7.66/3.62	YES
8.94/4.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
8.94/4.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.94/4.04	
8.94/4.04	
8.94/4.04	H-Termination with start terms of the given HASKELL could be proven:
8.94/4.04	
8.94/4.04	(0) HASKELL
8.94/4.04	(1) BR [EQUIVALENT, 0 ms]
8.94/4.04	(2) HASKELL
8.94/4.04	(3) COR [EQUIVALENT, 0 ms]
8.94/4.04	(4) HASKELL
8.94/4.04	(5) Narrow [SOUND, 0 ms]
8.94/4.04	(6) AND
8.94/4.04	    (7) QDP
8.94/4.04	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.94/4.04	        (9) YES
8.94/4.04	    (10) QDP
8.94/4.04	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.94/4.04	        (12) YES
8.94/4.04	
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(0)
8.94/4.04	Obligation:
8.94/4.04	mainModule Main 
8.94/4.04	module Main where {
8.94/4.04	import qualified Prelude;
8.94/4.04	}
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(1) BR (EQUIVALENT)
8.94/4.04	Replaced joker patterns by fresh variables and removed binding patterns.
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(2)
8.94/4.04	Obligation:
8.94/4.04	mainModule Main 
8.94/4.04	module Main where {
8.94/4.04	import qualified Prelude;
8.94/4.04	}
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(3) COR (EQUIVALENT)
8.94/4.04	Cond Reductions:
8.94/4.04	The following Function with conditions
8.94/4.04	"undefined |Falseundefined;
8.94/4.04	"
8.94/4.04	is transformed to
8.94/4.04	"undefined  = undefined1;
8.94/4.04	"
8.94/4.04	"undefined0 True = undefined;
8.94/4.04	"
8.94/4.04	"undefined1  = undefined0 False;
8.94/4.04	"
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(4)
8.94/4.04	Obligation:
8.94/4.04	mainModule Main 
8.94/4.04	module Main where {
8.94/4.04	import qualified Prelude;
8.94/4.04	}
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(5) Narrow (SOUND)
8.94/4.04	Haskell To QDPs
8.94/4.04	
8.94/4.04	digraph dp_graph {
8.94/4.04	node [outthreshold=100, inthreshold=100];1[label="concat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
8.94/4.04	3[label="concat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
8.94/4.04	4[label="foldr (++) [] vx3",fontsize=16,color="burlywood",shape="triangle"];18[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
8.94/4.04	18 -> 5[label="",style="solid", color="burlywood", weight=3];
8.94/4.04	19[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
8.94/4.04	19 -> 6[label="",style="solid", color="burlywood", weight=3];
8.94/4.04	5[label="foldr (++) [] (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
8.94/4.04	6[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
8.94/4.04	7 -> 9[label="",style="dashed", color="red", weight=0];
8.94/4.04	7[label="(++) vx30 foldr (++) [] vx31",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8.94/4.04	8[label="[]",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
8.94/4.04	10[label="foldr (++) [] vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
8.94/4.04	9[label="(++) vx30 vx4",fontsize=16,color="burlywood",shape="triangle"];20[label="vx30/vx300 : vx301",fontsize=10,color="white",style="solid",shape="box"];9 -> 20[label="",style="solid", color="burlywood", weight=9];
8.94/4.04	20 -> 12[label="",style="solid", color="burlywood", weight=3];
8.94/4.04	21[label="vx30/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 21[label="",style="solid", color="burlywood", weight=9];
8.94/4.04	21 -> 13[label="",style="solid", color="burlywood", weight=3];
8.94/4.04	11[label="vx31",fontsize=16,color="green",shape="box"];12[label="(++) (vx300 : vx301) vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
8.94/4.04	13[label="(++) [] vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
8.94/4.04	14[label="vx300 : vx301 ++ vx4",fontsize=16,color="green",shape="box"];14 -> 16[label="",style="dashed", color="green", weight=3];
8.94/4.04	15[label="vx4",fontsize=16,color="green",shape="box"];16 -> 9[label="",style="dashed", color="red", weight=0];
8.94/4.04	16[label="vx301 ++ vx4",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
8.94/4.04	17[label="vx301",fontsize=16,color="green",shape="box"];}
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(6)
8.94/4.04	Complex Obligation (AND)
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(7)
8.94/4.04	Obligation:
8.94/4.04	Q DP problem:
8.94/4.04	The TRS P consists of the following rules:
8.94/4.04	
8.94/4.04	   new_psPs(:(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)
8.94/4.04	
8.94/4.04	R is empty.
8.94/4.04	Q is empty.
8.94/4.04	We have to consider all minimal (P,Q,R)-chains.
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(8) QDPSizeChangeProof (EQUIVALENT)
8.94/4.04	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. 
8.94/4.04	
8.94/4.04	From the DPs we obtained the following set of size-change graphs:
8.94/4.04	*new_psPs(:(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)
8.94/4.04	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
8.94/4.04	
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(9)
8.94/4.04	YES
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(10)
8.94/4.04	Obligation:
8.94/4.04	Q DP problem:
8.94/4.04	The TRS P consists of the following rules:
8.94/4.04	
8.94/4.04	   new_foldr(:(vx30, vx31), h) -> new_foldr(vx31, h)
8.94/4.04	
8.94/4.04	R is empty.
8.94/4.04	Q is empty.
8.94/4.04	We have to consider all minimal (P,Q,R)-chains.
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(11) QDPSizeChangeProof (EQUIVALENT)
8.94/4.04	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. 
8.94/4.04	
8.94/4.04	From the DPs we obtained the following set of size-change graphs:
8.94/4.04	*new_foldr(:(vx30, vx31), h) -> new_foldr(vx31, h)
8.94/4.04	The graph contains the following edges 1 > 1, 2 >= 2
8.94/4.04	
8.94/4.04	
8.94/4.04	----------------------------------------
8.94/4.04	
8.94/4.04	(12)
8.94/4.04	YES
8.94/4.09	EOF