8.17/3.50	YES
9.89/4.01	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.89/4.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.89/4.01	
9.89/4.01	
9.89/4.01	H-Termination with start terms of the given HASKELL could be proven:
9.89/4.01	
9.89/4.01	(0) HASKELL
9.89/4.01	(1) LR [EQUIVALENT, 0 ms]
9.89/4.01	(2) HASKELL
9.89/4.01	(3) BR [EQUIVALENT, 0 ms]
9.89/4.01	(4) HASKELL
9.89/4.01	(5) COR [EQUIVALENT, 0 ms]
9.89/4.01	(6) HASKELL
9.89/4.01	(7) Narrow [SOUND, 0 ms]
9.89/4.01	(8) QDP
9.89/4.01	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.89/4.01	(10) YES
9.89/4.01	
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(0)
9.89/4.01	Obligation:
9.89/4.01	mainModule Main 
9.89/4.01	module Main where {
9.89/4.01	import qualified Prelude;
9.89/4.01	}
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(1) LR (EQUIVALENT)
9.89/4.01	Lambda Reductions:
9.89/4.01	The following Lambda expression
9.89/4.01	"\_->q"
9.89/4.01	is transformed to
9.89/4.01	"gtGt0 q _ = q;
9.89/4.01	"
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(2)
9.89/4.01	Obligation:
9.89/4.01	mainModule Main 
9.89/4.01	module Main where {
9.89/4.01	import qualified Prelude;
9.89/4.01	}
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(3) BR (EQUIVALENT)
9.89/4.01	Replaced joker patterns by fresh variables and removed binding patterns.
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(4)
9.89/4.01	Obligation:
9.89/4.01	mainModule Main 
9.89/4.01	module Main where {
9.89/4.01	import qualified Prelude;
9.89/4.01	}
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(5) COR (EQUIVALENT)
9.89/4.01	Cond Reductions:
9.89/4.01	The following Function with conditions
9.89/4.01	"undefined |Falseundefined;
9.89/4.01	"
9.89/4.01	is transformed to
9.89/4.01	"undefined  = undefined1;
9.89/4.01	"
9.89/4.01	"undefined0 True = undefined;
9.89/4.01	"
9.89/4.01	"undefined1  = undefined0 False;
9.89/4.01	"
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(6)
9.89/4.01	Obligation:
9.89/4.01	mainModule Main 
9.89/4.01	module Main where {
9.89/4.01	import qualified Prelude;
9.89/4.01	}
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(7) Narrow (SOUND)
9.89/4.01	Haskell To QDPs
9.89/4.01	
9.89/4.01	digraph dp_graph {
9.89/4.01	node [outthreshold=100, inthreshold=100];1[label="mapM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.89/4.01	3[label="mapM_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.89/4.01	4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.89/4.01	5[label="sequence_ . map vy3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.89/4.01	6[label="sequence_ (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.89/4.01	7[label="foldr (>>) (return ()) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];28[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.89/4.01	28 -> 8[label="",style="solid", color="burlywood", weight=3];
9.89/4.01	29[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 29[label="",style="solid", color="burlywood", weight=9];
9.89/4.01	29 -> 9[label="",style="solid", color="burlywood", weight=3];
9.89/4.01	8[label="foldr (>>) (return ()) (map vy3 (vy40 : vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.89/4.01	9[label="foldr (>>) (return ()) (map vy3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.89/4.01	10[label="foldr (>>) (return ()) (vy3 vy40 : map vy3 vy41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.89/4.01	11[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.89/4.01	12 -> 14[label="",style="dashed", color="red", weight=0];
9.89/4.01	12[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
9.89/4.01	13[label="return ()",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
9.89/4.01	15 -> 7[label="",style="dashed", color="red", weight=0];
9.89/4.01	15[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
9.89/4.01	14[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];14 -> 18[label="",style="solid", color="black", weight=3];
9.89/4.01	16[label="Just ()",fontsize=16,color="green",shape="box"];17[label="vy41",fontsize=16,color="green",shape="box"];18 -> 19[label="",style="dashed", color="red", weight=0];
9.89/4.01	18[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
9.89/4.01	20[label="vy3 vy40",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
9.89/4.01	19[label="vy6 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];30[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
9.89/4.01	30 -> 22[label="",style="solid", color="burlywood", weight=3];
9.89/4.01	31[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];19 -> 31[label="",style="solid", color="burlywood", weight=9];
9.89/4.01	31 -> 23[label="",style="solid", color="burlywood", weight=3];
9.89/4.01	24[label="vy40",fontsize=16,color="green",shape="box"];22[label="Nothing >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
9.89/4.01	23[label="Just vy60 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.89/4.01	25[label="Nothing",fontsize=16,color="green",shape="box"];26[label="gtGt0 vy5 vy60",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
9.89/4.01	27[label="vy5",fontsize=16,color="green",shape="box"];}
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(8)
9.89/4.01	Obligation:
9.89/4.01	Q DP problem:
9.89/4.01	The TRS P consists of the following rules:
9.89/4.01	
9.89/4.01	   new_foldr(vy3, :(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.89/4.01	
9.89/4.01	R is empty.
9.89/4.01	Q is empty.
9.89/4.01	We have to consider all minimal (P,Q,R)-chains.
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(9) QDPSizeChangeProof (EQUIVALENT)
9.89/4.01	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. 
9.89/4.01	
9.89/4.01	From the DPs we obtained the following set of size-change graphs:
9.89/4.01	*new_foldr(vy3, :(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.89/4.01	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.89/4.01	
9.89/4.01	
9.89/4.01	----------------------------------------
9.89/4.01	
9.89/4.01	(10)
9.89/4.01	YES
10.07/4.05	EOF