7.89/3.53	MAYBE
9.63/4.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.63/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.63/4.02	
9.63/4.02	
9.63/4.02	H-Termination with start terms of the given HASKELL could not be shown:
9.63/4.02	
9.63/4.02	(0) HASKELL
9.63/4.02	(1) BR [EQUIVALENT, 0 ms]
9.63/4.02	(2) HASKELL
9.63/4.02	(3) COR [EQUIVALENT, 0 ms]
9.63/4.02	(4) HASKELL
9.63/4.02	(5) Narrow [SOUND, 0 ms]
9.63/4.02	(6) QDP
9.63/4.02	    (7) NonTerminationLoopProof [COMPLETE, 0 ms]
9.63/4.02	    (8) NO
9.63/4.02	(9) Narrow [COMPLETE, 0 ms]
9.63/4.02	(10) QDP
9.63/4.02	    (11) PisEmptyProof [EQUIVALENT, 0 ms]
9.63/4.02	    (12) YES
9.63/4.02	
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(0)
9.63/4.02	Obligation:
9.63/4.02	mainModule Main 
9.63/4.02	module Main where {
9.63/4.02	import qualified Prelude;
9.63/4.02	data MyBool = MyTrue  | MyFalse ;
9.63/4.02	
9.63/4.02	until :: (a  ->  MyBool)  ->  (a  ->  a)  ->  a  ->  a;
9.63/4.02	until p f x = until0 x p f (p x);
9.63/4.02	
9.63/4.02	until0 x p f MyTrue = x;
9.63/4.02	until0 x p f MyFalse = until p f (f x);
9.63/4.02	
9.63/4.02	}
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(1) BR (EQUIVALENT)
9.63/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(2)
9.63/4.02	Obligation:
9.63/4.02	mainModule Main 
9.63/4.02	module Main where {
9.63/4.02	import qualified Prelude;
9.63/4.02	data MyBool = MyTrue  | MyFalse ;
9.63/4.02	
9.63/4.02	until :: (a  ->  MyBool)  ->  (a  ->  a)  ->  a  ->  a;
9.63/4.02	until p f x = until0 x p f (p x);
9.63/4.02	
9.63/4.02	until0 x p f MyTrue = x;
9.63/4.02	until0 x p f MyFalse = until p f (f x);
9.63/4.02	
9.63/4.02	}
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(3) COR (EQUIVALENT)
9.63/4.02	Cond Reductions:
9.63/4.02	The following Function with conditions
9.63/4.02	"undefined |Falseundefined;
9.63/4.02	"
9.63/4.02	is transformed to
9.63/4.02	"undefined  = undefined1;
9.63/4.02	"
9.63/4.02	"undefined0 True = undefined;
9.63/4.02	"
9.63/4.02	"undefined1  = undefined0 False;
9.63/4.02	"
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(4)
9.63/4.02	Obligation:
9.63/4.02	mainModule Main 
9.63/4.02	module Main where {
9.63/4.02	import qualified Prelude;
9.63/4.02	data MyBool = MyTrue  | MyFalse ;
9.63/4.02	
9.63/4.02	until :: (a  ->  MyBool)  ->  (a  ->  a)  ->  a  ->  a;
9.63/4.02	until p f x = until0 x p f (p x);
9.63/4.02	
9.63/4.02	until0 x p f MyTrue = x;
9.63/4.02	until0 x p f MyFalse = until p f (f x);
9.63/4.02	
9.63/4.02	}
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(5) Narrow (SOUND)
9.63/4.02	Haskell To QDPs
9.63/4.02	
9.63/4.02	digraph dp_graph {
9.63/4.02	node [outthreshold=100, inthreshold=100];1[label="until",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.63/4.02	3[label="until vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.63/4.02	4[label="until vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
9.63/4.02	5[label="until vx3 vx4 vx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
9.63/4.02	6 -> 7[label="",style="dashed", color="red", weight=0];
9.63/4.02	6[label="until0 vx5 vx3 vx4 (vx3 vx5)",fontsize=16,color="magenta"];6 -> 8[label="",style="dashed", color="magenta", weight=3];
9.63/4.02	8[label="vx3 vx5",fontsize=16,color="green",shape="box"];8 -> 12[label="",style="dashed", color="green", weight=3];
9.63/4.02	7[label="until0 vx5 vx3 vx4 vx6",fontsize=16,color="burlywood",shape="triangle"];17[label="vx6/MyTrue",fontsize=10,color="white",style="solid",shape="box"];7 -> 17[label="",style="solid", color="burlywood", weight=9];
9.63/4.02	17 -> 10[label="",style="solid", color="burlywood", weight=3];
9.63/4.02	18[label="vx6/MyFalse",fontsize=10,color="white",style="solid",shape="box"];7 -> 18[label="",style="solid", color="burlywood", weight=9];
9.63/4.02	18 -> 11[label="",style="solid", color="burlywood", weight=3];
9.63/4.02	12[label="vx5",fontsize=16,color="green",shape="box"];10[label="until0 vx5 vx3 vx4 MyTrue",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
9.63/4.02	11[label="until0 vx5 vx3 vx4 MyFalse",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
9.63/4.02	13[label="vx5",fontsize=16,color="green",shape="box"];14 -> 5[label="",style="dashed", color="red", weight=0];
9.63/4.02	14[label="until vx3 vx4 (vx4 vx5)",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
9.63/4.02	15[label="vx4 vx5",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3];
9.63/4.02	16[label="vx5",fontsize=16,color="green",shape="box"];}
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(6)
9.63/4.02	Obligation:
9.63/4.02	Q DP problem:
9.63/4.02	The TRS P consists of the following rules:
9.63/4.02	
9.63/4.02	   new_until(vx3, vx4, h) -> new_until0(vx3, vx4, h)
9.63/4.02	   new_until0(vx3, vx4, h) -> new_until(vx3, vx4, h)
9.63/4.02	
9.63/4.02	R is empty.
9.63/4.02	Q is empty.
9.63/4.02	We have to consider all minimal (P,Q,R)-chains.
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(7) NonTerminationLoopProof (COMPLETE)
9.63/4.02	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
9.63/4.02	Found a loop by narrowing to the left:
9.63/4.02	
9.63/4.02	s = new_until0(vx3', vx4', h') evaluates to  t =new_until0(vx3', vx4', h')
9.63/4.02	
9.63/4.02	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
9.63/4.02	* Matcher: [ ]
9.63/4.02	* Semiunifier: [ ]
9.63/4.02	
9.63/4.02	--------------------------------------------------------------------------------
9.63/4.02	Rewriting sequence
9.63/4.02	
9.63/4.02	new_until0(vx3', vx4', h') -> new_until(vx3', vx4', h')
9.63/4.02	with rule new_until0(vx3'', vx4'', h'') -> new_until(vx3'', vx4'', h'') at position [] and matcher [vx3'' / vx3', vx4'' / vx4', h'' / h']
9.63/4.02	
9.63/4.02	new_until(vx3', vx4', h') -> new_until0(vx3', vx4', h')
9.63/4.02	with rule new_until(vx3, vx4, h) -> new_until0(vx3, vx4, h)
9.63/4.02	
9.63/4.02	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
9.63/4.02	
9.63/4.02	
9.63/4.02	All these steps are and every following step will be a correct step w.r.t to Q.
9.63/4.02	
9.63/4.02	
9.63/4.02	
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(8)
9.63/4.02	NO
9.63/4.02	
9.63/4.02	----------------------------------------
9.63/4.02	
9.63/4.02	(9) Narrow (COMPLETE)
9.63/4.02	Haskell To QDPs
9.63/4.02	
9.63/4.02	digraph dp_graph {
9.63/4.02	node [outthreshold=100, inthreshold=100];1[label="until",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.63/4.02	3[label="until vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.63/4.02	4[label="until vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
9.63/4.02	5[label="until vx3 vx4 vx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
9.63/4.02	6 -> 7[label="",style="dashed", color="red", weight=0];
9.63/4.02	6[label="until0 vx5 vx3 vx4 (vx3 vx5)",fontsize=16,color="magenta"];6 -> 8[label="",style="dashed", color="magenta", weight=3];
9.63/4.03	8[label="vx3 vx5",fontsize=16,color="green",shape="box"];8 -> 12[label="",style="dashed", color="green", weight=3];
9.63/4.03	7[label="until0 vx5 vx3 vx4 vx6",fontsize=16,color="burlywood",shape="triangle"];17[label="vx6/MyTrue",fontsize=10,color="white",style="solid",shape="box"];7 -> 17[label="",style="solid", color="burlywood", weight=9];
9.63/4.03	17 -> 10[label="",style="solid", color="burlywood", weight=3];
9.63/4.03	18[label="vx6/MyFalse",fontsize=10,color="white",style="solid",shape="box"];7 -> 18[label="",style="solid", color="burlywood", weight=9];
9.63/4.03	18 -> 11[label="",style="solid", color="burlywood", weight=3];
9.63/4.03	12[label="vx5",fontsize=16,color="green",shape="box"];10[label="until0 vx5 vx3 vx4 MyTrue",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
9.63/4.03	11[label="until0 vx5 vx3 vx4 MyFalse",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
9.63/4.03	13[label="vx5",fontsize=16,color="green",shape="box"];14 -> 5[label="",style="dashed", color="red", weight=0];
9.63/4.03	14[label="until vx3 vx4 (vx4 vx5)",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
9.63/4.03	15[label="vx4 vx5",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3];
9.63/4.03	16[label="vx5",fontsize=16,color="green",shape="box"];}
9.63/4.03	
9.63/4.03	----------------------------------------
9.63/4.03	
9.63/4.03	(10)
9.63/4.03	Obligation:
9.63/4.03	Q DP problem:
9.63/4.03	P is empty.
9.63/4.03	R is empty.
9.63/4.03	Q is empty.
9.63/4.03	We have to consider all (P,Q,R)-chains.
9.63/4.03	----------------------------------------
9.63/4.03	
9.63/4.03	(11) PisEmptyProof (EQUIVALENT)
9.63/4.03	The TRS P is empty. Hence, there is no (P,Q,R) chain.
9.63/4.03	----------------------------------------
9.63/4.03	
9.63/4.03	(12)
9.63/4.03	YES
9.63/4.07	EOF