7.77/3.54	YES
9.38/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.38/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.38/3.98	
9.38/3.98	
9.38/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.38/3.98	
9.38/3.98	(0) HASKELL
9.38/3.98	(1) BR [EQUIVALENT, 0 ms]
9.38/3.98	(2) HASKELL
9.38/3.98	(3) COR [EQUIVALENT, 0 ms]
9.38/3.98	(4) HASKELL
9.38/3.98	(5) Narrow [SOUND, 0 ms]
9.38/3.98	(6) QDP
9.38/3.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.38/3.98	(8) YES
9.38/3.98	
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(0)
9.38/3.98	Obligation:
9.38/3.98	mainModule Main 
9.38/3.98	module Main where {
9.38/3.98	import qualified Prelude;
9.38/3.98	}
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(1) BR (EQUIVALENT)
9.38/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(2)
9.38/3.98	Obligation:
9.38/3.98	mainModule Main 
9.38/3.98	module Main where {
9.38/3.98	import qualified Prelude;
9.38/3.98	}
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(3) COR (EQUIVALENT)
9.38/3.98	Cond Reductions:
9.38/3.98	The following Function with conditions
9.38/3.98	"takeWhile p [] = [];
9.38/3.98	takeWhile p (x : xs)|p xx : takeWhile p xs|otherwise[];
9.38/3.98	"
9.38/3.98	is transformed to
9.38/3.98	"takeWhile p [] = takeWhile3 p [];
9.38/3.98	takeWhile p (x : xs) = takeWhile2 p (x : xs);
9.38/3.98	"
9.38/3.98	"takeWhile0 p x xs True = [];
9.38/3.98	"
9.38/3.98	"takeWhile1 p x xs True = x : takeWhile p xs;
9.38/3.98	takeWhile1 p x xs False = takeWhile0 p x xs otherwise;
9.38/3.98	"
9.38/3.98	"takeWhile2 p (x : xs) = takeWhile1 p x xs (p x);
9.38/3.98	"
9.38/3.98	"takeWhile3 p [] = [];
9.38/3.98	takeWhile3 vz wu = takeWhile2 vz wu;
9.38/3.98	"
9.38/3.98	The following Function with conditions
9.38/3.98	"undefined |Falseundefined;
9.38/3.98	"
9.38/3.98	is transformed to
9.38/3.98	"undefined  = undefined1;
9.38/3.98	"
9.38/3.98	"undefined0 True = undefined;
9.38/3.98	"
9.38/3.98	"undefined1  = undefined0 False;
9.38/3.98	"
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(4)
9.38/3.98	Obligation:
9.38/3.98	mainModule Main 
9.38/3.98	module Main where {
9.38/3.98	import qualified Prelude;
9.38/3.98	}
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(5) Narrow (SOUND)
9.38/3.98	Haskell To QDPs
9.38/3.98	
9.38/3.98	digraph dp_graph {
9.38/3.98	node [outthreshold=100, inthreshold=100];1[label="takeWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.38/3.98	3[label="takeWhile wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.38/3.98	4[label="takeWhile wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];23[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
9.38/3.98	23 -> 5[label="",style="solid", color="burlywood", weight=3];
9.38/3.98	24[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
9.38/3.98	24 -> 6[label="",style="solid", color="burlywood", weight=3];
9.38/3.98	5[label="takeWhile wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.38/3.98	6[label="takeWhile wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.38/3.98	7[label="takeWhile2 wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.38/3.98	8[label="takeWhile3 wv3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.38/3.98	9 -> 11[label="",style="dashed", color="red", weight=0];
9.38/3.98	9[label="takeWhile1 wv3 wv40 wv41 (wv3 wv40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9.38/3.98	10[label="[]",fontsize=16,color="green",shape="box"];12[label="wv3 wv40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
9.38/3.98	11[label="takeWhile1 wv3 wv40 wv41 wv5",fontsize=16,color="burlywood",shape="triangle"];25[label="wv5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
9.38/3.98	25 -> 14[label="",style="solid", color="burlywood", weight=3];
9.38/3.98	26[label="wv5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
9.38/3.98	26 -> 15[label="",style="solid", color="burlywood", weight=3];
9.38/3.98	16[label="wv40",fontsize=16,color="green",shape="box"];14[label="takeWhile1 wv3 wv40 wv41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.38/3.98	15[label="takeWhile1 wv3 wv40 wv41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.38/3.98	17[label="takeWhile0 wv3 wv40 wv41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
9.38/3.98	18[label="wv40 : takeWhile wv3 wv41",fontsize=16,color="green",shape="box"];18 -> 20[label="",style="dashed", color="green", weight=3];
9.38/3.98	19[label="takeWhile0 wv3 wv40 wv41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
9.38/3.98	20 -> 4[label="",style="dashed", color="red", weight=0];
9.38/3.98	20[label="takeWhile wv3 wv41",fontsize=16,color="magenta"];20 -> 22[label="",style="dashed", color="magenta", weight=3];
9.38/3.98	21[label="[]",fontsize=16,color="green",shape="box"];22[label="wv41",fontsize=16,color="green",shape="box"];}
9.38/3.98	
9.38/3.98	----------------------------------------
9.38/3.98	
9.38/3.98	(6)
9.38/3.98	Obligation:
9.38/3.98	Q DP problem:
9.38/3.98	The TRS P consists of the following rules:
9.38/3.98	
9.38/3.98	   new_takeWhile1(wv3, wv40, wv41, h) -> new_takeWhile(wv3, wv41, h)
9.38/3.98	   new_takeWhile(wv3, :(wv40, wv41), h) -> new_takeWhile1(wv3, wv40, wv41, h)
9.38/3.99	
9.38/3.99	R is empty.
9.38/3.99	Q is empty.
9.38/3.99	We have to consider all minimal (P,Q,R)-chains.
9.38/3.99	----------------------------------------
9.38/3.99	
9.38/3.99	(7) QDPSizeChangeProof (EQUIVALENT)
9.38/3.99	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. 
9.38/3.99	
9.38/3.99	From the DPs we obtained the following set of size-change graphs:
9.38/3.99	*new_takeWhile(wv3, :(wv40, wv41), h) -> new_takeWhile1(wv3, wv40, wv41, h)
9.38/3.99	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
9.38/3.99	
9.38/3.99	
9.38/3.99	*new_takeWhile1(wv3, wv40, wv41, h) -> new_takeWhile(wv3, wv41, h)
9.38/3.99	The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
9.38/3.99	
9.38/3.99	
9.38/3.99	----------------------------------------
9.38/3.99	
9.38/3.99	(8)
9.38/3.99	YES
9.38/4.03	EOF