7.76/3.61	YES
9.51/4.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.51/4.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.51/4.09	
9.51/4.09	
9.51/4.09	H-Termination with start terms of the given HASKELL could be proven:
9.51/4.09	
9.51/4.09	(0) HASKELL
9.51/4.09	(1) BR [EQUIVALENT, 0 ms]
9.51/4.09	(2) HASKELL
9.51/4.09	(3) COR [EQUIVALENT, 0 ms]
9.51/4.09	(4) HASKELL
9.51/4.09	(5) Narrow [SOUND, 0 ms]
9.51/4.09	(6) QDP
9.51/4.09	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.51/4.09	(8) YES
9.51/4.09	
9.51/4.09	
9.51/4.09	----------------------------------------
9.51/4.09	
9.51/4.09	(0)
9.51/4.09	Obligation:
9.51/4.09	mainModule Main 
9.51/4.09	module Main where {
9.51/4.09	import qualified Prelude;
9.51/4.09	data List a = Cons a (List a)  | Nil ;
9.51/4.09	
9.51/4.09	data Main.Maybe a = Nothing  | Just a ;
9.51/4.09	
9.51/4.09	data Tup0 = Tup0 ;
9.51/4.09	
9.51/4.09	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.51/4.09	foldr f z Nil = z;
9.51/4.09	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.09	
9.51/4.09	gtGt0 q vv = q;
9.51/4.09	
9.51/4.09	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.51/4.09	gtGtEsMaybe (Main.Just x) k = k x;
9.51/4.09	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.51/4.09	
9.51/4.09	gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
9.51/4.09	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.51/4.09	
9.51/4.09	map :: (a  ->  b)  ->  List a  ->  List b;
9.51/4.10	map f Nil = Nil;
9.51/4.10	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.10	
9.51/4.10	mapM_ f = pt sequence_Maybe (map f);
9.51/4.10	
9.51/4.10	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.51/4.10	pt f g x = f (g x);
9.51/4.10	
9.51/4.10	returnMaybe :: a  ->  Main.Maybe a;
9.51/4.10	returnMaybe = Main.Just;
9.51/4.10	
9.51/4.10	sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
9.51/4.10	sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);
9.51/4.10	
9.51/4.10	}
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(1) BR (EQUIVALENT)
9.51/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(2)
9.51/4.10	Obligation:
9.51/4.10	mainModule Main 
9.51/4.10	module Main where {
9.51/4.10	import qualified Prelude;
9.51/4.10	data List a = Cons a (List a)  | Nil ;
9.51/4.10	
9.51/4.10	data Main.Maybe a = Nothing  | Just a ;
9.51/4.10	
9.51/4.10	data Tup0 = Tup0 ;
9.51/4.10	
9.51/4.10	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.51/4.10	foldr f z Nil = z;
9.51/4.10	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.10	
9.51/4.10	gtGt0 q vv = q;
9.51/4.10	
9.51/4.10	gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
9.51/4.10	gtGtEsMaybe (Main.Just x) k = k x;
9.51/4.10	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.51/4.10	
9.51/4.10	gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
9.51/4.10	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.51/4.10	
9.51/4.10	map :: (b  ->  a)  ->  List b  ->  List a;
9.51/4.10	map f Nil = Nil;
9.51/4.10	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.10	
9.51/4.10	mapM_ f = pt sequence_Maybe (map f);
9.51/4.10	
9.51/4.10	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.51/4.10	pt f g x = f (g x);
9.51/4.10	
9.51/4.10	returnMaybe :: a  ->  Main.Maybe a;
9.51/4.10	returnMaybe = Main.Just;
9.51/4.10	
9.51/4.10	sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
9.51/4.10	sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);
9.51/4.10	
9.51/4.10	}
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(3) COR (EQUIVALENT)
9.51/4.10	Cond Reductions:
9.51/4.10	The following Function with conditions
9.51/4.10	"undefined |Falseundefined;
9.51/4.10	"
9.51/4.10	is transformed to
9.51/4.10	"undefined  = undefined1;
9.51/4.10	"
9.51/4.10	"undefined0 True = undefined;
9.51/4.10	"
9.51/4.10	"undefined1  = undefined0 False;
9.51/4.10	"
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(4)
9.51/4.10	Obligation:
9.51/4.10	mainModule Main 
9.51/4.10	module Main where {
9.51/4.10	import qualified Prelude;
9.51/4.10	data List a = Cons a (List a)  | Nil ;
9.51/4.10	
9.51/4.10	data Main.Maybe a = Nothing  | Just a ;
9.51/4.10	
9.51/4.10	data Tup0 = Tup0 ;
9.51/4.10	
9.51/4.10	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.51/4.10	foldr f z Nil = z;
9.51/4.10	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.10	
9.51/4.10	gtGt0 q vv = q;
9.51/4.10	
9.51/4.10	gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
9.51/4.10	gtGtEsMaybe (Main.Just x) k = k x;
9.51/4.10	gtGtEsMaybe Main.Nothing k = Main.Nothing;
9.51/4.10	
9.51/4.10	gtGtMaybe :: Main.Maybe a  ->  Main.Maybe b  ->  Main.Maybe b;
9.51/4.10	gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);
9.51/4.10	
9.51/4.10	map :: (a  ->  b)  ->  List a  ->  List b;
9.51/4.10	map f Nil = Nil;
9.51/4.10	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.10	
9.51/4.10	mapM_ f = pt sequence_Maybe (map f);
9.51/4.10	
9.51/4.10	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.51/4.10	pt f g x = f (g x);
9.51/4.10	
9.51/4.10	returnMaybe :: a  ->  Main.Maybe a;
9.51/4.10	returnMaybe = Main.Just;
9.51/4.10	
9.51/4.10	sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
9.51/4.10	sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);
9.51/4.10	
9.51/4.10	}
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(5) Narrow (SOUND)
9.51/4.10	Haskell To QDPs
9.51/4.10	
9.51/4.10	digraph dp_graph {
9.51/4.10	node [outthreshold=100, inthreshold=100];1[label="mapM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.51/4.10	3[label="mapM_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.51/4.10	4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.51/4.10	5[label="pt sequence_Maybe (map vy3) vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.51/4.10	6[label="sequence_Maybe (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.51/4.10	7[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];28[label="vy4/Cons vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.51/4.10	28 -> 8[label="",style="solid", color="burlywood", weight=3];
9.51/4.10	29[label="vy4/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 29[label="",style="solid", color="burlywood", weight=9];
9.51/4.10	29 -> 9[label="",style="solid", color="burlywood", weight=3];
9.51/4.10	8[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 (Cons vy40 vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.51/4.10	9[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 Nil)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.51/4.10	10[label="foldr gtGtMaybe (returnMaybe Tup0) (Cons (vy3 vy40) (map vy3 vy41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.51/4.10	11[label="foldr gtGtMaybe (returnMaybe Tup0) Nil",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.51/4.10	12 -> 14[label="",style="dashed", color="red", weight=0];
9.51/4.10	12[label="gtGtMaybe (vy3 vy40) (foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy41))",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
9.51/4.10	13[label="returnMaybe Tup0",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
9.51/4.10	15 -> 7[label="",style="dashed", color="red", weight=0];
9.51/4.10	15[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy41)",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
9.51/4.10	14[label="gtGtMaybe (vy3 vy40) vy5",fontsize=16,color="black",shape="triangle"];14 -> 18[label="",style="solid", color="black", weight=3];
9.51/4.10	16[label="Just Tup0",fontsize=16,color="green",shape="box"];17[label="vy41",fontsize=16,color="green",shape="box"];18 -> 19[label="",style="dashed", color="red", weight=0];
9.51/4.10	18[label="gtGtEsMaybe (vy3 vy40) (gtGt0 vy5)",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
9.51/4.10	20[label="vy3 vy40",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
9.51/4.10	19[label="gtGtEsMaybe vy6 (gtGt0 vy5)",fontsize=16,color="burlywood",shape="triangle"];30[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
9.51/4.10	30 -> 22[label="",style="solid", color="burlywood", weight=3];
9.51/4.10	31[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];19 -> 31[label="",style="solid", color="burlywood", weight=9];
9.51/4.10	31 -> 23[label="",style="solid", color="burlywood", weight=3];
9.51/4.10	24[label="vy40",fontsize=16,color="green",shape="box"];22[label="gtGtEsMaybe Nothing (gtGt0 vy5)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
9.51/4.10	23[label="gtGtEsMaybe (Just vy60) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.51/4.10	25[label="Nothing",fontsize=16,color="green",shape="box"];26[label="gtGt0 vy5 vy60",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
9.51/4.10	27[label="vy5",fontsize=16,color="green",shape="box"];}
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(6)
9.51/4.10	Obligation:
9.51/4.10	Q DP problem:
9.51/4.10	The TRS P consists of the following rules:
9.51/4.10	
9.51/4.10	   new_foldr(vy3, Cons(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.51/4.10	
9.51/4.10	R is empty.
9.51/4.10	Q is empty.
9.51/4.10	We have to consider all minimal (P,Q,R)-chains.
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(7) QDPSizeChangeProof (EQUIVALENT)
9.51/4.10	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. 
9.51/4.10	
9.51/4.10	From the DPs we obtained the following set of size-change graphs:
9.51/4.10	*new_foldr(vy3, Cons(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
9.51/4.10	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.51/4.10	
9.51/4.10	
9.51/4.10	----------------------------------------
9.51/4.10	
9.51/4.10	(8)
9.51/4.10	YES
9.51/4.16	EOF