8.02/3.73	MAYBE
9.75/4.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.75/4.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.75/4.21	
9.75/4.21	
9.75/4.21	H-Termination with start terms of the given HASKELL could not be shown:
9.75/4.21	
9.75/4.21	(0) HASKELL
9.75/4.21	(1) BR [EQUIVALENT, 0 ms]
9.75/4.21	(2) HASKELL
9.75/4.21	(3) COR [EQUIVALENT, 0 ms]
9.75/4.21	(4) HASKELL
9.75/4.21	(5) Narrow [SOUND, 0 ms]
9.75/4.21	(6) QDP
9.75/4.21	    (7) NonTerminationLoopProof [COMPLETE, 0 ms]
9.75/4.21	    (8) NO
9.75/4.21	(9) Narrow [COMPLETE, 0 ms]
9.75/4.21	(10) QDP
9.75/4.21	    (11) PisEmptyProof [EQUIVALENT, 0 ms]
9.75/4.21	    (12) YES
9.75/4.21	
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(0)
9.75/4.21	Obligation:
9.75/4.21	mainModule Main 
9.75/4.21	module Main where {
9.75/4.21	import qualified Prelude;
9.75/4.21	data List a = Cons a (List a)  | Nil ;
9.75/4.21	
9.75/4.21	iterate :: (a  ->  a)  ->  a  ->  List a;
9.75/4.21	iterate f x = Cons x (iterate f (f x));
9.75/4.21	
9.75/4.21	}
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(1) BR (EQUIVALENT)
9.75/4.21	Replaced joker patterns by fresh variables and removed binding patterns.
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(2)
9.75/4.21	Obligation:
9.75/4.21	mainModule Main 
9.75/4.21	module Main where {
9.75/4.21	import qualified Prelude;
9.75/4.21	data List a = Cons a (List a)  | Nil ;
9.75/4.21	
9.75/4.21	iterate :: (a  ->  a)  ->  a  ->  List a;
9.75/4.21	iterate f x = Cons x (iterate f (f x));
9.75/4.21	
9.75/4.21	}
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(3) COR (EQUIVALENT)
9.75/4.21	Cond Reductions:
9.75/4.21	The following Function with conditions
9.75/4.21	"undefined |Falseundefined;
9.75/4.21	"
9.75/4.21	is transformed to
9.75/4.21	"undefined  = undefined1;
9.75/4.21	"
9.75/4.21	"undefined0 True = undefined;
9.75/4.21	"
9.75/4.21	"undefined1  = undefined0 False;
9.75/4.21	"
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(4)
9.75/4.21	Obligation:
9.75/4.21	mainModule Main 
9.75/4.21	module Main where {
9.75/4.21	import qualified Prelude;
9.75/4.21	data List a = Cons a (List a)  | Nil ;
9.75/4.21	
9.75/4.21	iterate :: (a  ->  a)  ->  a  ->  List a;
9.75/4.21	iterate f x = Cons x (iterate f (f x));
9.75/4.21	
9.75/4.21	}
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(5) Narrow (SOUND)
9.75/4.21	Haskell To QDPs
9.75/4.21	
9.75/4.21	digraph dp_graph {
9.75/4.21	node [outthreshold=100, inthreshold=100];1[label="iterate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.75/4.21	3[label="iterate vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.75/4.21	4[label="iterate vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.75/4.21	5[label="Cons vx4 (iterate vx3 (vx3 vx4))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
9.75/4.21	6 -> 4[label="",style="dashed", color="red", weight=0];
9.75/4.21	6[label="iterate vx3 (vx3 vx4)",fontsize=16,color="magenta"];6 -> 7[label="",style="dashed", color="magenta", weight=3];
9.75/4.21	7[label="vx3 vx4",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="green", weight=3];
9.75/4.21	8[label="vx4",fontsize=16,color="green",shape="box"];}
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(6)
9.75/4.21	Obligation:
9.75/4.21	Q DP problem:
9.75/4.21	The TRS P consists of the following rules:
9.75/4.21	
9.75/4.21	   new_iterate(vx3, h) -> new_iterate(vx3, h)
9.75/4.21	
9.75/4.21	R is empty.
9.75/4.21	Q is empty.
9.75/4.21	We have to consider all minimal (P,Q,R)-chains.
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(7) NonTerminationLoopProof (COMPLETE)
9.75/4.21	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
9.75/4.21	Found a loop by semiunifying a rule from P directly.
9.75/4.21	
9.75/4.21	s = new_iterate(vx3, h) evaluates to  t =new_iterate(vx3, h)
9.75/4.21	
9.75/4.21	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
9.75/4.21	* Matcher: [ ]
9.75/4.21	* Semiunifier: [ ]
9.75/4.21	
9.75/4.21	--------------------------------------------------------------------------------
9.75/4.21	Rewriting sequence
9.75/4.21	
9.75/4.21	The DP semiunifies directly so there is only one rewrite step from new_iterate(vx3, h) to new_iterate(vx3, h).
9.75/4.21	
9.75/4.21	
9.75/4.21	
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(8)
9.75/4.21	NO
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(9) Narrow (COMPLETE)
9.75/4.21	Haskell To QDPs
9.75/4.21	
9.75/4.21	digraph dp_graph {
9.75/4.21	node [outthreshold=100, inthreshold=100];1[label="iterate",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.75/4.21	3[label="iterate vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.75/4.21	4[label="iterate vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.75/4.21	5[label="Cons vx4 (iterate vx3 (vx3 vx4))",fontsize=16,color="green",shape="box"];5 -> 6[label="",style="dashed", color="green", weight=3];
9.75/4.21	6 -> 4[label="",style="dashed", color="red", weight=0];
9.75/4.21	6[label="iterate vx3 (vx3 vx4)",fontsize=16,color="magenta"];6 -> 7[label="",style="dashed", color="magenta", weight=3];
9.75/4.21	7[label="vx3 vx4",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="green", weight=3];
9.75/4.21	8[label="vx4",fontsize=16,color="green",shape="box"];}
9.75/4.21	
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(10)
9.75/4.21	Obligation:
9.75/4.21	Q DP problem:
9.75/4.21	P is empty.
9.75/4.21	R is empty.
9.75/4.21	Q is empty.
9.75/4.21	We have to consider all (P,Q,R)-chains.
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(11) PisEmptyProof (EQUIVALENT)
9.75/4.21	The TRS P is empty. Hence, there is no (P,Q,R) chain.
9.75/4.21	----------------------------------------
9.75/4.21	
9.75/4.21	(12)
9.75/4.21	YES
9.87/4.25	EOF