7.60/3.50	YES
9.25/3.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.25/3.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.25/3.97	
9.25/3.97	
9.25/3.97	H-Termination with start terms of the given HASKELL could be proven:
9.25/3.97	
9.25/3.97	(0) HASKELL
9.25/3.97	(1) BR [EQUIVALENT, 0 ms]
9.25/3.97	(2) HASKELL
9.25/3.97	(3) COR [EQUIVALENT, 0 ms]
9.25/3.97	(4) HASKELL
9.25/3.97	(5) Narrow [SOUND, 0 ms]
9.25/3.97	(6) QDP
9.25/3.97	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.25/3.97	(8) YES
9.25/3.97	
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(0)
9.25/3.97	Obligation:
9.25/3.97	mainModule Main 
9.25/3.97	module Main where {
9.25/3.97	import qualified Prelude;
9.25/3.97	data List a = Cons a (List a)  | Nil ;
9.25/3.97	
9.25/3.97	data Main.Maybe a = Nothing  | Just a ;
9.25/3.97	
9.25/3.97	data Tup0 = Tup0 ;
9.25/3.97	
9.25/3.97	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.25/3.97	foldr f z Nil = z;
9.25/3.97	foldr f z (Cons x xs) = f x (foldr f z xs);
9.25/3.97	
9.25/3.97	gtGt0 q vv = q;
9.25/3.97	
9.25/3.97	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.25/3.97	gtGtEsMaybe (Main.Just x) k = k x;
9.25/3.97	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.25/3.97	
9.25/3.97	gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
9.25/3.97	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.25/3.97	
9.25/3.97	returnMaybe :: a  ->  Main.Maybe a;
9.25/3.97	returnMaybe = Main.Just;
9.25/3.97	
9.25/3.97	sequence_ = foldr gtGtMaybe (returnMaybe Tup0);
9.25/3.97	
9.25/3.97	}
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(1) BR (EQUIVALENT)
9.25/3.97	Replaced joker patterns by fresh variables and removed binding patterns.
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(2)
9.25/3.97	Obligation:
9.25/3.97	mainModule Main 
9.25/3.97	module Main where {
9.25/3.97	import qualified Prelude;
9.25/3.97	data List a = Cons a (List a)  | Nil ;
9.25/3.97	
9.25/3.97	data Main.Maybe a = Nothing  | Just a ;
9.25/3.97	
9.25/3.97	data Tup0 = Tup0 ;
9.25/3.97	
9.25/3.97	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.25/3.97	foldr f z Nil = z;
9.25/3.97	foldr f z (Cons x xs) = f x (foldr f z xs);
9.25/3.97	
9.25/3.97	gtGt0 q vv = q;
9.25/3.97	
9.25/3.97	gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
9.25/3.97	gtGtEsMaybe (Main.Just x) k = k x;
9.25/3.97	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.25/3.97	
9.25/3.97	gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
9.25/3.97	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.25/3.97	
9.25/3.97	returnMaybe :: a  ->  Main.Maybe a;
9.25/3.97	returnMaybe = Main.Just;
9.25/3.97	
9.25/3.97	sequence_ = foldr gtGtMaybe (returnMaybe Tup0);
9.25/3.97	
9.25/3.97	}
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(3) COR (EQUIVALENT)
9.25/3.97	Cond Reductions:
9.25/3.97	The following Function with conditions
9.25/3.97	"undefined |Falseundefined;
9.25/3.97	"
9.25/3.97	is transformed to
9.25/3.97	"undefined  = undefined1;
9.25/3.97	"
9.25/3.97	"undefined0 True = undefined;
9.25/3.97	"
9.25/3.97	"undefined1  = undefined0 False;
9.25/3.97	"
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(4)
9.25/3.97	Obligation:
9.25/3.97	mainModule Main 
9.25/3.97	module Main where {
9.25/3.97	import qualified Prelude;
9.25/3.97	data List a = Cons a (List a)  | Nil ;
9.25/3.97	
9.25/3.97	data Main.Maybe a = Nothing  | Just a ;
9.25/3.97	
9.25/3.97	data Tup0 = Tup0 ;
9.25/3.97	
9.25/3.97	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.25/3.97	foldr f z Nil = z;
9.25/3.97	foldr f z (Cons x xs) = f x (foldr f z xs);
9.25/3.97	
9.25/3.97	gtGt0 q vv = q;
9.25/3.97	
9.25/3.97	gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
9.25/3.97	gtGtEsMaybe (Main.Just x) k = k x;
9.25/3.97	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.25/3.97	
9.25/3.97	gtGtMaybe :: Main.Maybe a  ->  Main.Maybe b  ->  Main.Maybe b;
9.25/3.97	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.25/3.97	
9.25/3.97	returnMaybe :: a  ->  Main.Maybe a;
9.25/3.97	returnMaybe = Main.Just;
9.25/3.97	
9.25/3.97	sequence_ = foldr gtGtMaybe (returnMaybe Tup0);
9.25/3.97	
9.25/3.97	}
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(5) Narrow (SOUND)
9.25/3.97	Haskell To QDPs
9.25/3.97	
9.25/3.97	digraph dp_graph {
9.25/3.97	node [outthreshold=100, inthreshold=100];1[label="sequence_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.25/3.97	3[label="sequence_ vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.25/3.97	4[label="foldr gtGtMaybe (returnMaybe Tup0) vy3",fontsize=16,color="burlywood",shape="triangle"];19[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
9.25/3.97	19 -> 5[label="",style="solid", color="burlywood", weight=3];
9.25/3.97	20[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
9.25/3.97	20 -> 6[label="",style="solid", color="burlywood", weight=3];
9.25/3.97	5[label="foldr gtGtMaybe (returnMaybe Tup0) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.25/3.97	6[label="foldr gtGtMaybe (returnMaybe Tup0) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.25/3.97	7 -> 9[label="",style="dashed", color="red", weight=0];
9.25/3.97	7[label="gtGtMaybe vy30 (foldr gtGtMaybe (returnMaybe Tup0) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.25/3.97	8[label="returnMaybe Tup0",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.25/3.97	10 -> 4[label="",style="dashed", color="red", weight=0];
9.25/3.97	10[label="foldr gtGtMaybe (returnMaybe Tup0) vy31",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9.25/3.97	9[label="gtGtMaybe vy30 vy4",fontsize=16,color="black",shape="triangle"];9 -> 13[label="",style="solid", color="black", weight=3];
9.25/3.97	11[label="Just Tup0",fontsize=16,color="green",shape="box"];12[label="vy31",fontsize=16,color="green",shape="box"];13[label="gtGtEsMaybe vy30 (gtGt0 vy4)",fontsize=16,color="burlywood",shape="box"];21[label="vy30/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 21[label="",style="solid", color="burlywood", weight=9];
9.25/3.97	21 -> 14[label="",style="solid", color="burlywood", weight=3];
9.25/3.97	22[label="vy30/Just vy300",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9];
9.25/3.97	22 -> 15[label="",style="solid", color="burlywood", weight=3];
9.25/3.97	14[label="gtGtEsMaybe Nothing (gtGt0 vy4)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
9.25/3.97	15[label="gtGtEsMaybe (Just vy300) (gtGt0 vy4)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
9.25/3.97	16[label="Nothing",fontsize=16,color="green",shape="box"];17[label="gtGt0 vy4 vy300",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
9.25/3.97	18[label="vy4",fontsize=16,color="green",shape="box"];}
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(6)
9.25/3.97	Obligation:
9.25/3.97	Q DP problem:
9.25/3.97	The TRS P consists of the following rules:
9.25/3.97	
9.25/3.97	   new_foldr(Cons(vy30, vy31), h) -> new_foldr(vy31, h)
9.25/3.97	
9.25/3.97	R is empty.
9.25/3.97	Q is empty.
9.25/3.97	We have to consider all minimal (P,Q,R)-chains.
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(7) QDPSizeChangeProof (EQUIVALENT)
9.25/3.97	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. 
9.25/3.97	
9.25/3.97	From the DPs we obtained the following set of size-change graphs:
9.25/3.97	*new_foldr(Cons(vy30, vy31), h) -> new_foldr(vy31, h)
9.25/3.97	The graph contains the following edges 1 > 1, 2 >= 2
9.25/3.97	
9.25/3.97	
9.25/3.97	----------------------------------------
9.25/3.97	
9.25/3.97	(8)
9.25/3.97	YES
9.49/4.12	EOF