8.23/3.63	YES
10.03/4.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.03/4.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.03/4.10	
10.03/4.10	
10.03/4.10	H-Termination with start terms of the given HASKELL could be proven:
10.03/4.10	
10.03/4.10	(0) HASKELL
10.03/4.10	(1) BR [EQUIVALENT, 0 ms]
10.03/4.10	(2) HASKELL
10.03/4.10	(3) COR [EQUIVALENT, 0 ms]
10.03/4.10	(4) HASKELL
10.03/4.10	(5) Narrow [SOUND, 0 ms]
10.03/4.10	(6) QDP
10.03/4.10	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.03/4.10	(8) YES
10.03/4.10	
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(0)
10.03/4.10	Obligation:
10.03/4.10	mainModule Main 
10.03/4.10	module Main where {
10.03/4.10	import qualified Prelude;
10.03/4.10	}
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(1) BR (EQUIVALENT)
10.03/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(2)
10.03/4.10	Obligation:
10.03/4.10	mainModule Main 
10.03/4.10	module Main where {
10.03/4.10	import qualified Prelude;
10.03/4.10	}
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(3) COR (EQUIVALENT)
10.03/4.10	Cond Reductions:
10.03/4.10	The following Function with conditions
10.03/4.10	"undefined |Falseundefined;
10.03/4.10	"
10.03/4.10	is transformed to
10.03/4.10	"undefined  = undefined1;
10.03/4.10	"
10.03/4.10	"undefined0 True = undefined;
10.03/4.10	"
10.03/4.10	"undefined1  = undefined0 False;
10.03/4.10	"
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(4)
10.03/4.10	Obligation:
10.03/4.10	mainModule Main 
10.03/4.10	module Main where {
10.03/4.10	import qualified Prelude;
10.03/4.10	}
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(5) Narrow (SOUND)
10.03/4.10	Haskell To QDPs
10.03/4.10	
10.03/4.10	digraph dp_graph {
10.03/4.10	node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.03/4.10	3[label="notElem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.03/4.10	4[label="notElem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.03/4.10	5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.03/4.10	6[label="all ((/=) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.03/4.10	7[label="and . map ((/=) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.03/4.10	8[label="and (map ((/=) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.03/4.10	9[label="foldr (&&) True (map ((/=) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];25[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
10.03/4.10	25 -> 10[label="",style="solid", color="burlywood", weight=3];
10.03/4.10	26[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 26[label="",style="solid", color="burlywood", weight=9];
10.03/4.10	26 -> 11[label="",style="solid", color="burlywood", weight=3];
10.03/4.10	10[label="foldr (&&) True (map ((/=) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.03/4.10	11[label="foldr (&&) True (map ((/=) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.03/4.10	12[label="foldr (&&) True (((/=) vx3 vx40) : map ((/=) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.03/4.10	13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.03/4.10	14 -> 16[label="",style="dashed", color="red", weight=0];
10.03/4.10	14[label="(&&) (/=) vx3 vx40 foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
10.03/4.10	15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
10.03/4.10	17[label="foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
10.03/4.10	16[label="(&&) (/=) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
10.03/4.10	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vx3 == vx40) vx5",fontsize=16,color="burlywood",shape="box"];27[label="vx3/()",fontsize=10,color="white",style="solid",shape="box"];19 -> 27[label="",style="solid", color="burlywood", weight=9];
10.03/4.10	27 -> 20[label="",style="solid", color="burlywood", weight=3];
10.03/4.10	20[label="(&&) not (() == vx40) vx5",fontsize=16,color="burlywood",shape="box"];28[label="vx40/()",fontsize=10,color="white",style="solid",shape="box"];20 -> 28[label="",style="solid", color="burlywood", weight=9];
10.03/4.10	28 -> 21[label="",style="solid", color="burlywood", weight=3];
10.03/4.10	21[label="(&&) not (() == ()) vx5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
10.03/4.10	22[label="(&&) not True vx5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
10.03/4.10	23[label="(&&) False vx5",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
10.03/4.10	24[label="False",fontsize=16,color="green",shape="box"];}
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(6)
10.03/4.10	Obligation:
10.03/4.10	Q DP problem:
10.03/4.10	The TRS P consists of the following rules:
10.03/4.10	
10.03/4.10	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
10.03/4.10	
10.03/4.10	R is empty.
10.03/4.10	Q is empty.
10.03/4.10	We have to consider all minimal (P,Q,R)-chains.
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(7) QDPSizeChangeProof (EQUIVALENT)
10.03/4.10	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. 
10.03/4.10	
10.03/4.10	From the DPs we obtained the following set of size-change graphs:
10.03/4.10	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
10.03/4.10	The graph contains the following edges 1 >= 1, 2 > 2
10.03/4.10	
10.03/4.10	
10.03/4.10	----------------------------------------
10.03/4.10	
10.03/4.10	(8)
10.03/4.10	YES
10.10/4.17	EOF