7.96/3.64	YES
9.57/4.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.57/4.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.57/4.12	
9.57/4.12	
9.57/4.12	H-Termination with start terms of the given HASKELL could be proven:
9.57/4.12	
9.57/4.12	(0) HASKELL
9.57/4.12	(1) BR [EQUIVALENT, 0 ms]
9.57/4.12	(2) HASKELL
9.57/4.12	(3) COR [EQUIVALENT, 0 ms]
9.57/4.12	(4) HASKELL
9.57/4.12	(5) Narrow [SOUND, 0 ms]
9.57/4.12	(6) QDP
9.57/4.12	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.57/4.12	(8) YES
9.57/4.12	
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(0)
9.57/4.12	Obligation:
9.57/4.12	mainModule Main 
9.57/4.12	module Main where {
9.57/4.12	import qualified Prelude;
9.57/4.12	data List a = Cons a (List a)  | Nil ;
9.57/4.12	
9.57/4.12	data Main.Maybe a = Nothing  | Just a ;
9.57/4.12	
9.57/4.12	gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
9.57/4.12	gtGtEsMaybe (Main.Just x) k = k x;
9.57/4.12	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.57/4.12	
9.57/4.12	map :: (b  ->  a)  ->  List b  ->  List a;
9.57/4.12	map f Nil = Nil;
9.57/4.12	map f (Cons x xs) = Cons (f x) (map f xs);
9.57/4.12	
9.57/4.12	mapM f = pt sequence (map f);
9.57/4.12	
9.57/4.12	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.57/4.12	pt f g x = f (g x);
9.57/4.12	
9.57/4.12	returnMaybe :: a  ->  Main.Maybe a;
9.57/4.12	returnMaybe = Main.Just;
9.57/4.12	
9.57/4.12	sequence Nil = returnMaybe Nil;
9.57/4.12	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.57/4.12	
9.57/4.12	sequence0 x xs = returnMaybe (Cons x xs);
9.57/4.12	
9.57/4.12	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.57/4.12	
9.57/4.12	}
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(1) BR (EQUIVALENT)
9.57/4.12	Replaced joker patterns by fresh variables and removed binding patterns.
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(2)
9.57/4.12	Obligation:
9.57/4.12	mainModule Main 
9.57/4.12	module Main where {
9.57/4.12	import qualified Prelude;
9.57/4.12	data List a = Cons a (List a)  | Nil ;
9.57/4.12	
9.57/4.12	data Main.Maybe a = Nothing  | Just a ;
9.57/4.12	
9.57/4.12	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.57/4.12	gtGtEsMaybe (Main.Just x) k = k x;
9.57/4.12	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.57/4.12	
9.57/4.12	map :: (b  ->  a)  ->  List b  ->  List a;
9.57/4.12	map f Nil = Nil;
9.57/4.12	map f (Cons x xs) = Cons (f x) (map f xs);
9.57/4.12	
9.57/4.12	mapM f = pt sequence (map f);
9.57/4.12	
9.57/4.12	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
9.57/4.12	pt f g x = f (g x);
9.57/4.12	
9.57/4.12	returnMaybe :: a  ->  Main.Maybe a;
9.57/4.12	returnMaybe = Main.Just;
9.57/4.12	
9.57/4.12	sequence Nil = returnMaybe Nil;
9.57/4.12	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.57/4.12	
9.57/4.12	sequence0 x xs = returnMaybe (Cons x xs);
9.57/4.12	
9.57/4.12	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.57/4.12	
9.57/4.12	}
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(3) COR (EQUIVALENT)
9.57/4.12	Cond Reductions:
9.57/4.12	The following Function with conditions
9.57/4.12	"undefined |Falseundefined;
9.57/4.12	"
9.57/4.12	is transformed to
9.57/4.12	"undefined  = undefined1;
9.57/4.12	"
9.57/4.12	"undefined0 True = undefined;
9.57/4.12	"
9.57/4.12	"undefined1  = undefined0 False;
9.57/4.12	"
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(4)
9.57/4.12	Obligation:
9.57/4.12	mainModule Main 
9.57/4.12	module Main where {
9.57/4.12	import qualified Prelude;
9.57/4.12	data List a = Cons a (List a)  | Nil ;
9.57/4.12	
9.57/4.12	data Main.Maybe a = Nothing  | Just a ;
9.57/4.12	
9.57/4.12	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.57/4.12	gtGtEsMaybe (Main.Just x) k = k x;
9.57/4.12	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.57/4.12	
9.57/4.12	map :: (a  ->  b)  ->  List a  ->  List b;
9.57/4.12	map f Nil = Nil;
9.57/4.12	map f (Cons x xs) = Cons (f x) (map f xs);
9.57/4.12	
9.57/4.12	mapM f = pt sequence (map f);
9.57/4.12	
9.57/4.12	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
9.57/4.12	pt f g x = f (g x);
9.57/4.12	
9.57/4.12	returnMaybe :: a  ->  Main.Maybe a;
9.57/4.12	returnMaybe = Main.Just;
9.57/4.12	
9.57/4.12	sequence Nil = returnMaybe Nil;
9.57/4.12	sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);
9.57/4.12	
9.57/4.12	sequence0 x xs = returnMaybe (Cons x xs);
9.57/4.12	
9.57/4.12	sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);
9.57/4.12	
9.57/4.12	}
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(5) Narrow (SOUND)
9.57/4.12	Haskell To QDPs
9.57/4.12	
9.57/4.12	digraph dp_graph {
9.57/4.12	node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.57/4.12	3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.57/4.12	4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.57/4.12	5[label="pt sequence (map vx3) vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.57/4.12	6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];32[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 32[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	32 -> 7[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	33[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];6 -> 33[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	33 -> 8[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	7[label="sequence (map vx3 (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.57/4.12	8[label="sequence (map vx3 Nil)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.57/4.12	9[label="sequence (Cons (vx3 vx40) (map vx3 vx41))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.57/4.12	10[label="sequence Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.57/4.12	11 -> 13[label="",style="dashed", color="red", weight=0];
9.57/4.12	11[label="gtGtEsMaybe (vx3 vx40) (sequence1 (map vx3 vx41))",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.57/4.12	12[label="returnMaybe Nil",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
9.57/4.12	14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
9.57/4.12	13[label="gtGtEsMaybe vx5 (sequence1 (map vx3 vx41))",fontsize=16,color="burlywood",shape="triangle"];34[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 34[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	34 -> 17[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	35[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];13 -> 35[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	35 -> 18[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	15[label="Just Nil",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="gtGtEsMaybe Nothing (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
9.57/4.12	18[label="gtGtEsMaybe (Just vx50) (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.57/4.12	20[label="Nothing",fontsize=16,color="green",shape="box"];21[label="sequence1 (map vx3 vx41) vx50",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.57/4.12	22 -> 23[label="",style="dashed", color="red", weight=0];
9.57/4.12	22[label="gtGtEsMaybe (sequence (map vx3 vx41)) (sequence0 vx50)",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
9.57/4.12	24 -> 6[label="",style="dashed", color="red", weight=0];
9.57/4.12	24[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
9.57/4.12	23[label="gtGtEsMaybe vx6 (sequence0 vx50)",fontsize=16,color="burlywood",shape="triangle"];36[label="vx6/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 36[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	36 -> 26[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	37[label="vx6/Just vx60",fontsize=10,color="white",style="solid",shape="box"];23 -> 37[label="",style="solid", color="burlywood", weight=9];
9.57/4.12	37 -> 27[label="",style="solid", color="burlywood", weight=3];
9.57/4.12	25[label="vx41",fontsize=16,color="green",shape="box"];26[label="gtGtEsMaybe Nothing (sequence0 vx50)",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
9.57/4.12	27[label="gtGtEsMaybe (Just vx60) (sequence0 vx50)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
9.57/4.12	28[label="Nothing",fontsize=16,color="green",shape="box"];29[label="sequence0 vx50 vx60",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
9.57/4.12	30[label="returnMaybe (Cons vx50 vx60)",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
9.57/4.12	31[label="Just (Cons vx50 vx60)",fontsize=16,color="green",shape="box"];}
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(6)
9.57/4.12	Obligation:
9.57/4.12	Q DP problem:
9.57/4.12	The TRS P consists of the following rules:
9.57/4.12	
9.57/4.12	   new_sequence(vx3, Cons(vx40, vx41), h, ba) -> new_gtGtEsMaybe(vx3, vx41, h, ba)
9.57/4.12	   new_gtGtEsMaybe(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
9.57/4.12	
9.57/4.12	R is empty.
9.57/4.12	Q is empty.
9.57/4.12	We have to consider all minimal (P,Q,R)-chains.
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(7) QDPSizeChangeProof (EQUIVALENT)
9.57/4.12	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. 
9.57/4.12	
9.57/4.12	From the DPs we obtained the following set of size-change graphs:
9.57/4.12	*new_gtGtEsMaybe(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
9.57/4.12	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
9.57/4.12	
9.57/4.12	
9.57/4.12	*new_sequence(vx3, Cons(vx40, vx41), h, ba) -> new_gtGtEsMaybe(vx3, vx41, h, ba)
9.57/4.12	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.57/4.12	
9.57/4.12	
9.57/4.12	----------------------------------------
9.57/4.12	
9.57/4.12	(8)
9.57/4.12	YES
9.71/4.16	EOF