10.66/4.43	MAYBE
12.29/4.95	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.29/4.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.29/4.95	
12.29/4.95	
12.29/4.95	H-Termination with start terms of the given HASKELL could not be shown:
12.29/4.95	
12.29/4.95	(0) HASKELL
12.29/4.95	(1) CR [EQUIVALENT, 0 ms]
12.29/4.95	(2) HASKELL
12.29/4.95	(3) BR [EQUIVALENT, 0 ms]
12.29/4.95	(4) HASKELL
12.29/4.95	(5) COR [EQUIVALENT, 0 ms]
12.29/4.95	(6) HASKELL
12.29/4.95	(7) Narrow [SOUND, 0 ms]
12.29/4.95	(8) QDP
12.29/4.95	    (9) NonTerminationLoopProof [COMPLETE, 0 ms]
12.29/4.95	    (10) NO
12.29/4.95	(11) Narrow [COMPLETE, 0 ms]
12.29/4.95	(12) QDP
12.29/4.95	    (13) DependencyGraphProof [EQUIVALENT, 0 ms]
12.29/4.95	    (14) TRUE
12.29/4.95	
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(0)
12.29/4.95	Obligation:
12.29/4.95	mainModule Main 
12.29/4.95	module Maybe where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	module List where {
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	unfoldr :: (b  ->  Maybe (a,b))  ->  b  ->  [a];
12.29/4.95	unfoldr f b = case f b of { 
12.29/4.95	Just (a,new_b)-> a : unfoldr f new_b; 
12.29/4.95	Nothing-> []; 
12.29/4.95	 } ;
12.29/4.95	
12.29/4.95	}
12.29/4.95	module Main where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(1) CR (EQUIVALENT)
12.29/4.95	Case Reductions:
12.29/4.95	The following Case expression
12.29/4.95	"case f b of {
12.29/4.95	Just (a,new_b)  -> a : unfoldr f new_b;
12.29/4.95	Nothing  -> []}
12.29/4.95	"
12.29/4.95	is transformed to
12.29/4.95	"unfoldr0 f (Just (a,new_b)) = a : unfoldr f new_b;
12.29/4.95	unfoldr0 f Nothing = [];
12.29/4.95	"
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(2)
12.29/4.95	Obligation:
12.29/4.95	mainModule Main 
12.29/4.95	module Maybe where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	module List where {
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	unfoldr :: (a  ->  Maybe (b,a))  ->  a  ->  [b];
12.29/4.95	unfoldr f b = unfoldr0 f (f b);
12.29/4.95	
12.29/4.95	unfoldr0 f (Just (a,new_b)) = a : unfoldr f new_b;
12.29/4.95	unfoldr0 f Nothing = [];
12.29/4.95	
12.29/4.95	}
12.29/4.95	module Main where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(3) BR (EQUIVALENT)
12.29/4.95	Replaced joker patterns by fresh variables and removed binding patterns.
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(4)
12.29/4.95	Obligation:
12.29/4.95	mainModule Main 
12.29/4.95	module Maybe where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	module List where {
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	unfoldr :: (b  ->  Maybe (a,b))  ->  b  ->  [a];
12.29/4.95	unfoldr f b = unfoldr0 f (f b);
12.29/4.95	
12.29/4.95	unfoldr0 f (Just (a,new_b)) = a : unfoldr f new_b;
12.29/4.95	unfoldr0 f Nothing = [];
12.29/4.95	
12.29/4.95	}
12.29/4.95	module Main where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(5) COR (EQUIVALENT)
12.29/4.95	Cond Reductions:
12.29/4.95	The following Function with conditions
12.29/4.95	"undefined |Falseundefined;
12.29/4.95	"
12.29/4.95	is transformed to
12.29/4.95	"undefined  = undefined1;
12.29/4.95	"
12.29/4.95	"undefined0 True = undefined;
12.29/4.95	"
12.29/4.95	"undefined1  = undefined0 False;
12.29/4.95	"
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(6)
12.29/4.95	Obligation:
12.29/4.95	mainModule Main 
12.29/4.95	module Maybe where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	module List where {
12.29/4.95	import qualified Main;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	unfoldr :: (b  ->  Maybe (a,b))  ->  b  ->  [a];
12.29/4.95	unfoldr f b = unfoldr0 f (f b);
12.29/4.95	
12.29/4.95	unfoldr0 f (Just (a,new_b)) = a : unfoldr f new_b;
12.29/4.95	unfoldr0 f Nothing = [];
12.29/4.95	
12.29/4.95	}
12.29/4.95	module Main where {
12.29/4.95	import qualified List;
12.29/4.95	import qualified Maybe;
12.29/4.95	import qualified Prelude;
12.29/4.95	}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(7) Narrow (SOUND)
12.29/4.95	Haskell To QDPs
12.29/4.95	
12.29/4.95	digraph dp_graph {
12.29/4.95	node [outthreshold=100, inthreshold=100];1[label="List.unfoldr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.29/4.95	3[label="List.unfoldr vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.29/4.95	4[label="List.unfoldr vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.29/4.95	5 -> 6[label="",style="dashed", color="red", weight=0];
12.29/4.95	5[label="List.unfoldr0 vy3 (vy3 vy4)",fontsize=16,color="magenta"];5 -> 7[label="",style="dashed", color="magenta", weight=3];
12.29/4.95	7[label="vy3 vy4",fontsize=16,color="green",shape="box"];7 -> 11[label="",style="dashed", color="green", weight=3];
12.29/4.95	6[label="List.unfoldr0 vy3 vy5",fontsize=16,color="burlywood",shape="triangle"];17[label="vy5/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 17[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	17 -> 9[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	18[label="vy5/Just vy50",fontsize=10,color="white",style="solid",shape="box"];6 -> 18[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	18 -> 10[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	11[label="vy4",fontsize=16,color="green",shape="box"];9[label="List.unfoldr0 vy3 Nothing",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
12.29/4.95	10[label="List.unfoldr0 vy3 (Just vy50)",fontsize=16,color="burlywood",shape="box"];19[label="vy50/(vy500,vy501)",fontsize=10,color="white",style="solid",shape="box"];10 -> 19[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	19 -> 13[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	12[label="[]",fontsize=16,color="green",shape="box"];13[label="List.unfoldr0 vy3 (Just (vy500,vy501))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
12.29/4.95	14[label="vy500 : List.unfoldr vy3 vy501",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
12.29/4.95	15 -> 4[label="",style="dashed", color="red", weight=0];
12.29/4.95	15[label="List.unfoldr vy3 vy501",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
12.29/4.95	16[label="vy501",fontsize=16,color="green",shape="box"];}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(8)
12.29/4.95	Obligation:
12.29/4.95	Q DP problem:
12.29/4.95	The TRS P consists of the following rules:
12.29/4.95	
12.29/4.95	   new_unfoldr(vy3, ba, bb) -> new_unfoldr0(vy3, ba, bb)
12.29/4.95	   new_unfoldr0(vy3, ba, bb) -> new_unfoldr(vy3, ba, bb)
12.29/4.95	
12.29/4.95	R is empty.
12.29/4.95	Q is empty.
12.29/4.95	We have to consider all minimal (P,Q,R)-chains.
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(9) NonTerminationLoopProof (COMPLETE)
12.29/4.95	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
12.29/4.95	Found a loop by narrowing to the left:
12.29/4.95	
12.29/4.95	s = new_unfoldr0(vy3', ba', bb') evaluates to  t =new_unfoldr0(vy3', ba', bb')
12.29/4.95	
12.29/4.95	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
12.29/4.95	* Matcher: [ ]
12.29/4.95	* Semiunifier: [ ]
12.29/4.95	
12.29/4.95	--------------------------------------------------------------------------------
12.29/4.95	Rewriting sequence
12.29/4.95	
12.29/4.95	new_unfoldr0(vy3', ba', bb') -> new_unfoldr(vy3', ba', bb')
12.29/4.95	with rule new_unfoldr0(vy3'', ba'', bb'') -> new_unfoldr(vy3'', ba'', bb'') at position [] and matcher [vy3'' / vy3', ba'' / ba', bb'' / bb']
12.29/4.95	
12.29/4.95	new_unfoldr(vy3', ba', bb') -> new_unfoldr0(vy3', ba', bb')
12.29/4.95	with rule new_unfoldr(vy3, ba, bb) -> new_unfoldr0(vy3, ba, bb)
12.29/4.95	
12.29/4.95	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
12.29/4.95	
12.29/4.95	
12.29/4.95	All these steps are and every following step will be a correct step w.r.t to Q.
12.29/4.95	
12.29/4.95	
12.29/4.95	
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(10)
12.29/4.95	NO
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(11) Narrow (COMPLETE)
12.29/4.95	Haskell To QDPs
12.29/4.95	
12.29/4.95	digraph dp_graph {
12.29/4.95	node [outthreshold=100, inthreshold=100];1[label="List.unfoldr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.29/4.95	3[label="List.unfoldr vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.29/4.95	4[label="List.unfoldr vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.29/4.95	5 -> 6[label="",style="dashed", color="red", weight=0];
12.29/4.95	5[label="List.unfoldr0 vy3 (vy3 vy4)",fontsize=16,color="magenta"];5 -> 7[label="",style="dashed", color="magenta", weight=3];
12.29/4.95	7[label="vy3 vy4",fontsize=16,color="green",shape="box"];7 -> 11[label="",style="dashed", color="green", weight=3];
12.29/4.95	6[label="List.unfoldr0 vy3 vy5",fontsize=16,color="burlywood",shape="triangle"];17[label="vy5/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 17[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	17 -> 9[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	18[label="vy5/Just vy50",fontsize=10,color="white",style="solid",shape="box"];6 -> 18[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	18 -> 10[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	11[label="vy4",fontsize=16,color="green",shape="box"];9[label="List.unfoldr0 vy3 Nothing",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
12.29/4.95	10[label="List.unfoldr0 vy3 (Just vy50)",fontsize=16,color="burlywood",shape="box"];19[label="vy50/(vy500,vy501)",fontsize=10,color="white",style="solid",shape="box"];10 -> 19[label="",style="solid", color="burlywood", weight=9];
12.29/4.95	19 -> 13[label="",style="solid", color="burlywood", weight=3];
12.29/4.95	12[label="[]",fontsize=16,color="green",shape="box"];13[label="List.unfoldr0 vy3 (Just (vy500,vy501))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
12.29/4.95	14[label="vy500 : List.unfoldr vy3 vy501",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
12.29/4.95	15 -> 4[label="",style="dashed", color="red", weight=0];
12.29/4.95	15[label="List.unfoldr vy3 vy501",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
12.29/4.95	16[label="vy501",fontsize=16,color="green",shape="box"];}
12.29/4.95	
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(12)
12.29/4.95	Obligation:
12.29/4.95	Q DP problem:
12.29/4.95	The TRS P consists of the following rules:
12.29/4.95	
12.29/4.95	   new_unfoldr0(vy3, Just(@2(vy500, vy501)), ba, bb, []) -> new_unfoldr(vy3, vy501, ba, bb, [])
12.29/4.95	
12.29/4.95	R is empty.
12.29/4.95	Q is empty.
12.29/4.95	We have to consider all (P,Q,R)-chains.
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(13) DependencyGraphProof (EQUIVALENT)
12.29/4.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
12.29/4.95	----------------------------------------
12.29/4.95	
12.29/4.95	(14)
12.29/4.95	TRUE
12.57/7.81	EOF