8.11/3.55	YES
9.86/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.86/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.86/4.03	
9.86/4.03	
9.86/4.03	H-Termination with start terms of the given HASKELL could be proven:
9.86/4.03	
9.86/4.03	(0) HASKELL
9.86/4.03	(1) BR [EQUIVALENT, 0 ms]
9.86/4.03	(2) HASKELL
9.86/4.03	(3) COR [EQUIVALENT, 0 ms]
9.86/4.03	(4) HASKELL
9.86/4.03	(5) Narrow [SOUND, 0 ms]
9.86/4.03	(6) QDP
9.86/4.03	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.86/4.03	(8) YES
9.86/4.03	
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(0)
9.86/4.03	Obligation:
9.86/4.03	mainModule Main 
9.86/4.03	module Main where {
9.86/4.03	import qualified Prelude;
9.86/4.03	data List a = Cons a (List a)  | Nil ;
9.86/4.03	
9.86/4.03	data MyBool = MyTrue  | MyFalse ;
9.86/4.03	
9.86/4.03	dropWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.86/4.03	dropWhile p Nil = dropWhile3 p Nil;
9.86/4.03	dropWhile p (Cons vv vw) = dropWhile2 p (Cons vv vw);
9.86/4.03	
9.86/4.03	dropWhile0 p vv vw MyTrue = Cons vv vw;
9.86/4.03	
9.86/4.03	dropWhile1 p vv vw MyTrue = dropWhile p vw;
9.86/4.03	dropWhile1 p vv vw MyFalse = dropWhile0 p vv vw otherwise;
9.86/4.03	
9.86/4.03	dropWhile2 p (Cons vv vw) = dropWhile1 p vv vw (p vv);
9.86/4.03	
9.86/4.03	dropWhile3 p Nil = Nil;
9.86/4.03	dropWhile3 vz wu = dropWhile2 vz wu;
9.86/4.03	
9.86/4.03	otherwise :: MyBool;
9.86/4.03	otherwise = MyTrue;
9.86/4.03	
9.86/4.03	}
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(1) BR (EQUIVALENT)
9.86/4.03	Replaced joker patterns by fresh variables and removed binding patterns.
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(2)
9.86/4.03	Obligation:
9.86/4.03	mainModule Main 
9.86/4.03	module Main where {
9.86/4.03	import qualified Prelude;
9.86/4.03	data List a = Cons a (List a)  | Nil ;
9.86/4.03	
9.86/4.03	data MyBool = MyTrue  | MyFalse ;
9.86/4.03	
9.86/4.03	dropWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.86/4.03	dropWhile p Nil = dropWhile3 p Nil;
9.86/4.03	dropWhile p (Cons vv vw) = dropWhile2 p (Cons vv vw);
9.86/4.03	
9.86/4.03	dropWhile0 p vv vw MyTrue = Cons vv vw;
9.86/4.03	
9.86/4.03	dropWhile1 p vv vw MyTrue = dropWhile p vw;
9.86/4.03	dropWhile1 p vv vw MyFalse = dropWhile0 p vv vw otherwise;
9.86/4.03	
9.86/4.03	dropWhile2 p (Cons vv vw) = dropWhile1 p vv vw (p vv);
9.86/4.03	
9.86/4.03	dropWhile3 p Nil = Nil;
9.86/4.03	dropWhile3 vz wu = dropWhile2 vz wu;
9.86/4.03	
9.86/4.03	otherwise :: MyBool;
9.86/4.03	otherwise = MyTrue;
9.86/4.03	
9.86/4.03	}
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(3) COR (EQUIVALENT)
9.86/4.03	Cond Reductions:
9.86/4.03	The following Function with conditions
9.86/4.03	"undefined |Falseundefined;
9.86/4.03	"
9.86/4.03	is transformed to
9.86/4.03	"undefined  = undefined1;
9.86/4.03	"
9.86/4.03	"undefined0 True = undefined;
9.86/4.03	"
9.86/4.03	"undefined1  = undefined0 False;
9.86/4.03	"
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(4)
9.86/4.03	Obligation:
9.86/4.03	mainModule Main 
9.86/4.03	module Main where {
9.86/4.03	import qualified Prelude;
9.86/4.03	data List a = Cons a (List a)  | Nil ;
9.86/4.03	
9.86/4.03	data MyBool = MyTrue  | MyFalse ;
9.86/4.03	
9.86/4.03	dropWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.86/4.03	dropWhile p Nil = dropWhile3 p Nil;
9.86/4.03	dropWhile p (Cons vv vw) = dropWhile2 p (Cons vv vw);
9.86/4.03	
9.86/4.03	dropWhile0 p vv vw MyTrue = Cons vv vw;
9.86/4.03	
9.86/4.03	dropWhile1 p vv vw MyTrue = dropWhile p vw;
9.86/4.03	dropWhile1 p vv vw MyFalse = dropWhile0 p vv vw otherwise;
9.86/4.03	
9.86/4.03	dropWhile2 p (Cons vv vw) = dropWhile1 p vv vw (p vv);
9.86/4.03	
9.86/4.03	dropWhile3 p Nil = Nil;
9.86/4.03	dropWhile3 vz wu = dropWhile2 vz wu;
9.86/4.03	
9.86/4.03	otherwise :: MyBool;
9.86/4.03	otherwise = MyTrue;
9.86/4.03	
9.86/4.03	}
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(5) Narrow (SOUND)
9.86/4.03	Haskell To QDPs
9.86/4.03	
9.86/4.03	digraph dp_graph {
9.86/4.03	node [outthreshold=100, inthreshold=100];1[label="dropWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.86/4.03	3[label="dropWhile wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.86/4.03	4[label="dropWhile wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];22[label="wv4/Cons wv40 wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 22[label="",style="solid", color="burlywood", weight=9];
9.86/4.03	22 -> 5[label="",style="solid", color="burlywood", weight=3];
9.86/4.03	23[label="wv4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
9.86/4.03	23 -> 6[label="",style="solid", color="burlywood", weight=3];
9.86/4.03	5[label="dropWhile wv3 (Cons wv40 wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.86/4.03	6[label="dropWhile wv3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.86/4.03	7[label="dropWhile2 wv3 (Cons wv40 wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.86/4.03	8[label="dropWhile3 wv3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.86/4.03	9 -> 11[label="",style="dashed", color="red", weight=0];
9.86/4.03	9[label="dropWhile1 wv3 wv40 wv41 (wv3 wv40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9.86/4.03	10[label="Nil",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];
9.86/4.03	11[label="dropWhile1 wv3 wv40 wv41 wv5",fontsize=16,color="burlywood",shape="triangle"];24[label="wv5/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 24[label="",style="solid", color="burlywood", weight=9];
9.86/4.03	24 -> 14[label="",style="solid", color="burlywood", weight=3];
9.86/4.03	25[label="wv5/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
9.86/4.03	25 -> 15[label="",style="solid", color="burlywood", weight=3];
9.86/4.03	16[label="wv40",fontsize=16,color="green",shape="box"];14[label="dropWhile1 wv3 wv40 wv41 MyTrue",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.86/4.03	15[label="dropWhile1 wv3 wv40 wv41 MyFalse",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.86/4.03	17 -> 4[label="",style="dashed", color="red", weight=0];
9.86/4.03	17[label="dropWhile wv3 wv41",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
9.86/4.03	18[label="dropWhile0 wv3 wv40 wv41 otherwise",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
9.86/4.03	19[label="wv41",fontsize=16,color="green",shape="box"];20[label="dropWhile0 wv3 wv40 wv41 MyTrue",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.86/4.03	21[label="Cons wv40 wv41",fontsize=16,color="green",shape="box"];}
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(6)
9.86/4.03	Obligation:
9.86/4.03	Q DP problem:
9.86/4.03	The TRS P consists of the following rules:
9.86/4.03	
9.86/4.03	   new_dropWhile1(wv3, wv40, wv41, h) -> new_dropWhile(wv3, wv41, h)
9.86/4.03	   new_dropWhile(wv3, Cons(wv40, wv41), h) -> new_dropWhile1(wv3, wv40, wv41, h)
9.86/4.03	
9.86/4.03	R is empty.
9.86/4.03	Q is empty.
9.86/4.03	We have to consider all minimal (P,Q,R)-chains.
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(7) QDPSizeChangeProof (EQUIVALENT)
9.86/4.03	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. 
9.86/4.03	
9.86/4.03	From the DPs we obtained the following set of size-change graphs:
9.86/4.03	*new_dropWhile(wv3, Cons(wv40, wv41), h) -> new_dropWhile1(wv3, wv40, wv41, h)
9.86/4.03	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
9.86/4.03	
9.86/4.03	
9.86/4.03	*new_dropWhile1(wv3, wv40, wv41, h) -> new_dropWhile(wv3, wv41, h)
9.86/4.03	The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
9.86/4.03	
9.86/4.03	
9.86/4.03	----------------------------------------
9.86/4.03	
9.86/4.03	(8)
9.86/4.03	YES
9.91/4.06	EOF