7.80/3.54	YES
9.12/4.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.12/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.12/4.02	
9.12/4.02	
9.12/4.02	H-Termination with start terms of the given HASKELL could be proven:
9.12/4.02	
9.12/4.02	(0) HASKELL
9.12/4.02	(1) BR [EQUIVALENT, 0 ms]
9.12/4.02	(2) HASKELL
9.12/4.02	(3) COR [EQUIVALENT, 0 ms]
9.12/4.02	(4) HASKELL
9.12/4.02	(5) Narrow [SOUND, 0 ms]
9.12/4.02	(6) QDP
9.12/4.02	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.12/4.02	(8) YES
9.12/4.02	
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(0)
9.12/4.02	Obligation:
9.12/4.02	mainModule Main 
9.12/4.02	module Main where {
9.12/4.02	import qualified Prelude;
9.12/4.02	data List a = Cons a (List a)  | Nil ;
9.12/4.02	
9.12/4.02	data Main.Maybe a = Nothing  | Just a ;
9.12/4.02	
9.12/4.02	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.12/4.02	gtGtEsMaybe (Main.Just x) k = k x;
9.12/4.02	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.12/4.02	
9.12/4.02	returnMaybe :: a  ->  Main.Maybe a;
9.12/4.02	returnMaybe = Main.Just;
9.12/4.02	
9.12/4.02	sequence Nil = returnMaybe Nil;
9.12/4.02	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.12/4.02	
9.12/4.02	sequence0 x xs = returnMaybe (Cons x xs);
9.12/4.02	
9.12/4.02	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.12/4.02	
9.12/4.02	}
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(1) BR (EQUIVALENT)
9.12/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(2)
9.12/4.02	Obligation:
9.12/4.02	mainModule Main 
9.12/4.02	module Main where {
9.12/4.02	import qualified Prelude;
9.12/4.02	data List a = Cons a (List a)  | Nil ;
9.12/4.02	
9.12/4.02	data Main.Maybe a = Nothing  | Just a ;
9.12/4.02	
9.12/4.02	gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
9.12/4.02	gtGtEsMaybe (Main.Just x) k = k x;
9.12/4.02	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.12/4.02	
9.12/4.02	returnMaybe :: a  ->  Main.Maybe a;
9.12/4.02	returnMaybe = Main.Just;
9.12/4.02	
9.12/4.02	sequence Nil = returnMaybe Nil;
9.12/4.02	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.12/4.02	
9.12/4.02	sequence0 x xs = returnMaybe (Cons x xs);
9.12/4.02	
9.12/4.02	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.12/4.02	
9.12/4.02	}
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(3) COR (EQUIVALENT)
9.12/4.02	Cond Reductions:
9.12/4.02	The following Function with conditions
9.12/4.02	"undefined |Falseundefined;
9.12/4.02	"
9.12/4.02	is transformed to
9.12/4.02	"undefined  = undefined1;
9.12/4.02	"
9.12/4.02	"undefined0 True = undefined;
9.12/4.02	"
9.12/4.02	"undefined1  = undefined0 False;
9.12/4.02	"
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(4)
9.12/4.02	Obligation:
9.12/4.02	mainModule Main 
9.12/4.02	module Main where {
9.12/4.02	import qualified Prelude;
9.12/4.02	data List a = Cons a (List a)  | Nil ;
9.12/4.02	
9.12/4.02	data Main.Maybe a = Nothing  | Just a ;
9.12/4.02	
9.12/4.02	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.12/4.02	gtGtEsMaybe (Main.Just x) k = k x;
9.12/4.02	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.12/4.02	
9.12/4.02	returnMaybe :: a  ->  Main.Maybe a;
9.12/4.02	returnMaybe = Main.Just;
9.12/4.02	
9.12/4.02	sequence Nil = returnMaybe Nil;
9.12/4.02	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.12/4.02	
9.12/4.02	sequence0 x xs = returnMaybe (Cons x xs);
9.12/4.02	
9.12/4.02	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.12/4.02	
9.12/4.02	}
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(5) Narrow (SOUND)
9.12/4.02	Haskell To QDPs
9.12/4.02	
9.12/4.02	digraph dp_graph {
9.12/4.02	node [outthreshold=100, inthreshold=100];1[label="sequence",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.12/4.02	3[label="sequence vx3",fontsize=16,color="burlywood",shape="triangle"];23[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 23[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	23 -> 4[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	24[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];3 -> 24[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	24 -> 5[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	4[label="sequence (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
9.12/4.02	5[label="sequence Nil",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.12/4.02	6[label="gtGtEsMaybe vx30 (sequence1 vx31)",fontsize=16,color="burlywood",shape="box"];25[label="vx30/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 25[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	25 -> 8[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	26[label="vx30/Just vx300",fontsize=10,color="white",style="solid",shape="box"];6 -> 26[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	26 -> 9[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	7[label="returnMaybe Nil",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.12/4.02	8[label="gtGtEsMaybe Nothing (sequence1 vx31)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.12/4.02	9[label="gtGtEsMaybe (Just vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
9.12/4.02	10[label="Just Nil",fontsize=16,color="green",shape="box"];11[label="Nothing",fontsize=16,color="green",shape="box"];12[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
9.12/4.02	13 -> 14[label="",style="dashed", color="red", weight=0];
9.12/4.02	13[label="gtGtEsMaybe (sequence vx31) (sequence0 vx300)",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
9.12/4.02	15 -> 3[label="",style="dashed", color="red", weight=0];
9.12/4.02	15[label="sequence vx31",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
9.12/4.02	14[label="gtGtEsMaybe vx4 (sequence0 vx300)",fontsize=16,color="burlywood",shape="triangle"];27[label="vx4/Nothing",fontsize=10,color="white",style="solid",shape="box"];14 -> 27[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	27 -> 17[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	28[label="vx4/Just vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 28[label="",style="solid", color="burlywood", weight=9];
9.12/4.02	28 -> 18[label="",style="solid", color="burlywood", weight=3];
9.12/4.02	16[label="vx31",fontsize=16,color="green",shape="box"];17[label="gtGtEsMaybe Nothing (sequence0 vx300)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
9.12/4.02	18[label="gtGtEsMaybe (Just vx40) (sequence0 vx300)",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
9.12/4.02	19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="sequence0 vx300 vx40",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.12/4.02	21[label="returnMaybe (Cons vx300 vx40)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.12/4.02	22[label="Just (Cons vx300 vx40)",fontsize=16,color="green",shape="box"];}
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(6)
9.12/4.02	Obligation:
9.12/4.02	Q DP problem:
9.12/4.02	The TRS P consists of the following rules:
9.12/4.02	
9.12/4.02	   new_sequence(Cons(Main.Just(vx300), vx31), h) -> new_sequence(vx31, h)
9.12/4.02	
9.12/4.02	R is empty.
9.12/4.02	Q is empty.
9.12/4.02	We have to consider all minimal (P,Q,R)-chains.
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(7) QDPSizeChangeProof (EQUIVALENT)
9.12/4.02	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. 
9.12/4.02	
9.12/4.02	From the DPs we obtained the following set of size-change graphs:
9.12/4.02	*new_sequence(Cons(Main.Just(vx300), vx31), h) -> new_sequence(vx31, h)
9.12/4.02	The graph contains the following edges 1 > 1, 2 >= 2
9.12/4.02	
9.12/4.02	
9.12/4.02	----------------------------------------
9.12/4.02	
9.12/4.02	(8)
9.12/4.02	YES
9.46/4.05	EOF