evaluates to  t =new_gtGtEs0(vx3, vx41, h, ba)
11.43/4.51	
11.43/4.51	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.43/4.51	* Matcher: [ ]
11.43/4.51	* Semiunifier: [ ]
11.43/4.51	
11.43/4.51	--------------------------------------------------------------------------------
11.43/4.51	Rewriting sequence
11.43/4.51	
11.43/4.51	The DP semiunifies directly so there is only one rewrite step from new_gtGtEs0(vx3, vx41, h, ba) to new_gtGtEs0(vx3, vx41, h, ba).
11.43/4.51	
11.43/4.51	
11.43/4.51	
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(20)
11.43/4.51	NO
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(21)
11.43/4.51	Obligation:
11.43/4.51	Q DP problem:
11.43/4.51	The TRS P consists of the following rules:
11.43/4.51	
11.43/4.51	   new_psPs(:(vx80, vx81), vx6, h) -> new_psPs(vx81, vx6, h)
11.43/4.51	
11.43/4.51	R is empty.
11.43/4.51	Q is empty.
11.43/4.51	We have to consider all minimal (P,Q,R)-chains.
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(22) QDPSizeChangeProof (EQUIVALENT)
11.43/4.51	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. 
11.43/4.51	
11.43/4.51	From the DPs we obtained the following set of size-change graphs:
11.43/4.51	*new_psPs(:(vx80, vx81), vx6, h) -> new_psPs(vx81, vx6, h)
11.43/4.51	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.43/4.51	
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(23)
11.43/4.51	YES
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(24) Narrow (COMPLETE)
11.43/4.51	Haskell To QDPs
11.43/4.51	
11.43/4.51	digraph dp_graph {
11.43/4.51	node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.43/4.51	3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.43/4.51	4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.43/4.51	5[label="sequence . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.43/4.51	6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];57[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 57[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	57 -> 7[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	58[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 58[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	58 -> 8[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	7[label="sequence (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.43/4.51	8[label="sequence (map vx3 [])",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.43/4.51	9[label="sequence (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
11.43/4.51	10[label="sequence []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11.43/4.51	11 -> 13[label="",style="dashed", color="red", weight=0];
11.43/4.51	11[label="vx3 vx40 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	12[label="return []",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
11.43/4.51	14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
11.43/4.51	13[label="vx5 >>= sequence1 (map vx3 vx41)",fontsize=16,color="burlywood",shape="triangle"];59[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];13 -> 59[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	59 -> 17[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	60[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 60[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	60 -> 18[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	15[label="[] : []",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="vx50 : vx51 >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
11.43/4.51	18[label="[] >>= sequence1 (map vx3 vx41)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
11.43/4.51	20 -> 22[label="",style="dashed", color="red", weight=0];
11.43/4.51	20[label="sequence1 (map vx3 vx41) vx50 ++ (vx51 >>= sequence1 (map vx3 vx41))",fontsize=16,color="magenta"];20 -> 23[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	21[label="[]",fontsize=16,color="green",shape="box"];23 -> 13[label="",style="dashed", color="red", weight=0];
11.43/4.51	23[label="vx51 >>= sequence1 (map vx3 vx41)",fontsize=16,color="magenta"];23 -> 24[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	22[label="sequence1 (map vx3 vx41) vx50 ++ vx6",fontsize=16,color="black",shape="triangle"];22 -> 25[label="",style="solid", color="black", weight=3];
11.43/4.51	24[label="vx51",fontsize=16,color="green",shape="box"];25 -> 26[label="",style="dashed", color="red", weight=0];
11.43/4.51	25[label="(sequence (map vx3 vx41) >>= sequence0 vx50) ++ vx6",fontsize=16,color="magenta"];25 -> 27[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	27 -> 6[label="",style="dashed", color="red", weight=0];
11.43/4.51	27[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];27 -> 28[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	26[label="(vx7 >>= sequence0 vx50) ++ vx6",fontsize=16,color="burlywood",shape="triangle"];61[label="vx7/vx70 : vx71",fontsize=10,color="white",style="solid",shape="box"];26 -> 61[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	61 -> 29[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	62[label="vx7/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 62[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	62 -> 30[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	28[label="vx41",fontsize=16,color="green",shape="box"];29[label="(vx70 : vx71 >>= sequence0 vx50) ++ vx6",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
11.43/4.51	30[label="([] >>= sequence0 vx50) ++ vx6",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
11.43/4.51	31[label="(sequence0 vx50 vx70 ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
11.43/4.51	32[label="[] ++ vx6",fontsize=16,color="black",shape="triangle"];32 -> 34[label="",style="solid", color="black", weight=3];
11.43/4.51	33[label="(return (vx50 : vx70) ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
11.43/4.51	34[label="vx6",fontsize=16,color="green",shape="box"];35[label="(((vx50 : vx70) : []) ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
11.43/4.51	36 -> 37[label="",style="dashed", color="red", weight=0];
11.43/4.51	36[label="((vx50 : vx70) : [] ++ (vx71 >>= sequence0 vx50)) ++ vx6",fontsize=16,color="magenta"];36 -> 38[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	38 -> 32[label="",style="dashed", color="red", weight=0];
11.43/4.51	38[label="[] ++ (vx71 >>= sequence0 vx50)",fontsize=16,color="magenta"];38 -> 39[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	37[label="((vx50 : vx70) : vx8) ++ vx6",fontsize=16,color="black",shape="triangle"];37 -> 40[label="",style="solid", color="black", weight=3];
11.43/4.51	39[label="vx71 >>= sequence0 vx50",fontsize=16,color="burlywood",shape="triangle"];63[label="vx71/vx710 : vx711",fontsize=10,color="white",style="solid",shape="box"];39 -> 63[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	63 -> 41[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	64[label="vx71/[]",fontsize=10,color="white",style="solid",shape="box"];39 -> 64[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	64 -> 42[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	40[label="(vx50 : vx70) : vx8 ++ vx6",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3];
11.43/4.51	41[label="vx710 : vx711 >>= sequence0 vx50",fontsize=16,color="black",shape="box"];41 -> 44[label="",style="solid", color="black", weight=3];
11.43/4.51	42[label="[] >>= sequence0 vx50",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3];
11.43/4.51	43[label="vx8 ++ vx6",fontsize=16,color="burlywood",shape="triangle"];65[label="vx8/vx80 : vx81",fontsize=10,color="white",style="solid",shape="box"];43 -> 65[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	65 -> 46[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	66[label="vx8/[]",fontsize=10,color="white",style="solid",shape="box"];43 -> 66[label="",style="solid", color="burlywood", weight=9];
11.43/4.51	66 -> 47[label="",style="solid", color="burlywood", weight=3];
11.43/4.51	44 -> 43[label="",style="dashed", color="red", weight=0];
11.43/4.51	44[label="sequence0 vx50 vx710 ++ (vx711 >>= sequence0 vx50)",fontsize=16,color="magenta"];44 -> 48[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	44 -> 49[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	45[label="[]",fontsize=16,color="green",shape="box"];46[label="(vx80 : vx81) ++ vx6",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
11.43/4.51	47[label="[] ++ vx6",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
11.43/4.51	48[label="sequence0 vx50 vx710",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3];
11.43/4.51	49 -> 39[label="",style="dashed", color="red", weight=0];
11.43/4.51	49[label="vx711 >>= sequence0 vx50",fontsize=16,color="magenta"];49 -> 53[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	50[label="vx80 : vx81 ++ vx6",fontsize=16,color="green",shape="box"];50 -> 54[label="",style="dashed", color="green", weight=3];
11.43/4.51	51[label="vx6",fontsize=16,color="green",shape="box"];52[label="return (vx50 : vx710)",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
11.43/4.51	53[label="vx711",fontsize=16,color="green",shape="box"];54 -> 43[label="",style="dashed", color="red", weight=0];
11.43/4.51	54[label="vx81 ++ vx6",fontsize=16,color="magenta"];54 -> 56[label="",style="dashed", color="magenta", weight=3];
11.43/4.51	55[label="(vx50 : vx710) : []",fontsize=16,color="green",shape="box"];56[label="vx81",fontsize=16,color="green",shape="box"];}
11.43/4.51	
11.43/4.51	----------------------------------------
11.43/4.51	
11.43/4.51	(25)
11.43/4.51	TRUE
11.54/4.55	EOF