7.74/3.53	YES
9.36/3.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.36/3.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.36/3.99	
9.36/3.99	
9.36/3.99	H-Termination with start terms of the given HASKELL could be proven:
9.36/3.99	
9.36/3.99	(0) HASKELL
9.36/3.99	(1) BR [EQUIVALENT, 0 ms]
9.36/3.99	(2) HASKELL
9.36/3.99	(3) COR [EQUIVALENT, 0 ms]
9.36/3.99	(4) HASKELL
9.36/3.99	(5) Narrow [SOUND, 0 ms]
9.36/3.99	(6) QDP
9.36/3.99	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.36/3.99	(8) YES
9.36/3.99	
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(0)
9.36/3.99	Obligation:
9.36/3.99	mainModule Main 
9.36/3.99	module Main where {
9.36/3.99	import qualified Prelude;
9.36/3.99	}
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(1) BR (EQUIVALENT)
9.36/3.99	Replaced joker patterns by fresh variables and removed binding patterns.
9.36/3.99	
9.36/3.99	Binding Reductions:
9.36/3.99	The bind variable of the following binding Pattern
9.36/3.99	"xs@(vw : vx)"
9.36/3.99	is replaced by the following term
9.36/3.99	"vw : vx"
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(2)
9.36/3.99	Obligation:
9.36/3.99	mainModule Main 
9.36/3.99	module Main where {
9.36/3.99	import qualified Prelude;
9.36/3.99	}
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(3) COR (EQUIVALENT)
9.36/3.99	Cond Reductions:
9.36/3.99	The following Function with conditions
9.36/3.99	"dropWhile p [] = [];
9.36/3.99	dropWhile p (vw : vx)|p vwdropWhile p vx|otherwisevw : vx;
9.36/3.99	"
9.36/3.99	is transformed to
9.36/3.99	"dropWhile p [] = dropWhile3 p [];
9.36/3.99	dropWhile p (vw : vx) = dropWhile2 p (vw : vx);
9.36/3.99	"
9.36/3.99	"dropWhile0 p vw vx True = vw : vx;
9.36/3.99	"
9.36/3.99	"dropWhile1 p vw vx True = dropWhile p vx;
9.36/3.99	dropWhile1 p vw vx False = dropWhile0 p vw vx otherwise;
9.36/3.99	"
9.36/3.99	"dropWhile2 p (vw : vx) = dropWhile1 p vw vx (p vw);
9.36/3.99	"
9.36/3.99	"dropWhile3 p [] = [];
9.36/3.99	dropWhile3 wv ww = dropWhile2 wv ww;
9.36/3.99	"
9.36/3.99	The following Function with conditions
9.36/3.99	"undefined |Falseundefined;
9.36/3.99	"
9.36/3.99	is transformed to
9.36/3.99	"undefined  = undefined1;
9.36/3.99	"
9.36/3.99	"undefined0 True = undefined;
9.36/3.99	"
9.36/3.99	"undefined1  = undefined0 False;
9.36/3.99	"
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(4)
9.36/3.99	Obligation:
9.36/3.99	mainModule Main 
9.36/3.99	module Main where {
9.36/3.99	import qualified Prelude;
9.36/3.99	}
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(5) Narrow (SOUND)
9.36/3.99	Haskell To QDPs
9.36/3.99	
9.36/3.99	digraph dp_graph {
9.36/3.99	node [outthreshold=100, inthreshold=100];1[label="dropWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.36/3.99	3[label="dropWhile wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.36/3.99	4[label="dropWhile wx3 wx4",fontsize=16,color="burlywood",shape="triangle"];22[label="wx4/wx40 : wx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 22[label="",style="solid", color="burlywood", weight=9];
9.36/3.99	22 -> 5[label="",style="solid", color="burlywood", weight=3];
9.36/3.99	23[label="wx4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
9.36/3.99	23 -> 6[label="",style="solid", color="burlywood", weight=3];
9.36/3.99	5[label="dropWhile wx3 (wx40 : wx41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.36/3.99	6[label="dropWhile wx3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.36/3.99	7[label="dropWhile2 wx3 (wx40 : wx41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.36/3.99	8[label="dropWhile3 wx3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.36/3.99	9 -> 11[label="",style="dashed", color="red", weight=0];
9.36/3.99	9[label="dropWhile1 wx3 wx40 wx41 (wx3 wx40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9.36/3.99	10[label="[]",fontsize=16,color="green",shape="box"];12[label="wx3 wx40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
9.36/3.99	11[label="dropWhile1 wx3 wx40 wx41 wx5",fontsize=16,color="burlywood",shape="triangle"];24[label="wx5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 24[label="",style="solid", color="burlywood", weight=9];
9.36/3.99	24 -> 14[label="",style="solid", color="burlywood", weight=3];
9.36/3.99	25[label="wx5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
9.36/3.99	25 -> 15[label="",style="solid", color="burlywood", weight=3];
9.36/3.99	16[label="wx40",fontsize=16,color="green",shape="box"];14[label="dropWhile1 wx3 wx40 wx41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.36/3.99	15[label="dropWhile1 wx3 wx40 wx41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.36/3.99	17[label="dropWhile0 wx3 wx40 wx41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
9.36/3.99	18 -> 4[label="",style="dashed", color="red", weight=0];
9.36/3.99	18[label="dropWhile wx3 wx41",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
9.36/3.99	19[label="dropWhile0 wx3 wx40 wx41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
9.36/3.99	20[label="wx41",fontsize=16,color="green",shape="box"];21[label="wx40 : wx41",fontsize=16,color="green",shape="box"];}
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(6)
9.36/3.99	Obligation:
9.36/3.99	Q DP problem:
9.36/3.99	The TRS P consists of the following rules:
9.36/3.99	
9.36/3.99	   new_dropWhile1(wx3, wx40, wx41, h) -> new_dropWhile(wx3, wx41, h)
9.36/3.99	   new_dropWhile(wx3, :(wx40, wx41), h) -> new_dropWhile1(wx3, wx40, wx41, h)
9.36/3.99	
9.36/3.99	R is empty.
9.36/3.99	Q is empty.
9.36/3.99	We have to consider all minimal (P,Q,R)-chains.
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(7) QDPSizeChangeProof (EQUIVALENT)
9.36/3.99	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. 
9.36/3.99	
9.36/3.99	From the DPs we obtained the following set of size-change graphs:
9.36/3.99	*new_dropWhile(wx3, :(wx40, wx41), h) -> new_dropWhile1(wx3, wx40, wx41, h)
9.36/3.99	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
9.36/3.99	
9.36/3.99	
9.36/3.99	*new_dropWhile1(wx3, wx40, wx41, h) -> new_dropWhile(wx3, wx41, h)
9.36/3.99	The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
9.36/3.99	
9.36/3.99	
9.36/3.99	----------------------------------------
9.36/3.99	
9.36/3.99	(8)
9.36/3.99	YES
9.62/4.05	EOF