7.95/3.53	YES
9.68/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.68/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.68/4.03	
9.68/4.03	
9.68/4.03	H-Termination with start terms of the given HASKELL could be proven:
9.68/4.03	
9.68/4.03	(0) HASKELL
9.68/4.03	(1) BR [EQUIVALENT, 0 ms]
9.68/4.03	(2) HASKELL
9.68/4.03	(3) COR [EQUIVALENT, 0 ms]
9.68/4.03	(4) HASKELL
9.68/4.03	(5) Narrow [SOUND, 0 ms]
9.68/4.03	(6) QDP
9.68/4.03	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.68/4.03	(8) YES
9.68/4.03	
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(0)
9.68/4.03	Obligation:
9.68/4.03	mainModule Main 
9.68/4.03	module Main where {
9.68/4.03	import qualified Prelude;
9.68/4.03	data List a = Cons a (List a)  | Nil ;
9.68/4.03	
9.68/4.03	data MyBool = MyTrue  | MyFalse ;
9.68/4.03	
9.68/4.03	data Tup0 = Tup0 ;
9.68/4.03	
9.68/4.03	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.68/4.03	all p = pt and (map p);
9.68/4.03	
9.68/4.03	and :: List MyBool  ->  MyBool;
9.68/4.03	and = foldr asAs MyTrue;
9.68/4.03	
9.68/4.03	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.68/4.03	asAs MyFalse x = MyFalse;
9.68/4.03	asAs MyTrue x = x;
9.68/4.03	
9.68/4.03	esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	esEsTup0 Tup0 Tup0 = MyTrue;
9.68/4.03	
9.68/4.03	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.68/4.03	foldr f z Nil = z;
9.68/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.68/4.03	
9.68/4.03	fsEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	fsEsTup0 x y = not (esEsTup0 x y);
9.68/4.03	
9.68/4.03	map :: (b  ->  a)  ->  List b  ->  List a;
9.68/4.03	map f Nil = Nil;
9.68/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.68/4.03	
9.68/4.03	not :: MyBool  ->  MyBool;
9.68/4.03	not MyTrue = MyFalse;
9.68/4.03	not MyFalse = MyTrue;
9.68/4.03	
9.68/4.03	notElemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
9.68/4.03	notElemTup0 = pt all fsEsTup0;
9.68/4.03	
9.68/4.03	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
9.68/4.03	pt f g x = f (g x);
9.68/4.03	
9.68/4.03	}
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(1) BR (EQUIVALENT)
9.68/4.03	Replaced joker patterns by fresh variables and removed binding patterns.
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(2)
9.68/4.03	Obligation:
9.68/4.03	mainModule Main 
9.68/4.03	module Main where {
9.68/4.03	import qualified Prelude;
9.68/4.03	data List a = Cons a (List a)  | Nil ;
9.68/4.03	
9.68/4.03	data MyBool = MyTrue  | MyFalse ;
9.68/4.03	
9.68/4.03	data Tup0 = Tup0 ;
9.68/4.03	
9.68/4.03	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.68/4.03	all p = pt and (map p);
9.68/4.03	
9.68/4.03	and :: List MyBool  ->  MyBool;
9.68/4.03	and = foldr asAs MyTrue;
9.68/4.03	
9.68/4.03	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.68/4.03	asAs MyFalse x = MyFalse;
9.68/4.03	asAs MyTrue x = x;
9.68/4.03	
9.68/4.03	esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	esEsTup0 Tup0 Tup0 = MyTrue;
9.68/4.03	
9.68/4.03	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.68/4.03	foldr f z Nil = z;
9.68/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.68/4.03	
9.68/4.03	fsEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	fsEsTup0 x y = not (esEsTup0 x y);
9.68/4.03	
9.68/4.03	map :: (a  ->  b)  ->  List a  ->  List b;
9.68/4.03	map f Nil = Nil;
9.68/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.68/4.03	
9.68/4.03	not :: MyBool  ->  MyBool;
9.68/4.03	not MyTrue = MyFalse;
9.68/4.03	not MyFalse = MyTrue;
9.68/4.03	
9.68/4.03	notElemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
9.68/4.03	notElemTup0 = pt all fsEsTup0;
9.68/4.03	
9.68/4.03	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
9.68/4.03	pt f g x = f (g x);
9.68/4.03	
9.68/4.03	}
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(3) COR (EQUIVALENT)
9.68/4.03	Cond Reductions:
9.68/4.03	The following Function with conditions
9.68/4.03	"undefined |Falseundefined;
9.68/4.03	"
9.68/4.03	is transformed to
9.68/4.03	"undefined  = undefined1;
9.68/4.03	"
9.68/4.03	"undefined0 True = undefined;
9.68/4.03	"
9.68/4.03	"undefined1  = undefined0 False;
9.68/4.03	"
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(4)
9.68/4.03	Obligation:
9.68/4.03	mainModule Main 
9.68/4.03	module Main where {
9.68/4.03	import qualified Prelude;
9.68/4.03	data List a = Cons a (List a)  | Nil ;
9.68/4.03	
9.68/4.03	data MyBool = MyTrue  | MyFalse ;
9.68/4.03	
9.68/4.03	data Tup0 = Tup0 ;
9.68/4.03	
9.68/4.03	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.68/4.03	all p = pt and (map p);
9.68/4.03	
9.68/4.03	and :: List MyBool  ->  MyBool;
9.68/4.03	and = foldr asAs MyTrue;
9.68/4.03	
9.68/4.03	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.68/4.03	asAs MyFalse x = MyFalse;
9.68/4.03	asAs MyTrue x = x;
9.68/4.03	
9.68/4.03	esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	esEsTup0 Tup0 Tup0 = MyTrue;
9.68/4.03	
9.68/4.03	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.68/4.03	foldr f z Nil = z;
9.68/4.03	foldr f z (Cons x xs) = f x (foldr f z xs);
9.68/4.03	
9.68/4.03	fsEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.68/4.03	fsEsTup0 x y = not (esEsTup0 x y);
9.68/4.03	
9.68/4.03	map :: (b  ->  a)  ->  List b  ->  List a;
9.68/4.03	map f Nil = Nil;
9.68/4.03	map f (Cons x xs) = Cons (f x) (map f xs);
9.68/4.03	
9.68/4.03	not :: MyBool  ->  MyBool;
9.68/4.03	not MyTrue = MyFalse;
9.68/4.03	not MyFalse = MyTrue;
9.68/4.03	
9.68/4.03	notElemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
9.68/4.03	notElemTup0 = pt all fsEsTup0;
9.68/4.03	
9.68/4.03	pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
9.68/4.03	pt f g x = f (g x);
9.68/4.03	
9.68/4.03	}
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(5) Narrow (SOUND)
9.68/4.03	Haskell To QDPs
9.68/4.03	
9.68/4.03	digraph dp_graph {
9.68/4.03	node [outthreshold=100, inthreshold=100];1[label="notElemTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.68/4.03	3[label="notElemTup0 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.68/4.03	4[label="notElemTup0 vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.68/4.03	5[label="pt all fsEsTup0 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.68/4.03	6[label="all (fsEsTup0 vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.68/4.03	7[label="pt and (map (fsEsTup0 vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.68/4.03	8[label="and (map (fsEsTup0 vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.68/4.03	9[label="foldr asAs MyTrue (map (fsEsTup0 vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];25[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
9.68/4.03	25 -> 10[label="",style="solid", color="burlywood", weight=3];
9.68/4.03	26[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 26[label="",style="solid", color="burlywood", weight=9];
9.68/4.03	26 -> 11[label="",style="solid", color="burlywood", weight=3];
9.68/4.03	10[label="foldr asAs MyTrue (map (fsEsTup0 vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.68/4.03	11[label="foldr asAs MyTrue (map (fsEsTup0 vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.68/4.03	12[label="foldr asAs MyTrue (Cons (fsEsTup0 vx3 vx40) (map (fsEsTup0 vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.68/4.03	13[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.68/4.03	14 -> 16[label="",style="dashed", color="red", weight=0];
9.68/4.03	14[label="asAs (fsEsTup0 vx3 vx40) (foldr asAs MyTrue (map (fsEsTup0 vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.68/4.03	15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.68/4.03	17[label="foldr asAs MyTrue (map (fsEsTup0 vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.68/4.03	16[label="asAs (fsEsTup0 vx3 vx40) vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.68/4.03	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsTup0 vx3 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];27[label="vx3/Tup0",fontsize=10,color="white",style="solid",shape="box"];19 -> 27[label="",style="solid", color="burlywood", weight=9];
9.68/4.03	27 -> 20[label="",style="solid", color="burlywood", weight=3];
9.68/4.03	20[label="asAs (not (esEsTup0 Tup0 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];28[label="vx40/Tup0",fontsize=10,color="white",style="solid",shape="box"];20 -> 28[label="",style="solid", color="burlywood", weight=9];
9.68/4.03	28 -> 21[label="",style="solid", color="burlywood", weight=3];
9.68/4.03	21[label="asAs (not (esEsTup0 Tup0 Tup0)) vx5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.68/4.03	22[label="asAs (not MyTrue) vx5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
9.68/4.03	23[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
9.68/4.03	24[label="MyFalse",fontsize=16,color="green",shape="box"];}
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(6)
9.68/4.03	Obligation:
9.68/4.03	Q DP problem:
9.68/4.03	The TRS P consists of the following rules:
9.68/4.03	
9.68/4.03	   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.68/4.03	
9.68/4.03	R is empty.
9.68/4.03	Q is empty.
9.68/4.03	We have to consider all minimal (P,Q,R)-chains.
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(7) QDPSizeChangeProof (EQUIVALENT)
9.68/4.03	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. 
9.68/4.03	
9.68/4.03	From the DPs we obtained the following set of size-change graphs:
9.68/4.03	*new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.68/4.03	The graph contains the following edges 1 >= 1, 2 > 2
9.68/4.03	
9.68/4.03	
9.68/4.03	----------------------------------------
9.68/4.03	
9.68/4.03	(8)
9.68/4.03	YES
9.94/4.15	EOF