7.88/3.58	YES
9.73/4.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.73/4.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.73/4.08	
9.73/4.08	
9.73/4.08	H-Termination with start terms of the given HASKELL could be proven:
9.73/4.08	
9.73/4.08	(0) HASKELL
9.73/4.08	(1) LR [EQUIVALENT, 0 ms]
9.73/4.08	(2) HASKELL
9.73/4.08	(3) BR [EQUIVALENT, 0 ms]
9.73/4.08	(4) HASKELL
9.73/4.08	(5) COR [EQUIVALENT, 0 ms]
9.73/4.08	(6) HASKELL
9.73/4.08	(7) Narrow [SOUND, 0 ms]
9.73/4.08	(8) AND
9.73/4.08	    (9) QDP
9.73/4.08	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.73/4.08	        (11) YES
9.73/4.08	    (12) QDP
9.73/4.08	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.73/4.08	        (14) YES
9.73/4.08	    (15) QDP
9.73/4.08	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.73/4.08	        (17) YES
9.73/4.08	
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(0)
9.73/4.08	Obligation:
9.73/4.08	mainModule Main 
9.73/4.08	module Main where {
9.73/4.08	import qualified Prelude;
9.73/4.08	}
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(1) LR (EQUIVALENT)
9.73/4.08	Lambda Reductions:
9.73/4.08	The following Lambda expression
9.73/4.08	"\_->q"
9.73/4.08	is transformed to
9.73/4.08	"gtGt0 q _ = q;
9.73/4.08	"
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(2)
9.73/4.08	Obligation:
9.73/4.08	mainModule Main 
9.73/4.08	module Main where {
9.73/4.08	import qualified Prelude;
9.73/4.08	}
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(3) BR (EQUIVALENT)
9.73/4.08	Replaced joker patterns by fresh variables and removed binding patterns.
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(4)
9.73/4.08	Obligation:
9.73/4.08	mainModule Main 
9.73/4.08	module Main where {
9.73/4.08	import qualified Prelude;
9.73/4.08	}
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(5) COR (EQUIVALENT)
9.73/4.08	Cond Reductions:
9.73/4.08	The following Function with conditions
9.73/4.08	"undefined |Falseundefined;
9.73/4.08	"
9.73/4.08	is transformed to
9.73/4.08	"undefined  = undefined1;
9.73/4.08	"
9.73/4.08	"undefined0 True = undefined;
9.73/4.08	"
9.73/4.08	"undefined1  = undefined0 False;
9.73/4.08	"
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(6)
9.73/4.08	Obligation:
9.73/4.08	mainModule Main 
9.73/4.08	module Main where {
9.73/4.08	import qualified Prelude;
9.73/4.08	}
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(7) Narrow (SOUND)
9.73/4.08	Haskell To QDPs
9.73/4.08	
9.73/4.08	digraph dp_graph {
9.73/4.08	node [outthreshold=100, inthreshold=100];1[label="mapM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.73/4.08	3[label="mapM_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.73/4.08	4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.73/4.08	5[label="sequence_ . map vy3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.73/4.08	6[label="sequence_ (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.73/4.08	7[label="foldr (>>) (return ()) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];37[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 37[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	37 -> 8[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	38[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 38[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	38 -> 9[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	8[label="foldr (>>) (return ()) (map vy3 (vy40 : vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.73/4.08	9[label="foldr (>>) (return ()) (map vy3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.73/4.08	10[label="foldr (>>) (return ()) (vy3 vy40 : map vy3 vy41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.73/4.08	11[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.73/4.08	12 -> 14[label="",style="dashed", color="red", weight=0];
9.73/4.08	12[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	13[label="return ()",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
9.73/4.08	15 -> 7[label="",style="dashed", color="red", weight=0];
9.73/4.08	15[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	14[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];14 -> 18[label="",style="solid", color="black", weight=3];
9.73/4.08	16[label="() : []",fontsize=16,color="green",shape="box"];17[label="vy41",fontsize=16,color="green",shape="box"];18 -> 19[label="",style="dashed", color="red", weight=0];
9.73/4.08	18[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	20[label="vy3 vy40",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
9.73/4.08	19[label="vy6 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];39[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];19 -> 39[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	39 -> 22[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	40[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 40[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	40 -> 23[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	24[label="vy40",fontsize=16,color="green",shape="box"];22[label="vy60 : vy61 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
9.73/4.08	23[label="[] >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.73/4.08	25 -> 27[label="",style="dashed", color="red", weight=0];
9.73/4.08	25[label="gtGt0 vy5 vy60 ++ (vy61 >>= gtGt0 vy5)",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	26[label="[]",fontsize=16,color="green",shape="box"];28 -> 19[label="",style="dashed", color="red", weight=0];
9.73/4.08	28[label="vy61 >>= gtGt0 vy5",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	27[label="gtGt0 vy5 vy60 ++ vy7",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
9.73/4.08	29[label="vy61",fontsize=16,color="green",shape="box"];30[label="vy5 ++ vy7",fontsize=16,color="burlywood",shape="triangle"];41[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];30 -> 41[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	41 -> 31[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	42[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];30 -> 42[label="",style="solid", color="burlywood", weight=9];
9.73/4.08	42 -> 32[label="",style="solid", color="burlywood", weight=3];
9.73/4.08	31[label="(vy50 : vy51) ++ vy7",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
9.73/4.08	32[label="[] ++ vy7",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
9.73/4.08	33[label="vy50 : vy51 ++ vy7",fontsize=16,color="green",shape="box"];33 -> 35[label="",style="dashed", color="green", weight=3];
9.73/4.08	34[label="vy7",fontsize=16,color="green",shape="box"];35 -> 30[label="",style="dashed", color="red", weight=0];
9.73/4.08	35[label="vy51 ++ vy7",fontsize=16,color="magenta"];35 -> 36[label="",style="dashed", color="magenta", weight=3];
9.73/4.08	36[label="vy51",fontsize=16,color="green",shape="box"];}
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(8)
9.73/4.08	Complex Obligation (AND)
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(9)
9.73/4.08	Obligation:
9.73/4.08	Q DP problem:
9.73/4.08	The TRS P consists of the following rules:
9.73/4.08	
9.73/4.08	   new_gtGtEs(:(vy60, vy61), vy5, h) -> new_gtGtEs(vy61, vy5, h)
9.73/4.08	
9.73/4.08	R is empty.
9.73/4.08	Q is empty.
9.73/4.08	We have to consider all minimal (P,Q,R)-chains.
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(10) QDPSizeChangeProof (EQUIVALENT)
9.73/4.08	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. 
9.73/4.08	
9.73/4.08	From the DPs we obtained the following set of size-change graphs:
9.73/4.08	*new_gtGtEs(:(vy60, vy61), vy5, h) -> new_gtGtEs(vy61, vy5, h)
9.73/4.08	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.73/4.08	
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(11)
9.73/4.08	YES
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(12)
9.73/4.08	Obligation:
9.73/4.08	Q DP problem:
9.73/4.08	The TRS P consists of the following rules:
9.73/4.08	
9.73/4.08	   new_foldr(vy3, :(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.73/4.08	
9.73/4.08	R is empty.
9.73/4.08	Q is empty.
9.73/4.08	We have to consider all minimal (P,Q,R)-chains.
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(13) QDPSizeChangeProof (EQUIVALENT)
9.73/4.08	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. 
9.73/4.08	
9.73/4.08	From the DPs we obtained the following set of size-change graphs:
9.73/4.08	*new_foldr(vy3, :(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.73/4.08	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.73/4.08	
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(14)
9.73/4.08	YES
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(15)
9.73/4.08	Obligation:
9.73/4.08	Q DP problem:
9.73/4.08	The TRS P consists of the following rules:
9.73/4.08	
9.73/4.08	   new_psPs(:(vy50, vy51), vy7) -> new_psPs(vy51, vy7)
9.73/4.08	
9.73/4.08	R is empty.
9.73/4.08	Q is empty.
9.73/4.08	We have to consider all minimal (P,Q,R)-chains.
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(16) QDPSizeChangeProof (EQUIVALENT)
9.73/4.08	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. 
9.73/4.08	
9.73/4.08	From the DPs we obtained the following set of size-change graphs:
9.73/4.08	*new_psPs(:(vy50, vy51), vy7) -> new_psPs(vy51, vy7)
9.73/4.08	The graph contains the following edges 1 > 1, 2 >= 2
9.73/4.08	
9.73/4.08	
9.73/4.08	----------------------------------------
9.73/4.08	
9.73/4.08	(17)
9.73/4.08	YES
10.04/4.15	EOF