7.75/3.53	YES
9.93/4.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.93/4.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.93/4.10	
9.93/4.10	
9.93/4.10	H-Termination with start terms of the given HASKELL could be proven:
9.93/4.10	
9.93/4.10	(0) HASKELL
9.93/4.10	(1) LR [EQUIVALENT, 0 ms]
9.93/4.10	(2) HASKELL
9.93/4.10	(3) BR [EQUIVALENT, 0 ms]
9.93/4.10	(4) HASKELL
9.93/4.10	(5) COR [EQUIVALENT, 0 ms]
9.93/4.10	(6) HASKELL
9.93/4.10	(7) Narrow [SOUND, 0 ms]
9.93/4.10	(8) QDP
9.93/4.10	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.93/4.10	(10) YES
9.93/4.10	
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(0)
9.93/4.10	Obligation:
9.93/4.10	mainModule Main 
9.93/4.10	module Main where {
9.93/4.10	import qualified Prelude;
9.93/4.10	}
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(1) LR (EQUIVALENT)
9.93/4.10	Lambda Reductions:
9.93/4.10	The following Lambda expression
9.93/4.10	"\_->q"
9.93/4.10	is transformed to
9.93/4.10	"gtGt0 q _ = q;
9.93/4.10	"
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(2)
9.93/4.10	Obligation:
9.93/4.10	mainModule Main 
9.93/4.10	module Main where {
9.93/4.10	import qualified Prelude;
9.93/4.10	}
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(3) BR (EQUIVALENT)
9.93/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(4)
9.93/4.10	Obligation:
9.93/4.10	mainModule Main 
9.93/4.10	module Main where {
9.93/4.10	import qualified Prelude;
9.93/4.10	}
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(5) COR (EQUIVALENT)
9.93/4.10	Cond Reductions:
9.93/4.10	The following Function with conditions
9.93/4.10	"undefined |Falseundefined;
9.93/4.10	"
9.93/4.10	is transformed to
9.93/4.10	"undefined  = undefined1;
9.93/4.10	"
9.93/4.10	"undefined0 True = undefined;
9.93/4.10	"
9.93/4.10	"undefined1  = undefined0 False;
9.93/4.10	"
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(6)
9.93/4.10	Obligation:
9.93/4.10	mainModule Main 
9.93/4.10	module Main where {
9.93/4.10	import qualified Prelude;
9.93/4.10	}
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(7) Narrow (SOUND)
9.93/4.10	Haskell To QDPs
9.93/4.10	
9.93/4.10	digraph dp_graph {
9.93/4.10	node [outthreshold=100, inthreshold=100];1[label="sequence_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.93/4.10	3[label="sequence_ vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.93/4.10	4[label="foldr (>>) (return ()) vy3",fontsize=16,color="burlywood",shape="triangle"];19[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
9.93/4.10	19 -> 5[label="",style="solid", color="burlywood", weight=3];
9.93/4.10	20[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
9.93/4.10	20 -> 6[label="",style="solid", color="burlywood", weight=3];
9.93/4.10	5[label="foldr (>>) (return ()) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.93/4.10	6[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.93/4.10	7 -> 9[label="",style="dashed", color="red", weight=0];
9.93/4.10	7[label="(>>) vy30 foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.93/4.10	8[label="return ()",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.93/4.10	10 -> 4[label="",style="dashed", color="red", weight=0];
9.93/4.10	10[label="foldr (>>) (return ()) vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9.93/4.10	9[label="(>>) vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
9.93/4.10	11[label="Just ()",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="vy30 >>= gtGt0 vy4",fontsize=16,color="burlywood",shape="box"];21[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 21[label="",style="solid", color="burlywood", weight=9];
9.93/4.10	21 -> 14[label="",style="solid", color="burlywood", weight=3];
9.93/4.10	22[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9];
9.93/4.10	22 -> 15[label="",style="solid", color="burlywood", weight=3];
9.93/4.10	14[label="Nothing >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
9.93/4.10	15[label="Just vy300 >>= gtGt0 vy4",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
9.93/4.10	16[label="Nothing",fontsize=16,color="green",shape="box"];17[label="gtGt0 vy4 vy300",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
9.93/4.10	18[label="vy4",fontsize=16,color="green",shape="box"];}
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(8)
9.93/4.10	Obligation:
9.93/4.10	Q DP problem:
9.93/4.10	The TRS P consists of the following rules:
9.93/4.10	
9.93/4.10	   new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
9.93/4.10	
9.93/4.10	R is empty.
9.93/4.10	Q is empty.
9.93/4.10	We have to consider all minimal (P,Q,R)-chains.
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(9) QDPSizeChangeProof (EQUIVALENT)
9.93/4.10	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. 
9.93/4.10	
9.93/4.10	From the DPs we obtained the following set of size-change graphs:
9.93/4.10	*new_foldr(:(vy30, vy31), h) -> new_foldr(vy31, h)
9.93/4.10	The graph contains the following edges 1 > 1, 2 >= 2
9.93/4.10	
9.93/4.10	
9.93/4.10	----------------------------------------
9.93/4.10	
9.93/4.10	(10)
9.93/4.10	YES
9.93/4.15	EOF