7.89/3.57	YES
9.55/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.55/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.55/4.06	
9.55/4.06	
9.55/4.06	H-Termination with start terms of the given HASKELL could be proven:
9.55/4.06	
9.55/4.06	(0) HASKELL
9.55/4.06	(1) BR [EQUIVALENT, 0 ms]
9.55/4.06	(2) HASKELL
9.55/4.06	(3) COR [EQUIVALENT, 0 ms]
9.55/4.06	(4) HASKELL
9.55/4.06	(5) Narrow [SOUND, 0 ms]
9.55/4.06	(6) QDP
9.55/4.06	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.55/4.06	(8) YES
9.55/4.06	
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(0)
9.55/4.06	Obligation:
9.55/4.06	mainModule Main 
9.55/4.06	module Main where {
9.55/4.06	import qualified Prelude;
9.55/4.06	data List a = Cons a (List a)  | Nil ;
9.55/4.06	
9.55/4.06	data MyBool = MyTrue  | MyFalse ;
9.55/4.06	
9.55/4.06	otherwise :: MyBool;
9.55/4.06	otherwise = MyTrue;
9.55/4.06	
9.55/4.06	takeWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.55/4.06	takeWhile p Nil = takeWhile3 p Nil;
9.55/4.06	takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs);
9.55/4.06	
9.55/4.06	takeWhile0 p x xs MyTrue = Nil;
9.55/4.06	
9.55/4.06	takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs);
9.55/4.06	takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise;
9.55/4.06	
9.55/4.06	takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x);
9.55/4.06	
9.55/4.06	takeWhile3 p Nil = Nil;
9.55/4.06	takeWhile3 vx vy = takeWhile2 vx vy;
9.55/4.06	
9.55/4.06	}
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(1) BR (EQUIVALENT)
9.55/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(2)
9.55/4.06	Obligation:
9.55/4.06	mainModule Main 
9.55/4.06	module Main where {
9.55/4.06	import qualified Prelude;
9.55/4.06	data List a = Cons a (List a)  | Nil ;
9.55/4.06	
9.55/4.06	data MyBool = MyTrue  | MyFalse ;
9.55/4.06	
9.55/4.06	otherwise :: MyBool;
9.55/4.06	otherwise = MyTrue;
9.55/4.06	
9.55/4.06	takeWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.55/4.06	takeWhile p Nil = takeWhile3 p Nil;
9.55/4.06	takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs);
9.55/4.06	
9.55/4.06	takeWhile0 p x xs MyTrue = Nil;
9.55/4.06	
9.55/4.06	takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs);
9.55/4.06	takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise;
9.55/4.06	
9.55/4.06	takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x);
9.55/4.06	
9.55/4.06	takeWhile3 p Nil = Nil;
9.55/4.06	takeWhile3 vx vy = takeWhile2 vx vy;
9.55/4.06	
9.55/4.06	}
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(3) COR (EQUIVALENT)
9.55/4.06	Cond Reductions:
9.55/4.06	The following Function with conditions
9.55/4.06	"undefined |Falseundefined;
9.55/4.06	"
9.55/4.06	is transformed to
9.55/4.06	"undefined  = undefined1;
9.55/4.06	"
9.55/4.06	"undefined0 True = undefined;
9.55/4.06	"
9.55/4.06	"undefined1  = undefined0 False;
9.55/4.06	"
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(4)
9.55/4.06	Obligation:
9.55/4.06	mainModule Main 
9.55/4.06	module Main where {
9.55/4.06	import qualified Prelude;
9.55/4.06	data List a = Cons a (List a)  | Nil ;
9.55/4.06	
9.55/4.06	data MyBool = MyTrue  | MyFalse ;
9.55/4.06	
9.55/4.06	otherwise :: MyBool;
9.55/4.06	otherwise = MyTrue;
9.55/4.06	
9.55/4.06	takeWhile :: (a  ->  MyBool)  ->  List a  ->  List a;
9.55/4.06	takeWhile p Nil = takeWhile3 p Nil;
9.55/4.06	takeWhile p (Cons x xs) = takeWhile2 p (Cons x xs);
9.55/4.06	
9.55/4.06	takeWhile0 p x xs MyTrue = Nil;
9.55/4.06	
9.55/4.06	takeWhile1 p x xs MyTrue = Cons x (takeWhile p xs);
9.55/4.06	takeWhile1 p x xs MyFalse = takeWhile0 p x xs otherwise;
9.55/4.06	
9.55/4.06	takeWhile2 p (Cons x xs) = takeWhile1 p x xs (p x);
9.55/4.06	
9.55/4.06	takeWhile3 p Nil = Nil;
9.55/4.06	takeWhile3 vx vy = takeWhile2 vx vy;
9.55/4.06	
9.55/4.06	}
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(5) Narrow (SOUND)
9.55/4.06	Haskell To QDPs
9.55/4.06	
9.55/4.06	digraph dp_graph {
9.55/4.06	node [outthreshold=100, inthreshold=100];1[label="takeWhile",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.55/4.06	3[label="takeWhile vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.55/4.06	4[label="takeWhile vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];23[label="vz4/Cons vz40 vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
9.55/4.06	23 -> 5[label="",style="solid", color="burlywood", weight=3];
9.55/4.06	24[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
9.55/4.06	24 -> 6[label="",style="solid", color="burlywood", weight=3];
9.55/4.06	5[label="takeWhile vz3 (Cons vz40 vz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.55/4.06	6[label="takeWhile vz3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.55/4.06	7[label="takeWhile2 vz3 (Cons vz40 vz41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.55/4.06	8[label="takeWhile3 vz3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.55/4.06	9 -> 11[label="",style="dashed", color="red", weight=0];
9.55/4.06	9[label="takeWhile1 vz3 vz40 vz41 (vz3 vz40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9.55/4.06	10[label="Nil",fontsize=16,color="green",shape="box"];12[label="vz3 vz40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
9.55/4.06	11[label="takeWhile1 vz3 vz40 vz41 vz5",fontsize=16,color="burlywood",shape="triangle"];25[label="vz5/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 25[label="",style="solid", color="burlywood", weight=9];
9.55/4.06	25 -> 14[label="",style="solid", color="burlywood", weight=3];
9.55/4.06	26[label="vz5/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
9.55/4.06	26 -> 15[label="",style="solid", color="burlywood", weight=3];
9.55/4.06	16[label="vz40",fontsize=16,color="green",shape="box"];14[label="takeWhile1 vz3 vz40 vz41 MyTrue",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.55/4.06	15[label="takeWhile1 vz3 vz40 vz41 MyFalse",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.55/4.06	17[label="Cons vz40 (takeWhile vz3 vz41)",fontsize=16,color="green",shape="box"];17 -> 19[label="",style="dashed", color="green", weight=3];
9.55/4.06	18[label="takeWhile0 vz3 vz40 vz41 otherwise",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
9.55/4.06	19 -> 4[label="",style="dashed", color="red", weight=0];
9.55/4.06	19[label="takeWhile vz3 vz41",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
9.55/4.06	20[label="takeWhile0 vz3 vz40 vz41 MyTrue",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
9.55/4.06	21[label="vz41",fontsize=16,color="green",shape="box"];22[label="Nil",fontsize=16,color="green",shape="box"];}
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(6)
9.55/4.06	Obligation:
9.55/4.06	Q DP problem:
9.55/4.06	The TRS P consists of the following rules:
9.55/4.06	
9.55/4.06	   new_takeWhile1(vz3, vz40, vz41, h) -> new_takeWhile(vz3, vz41, h)
9.55/4.06	   new_takeWhile(vz3, Cons(vz40, vz41), h) -> new_takeWhile1(vz3, vz40, vz41, h)
9.55/4.06	
9.55/4.06	R is empty.
9.55/4.06	Q is empty.
9.55/4.06	We have to consider all minimal (P,Q,R)-chains.
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(7) QDPSizeChangeProof (EQUIVALENT)
9.55/4.06	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. 
9.55/4.06	
9.55/4.06	From the DPs we obtained the following set of size-change graphs:
9.55/4.06	*new_takeWhile(vz3, Cons(vz40, vz41), h) -> new_takeWhile1(vz3, vz40, vz41, h)
9.55/4.06	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
9.55/4.06	
9.55/4.06	
9.55/4.06	*new_takeWhile1(vz3, vz40, vz41, h) -> new_takeWhile(vz3, vz41, h)
9.55/4.06	The graph contains the following edges 1 >= 1, 3 >= 2, 4 >= 3
9.55/4.06	
9.55/4.06	
9.55/4.06	----------------------------------------
9.55/4.06	
9.55/4.06	(8)
9.55/4.06	YES
9.92/4.20	EOF