8.22/3.68	YES
9.88/4.13	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.88/4.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.88/4.13	
9.88/4.13	
9.88/4.13	H-Termination with start terms of the given HASKELL could be proven:
9.88/4.13	
9.88/4.13	(0) HASKELL
9.88/4.13	(1) BR [EQUIVALENT, 0 ms]
9.88/4.13	(2) HASKELL
9.88/4.13	(3) COR [EQUIVALENT, 0 ms]
9.88/4.13	(4) HASKELL
9.88/4.13	(5) Narrow [SOUND, 0 ms]
9.88/4.13	(6) QDP
9.88/4.13	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.88/4.13	(8) YES
9.88/4.13	
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(0)
9.88/4.13	Obligation:
9.88/4.13	mainModule Main 
9.88/4.13	module Main where {
9.88/4.13	import qualified Prelude;
9.88/4.13	}
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(1) BR (EQUIVALENT)
9.88/4.13	Replaced joker patterns by fresh variables and removed binding patterns.
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(2)
9.88/4.13	Obligation:
9.88/4.13	mainModule Main 
9.88/4.13	module Main where {
9.88/4.13	import qualified Prelude;
9.88/4.13	}
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(3) COR (EQUIVALENT)
9.88/4.13	Cond Reductions:
9.88/4.13	The following Function with conditions
9.88/4.13	"undefined |Falseundefined;
9.88/4.13	"
9.88/4.13	is transformed to
9.88/4.13	"undefined  = undefined1;
9.88/4.13	"
9.88/4.13	"undefined0 True = undefined;
9.88/4.13	"
9.88/4.13	"undefined1  = undefined0 False;
9.88/4.13	"
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(4)
9.88/4.13	Obligation:
9.88/4.13	mainModule Main 
9.88/4.13	module Main where {
9.88/4.13	import qualified Prelude;
9.88/4.13	}
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(5) Narrow (SOUND)
9.88/4.13	Haskell To QDPs
9.88/4.13	
9.88/4.13	digraph dp_graph {
9.88/4.13	node [outthreshold=100, inthreshold=100];1[label="or",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.88/4.13	3[label="or vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.88/4.13	4[label="foldr (||) False vx3",fontsize=16,color="burlywood",shape="triangle"];16[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
9.88/4.13	16 -> 5[label="",style="solid", color="burlywood", weight=3];
9.88/4.13	17[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
9.88/4.13	17 -> 6[label="",style="solid", color="burlywood", weight=3];
9.88/4.13	5[label="foldr (||) False (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.88/4.13	6[label="foldr (||) False []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.88/4.13	7 -> 9[label="",style="dashed", color="red", weight=0];
9.88/4.13	7[label="(||) vx30 foldr (||) False vx31",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.88/4.13	8[label="False",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.88/4.13	10[label="foldr (||) False vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.88/4.13	9[label="(||) vx30 vx4",fontsize=16,color="burlywood",shape="triangle"];18[label="vx30/False",fontsize=10,color="white",style="solid",shape="box"];9 -> 18[label="",style="solid", color="burlywood", weight=9];
9.88/4.13	18 -> 12[label="",style="solid", color="burlywood", weight=3];
9.88/4.13	19[label="vx30/True",fontsize=10,color="white",style="solid",shape="box"];9 -> 19[label="",style="solid", color="burlywood", weight=9];
9.88/4.13	19 -> 13[label="",style="solid", color="burlywood", weight=3];
9.88/4.13	11[label="vx31",fontsize=16,color="green",shape="box"];12[label="(||) False vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.88/4.13	13[label="(||) True vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.88/4.13	14[label="vx4",fontsize=16,color="green",shape="box"];15[label="True",fontsize=16,color="green",shape="box"];}
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(6)
9.88/4.13	Obligation:
9.88/4.13	Q DP problem:
9.88/4.13	The TRS P consists of the following rules:
9.88/4.13	
9.88/4.13	   new_foldr(:(vx30, vx31)) -> new_foldr(vx31)
9.88/4.13	
9.88/4.13	R is empty.
9.88/4.13	Q is empty.
9.88/4.13	We have to consider all minimal (P,Q,R)-chains.
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(7) QDPSizeChangeProof (EQUIVALENT)
9.88/4.13	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. 
9.88/4.13	
9.88/4.13	From the DPs we obtained the following set of size-change graphs:
9.88/4.13	*new_foldr(:(vx30, vx31)) -> new_foldr(vx31)
9.88/4.13	The graph contains the following edges 1 > 1
9.88/4.13	
9.88/4.13	
9.88/4.13	----------------------------------------
9.88/4.13	
9.88/4.13	(8)
9.88/4.13	YES
10.06/4.18	EOF