8.19/3.63	YES
10.09/4.17	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.09/4.17	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.09/4.17	
10.09/4.17	
10.09/4.17	H-Termination with start terms of the given HASKELL could be proven:
10.09/4.17	
10.09/4.17	(0) HASKELL
10.09/4.17	(1) BR [EQUIVALENT, 0 ms]
10.09/4.17	(2) HASKELL
10.09/4.17	(3) COR [EQUIVALENT, 0 ms]
10.09/4.17	(4) HASKELL
10.09/4.17	(5) Narrow [SOUND, 0 ms]
10.09/4.17	(6) QDP
10.09/4.17	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.09/4.17	(8) YES
10.09/4.17	
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(0)
10.09/4.17	Obligation:
10.09/4.17	mainModule Main 
10.09/4.17	module Main where {
10.09/4.17	import qualified Prelude;
10.09/4.17	data List a = Cons a (List a)  | Nil ;
10.09/4.17	
10.09/4.17	data MyBool = MyTrue  | MyFalse ;
10.09/4.17	
10.09/4.17	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.09/4.17	all p = pt and (map p);
10.09/4.17	
10.09/4.17	and :: List MyBool  ->  MyBool;
10.09/4.17	and = foldr asAs MyTrue;
10.09/4.17	
10.09/4.17	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	asAs MyFalse x = MyFalse;
10.09/4.17	asAs MyTrue x = x;
10.09/4.17	
10.09/4.17	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	esEsMyBool MyFalse MyFalse = MyTrue;
10.09/4.17	esEsMyBool MyFalse MyTrue = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyFalse = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyTrue = MyTrue;
10.09/4.17	
10.09/4.17	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.09/4.17	foldr f z Nil = z;
10.09/4.17	foldr f z (Cons x xs) = f x (foldr f z xs);
10.09/4.17	
10.09/4.17	fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	fsEsMyBool x y = not (esEsMyBool x y);
10.09/4.17	
10.09/4.17	map :: (b  ->  a)  ->  List b  ->  List a;
10.09/4.17	map f Nil = Nil;
10.09/4.17	map f (Cons x xs) = Cons (f x) (map f xs);
10.09/4.17	
10.09/4.17	not :: MyBool  ->  MyBool;
10.09/4.17	not MyTrue = MyFalse;
10.09/4.17	not MyFalse = MyTrue;
10.09/4.17	
10.09/4.17	notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
10.09/4.17	notElemMyBool = pt all fsEsMyBool;
10.09/4.17	
10.09/4.17	pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
10.09/4.17	pt f g x = f (g x);
10.09/4.17	
10.09/4.17	}
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(1) BR (EQUIVALENT)
10.09/4.17	Replaced joker patterns by fresh variables and removed binding patterns.
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(2)
10.09/4.17	Obligation:
10.09/4.17	mainModule Main 
10.09/4.17	module Main where {
10.09/4.17	import qualified Prelude;
10.09/4.17	data List a = Cons a (List a)  | Nil ;
10.09/4.17	
10.09/4.17	data MyBool = MyTrue  | MyFalse ;
10.09/4.17	
10.09/4.17	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.09/4.17	all p = pt and (map p);
10.09/4.17	
10.09/4.17	and :: List MyBool  ->  MyBool;
10.09/4.17	and = foldr asAs MyTrue;
10.09/4.17	
10.09/4.17	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	asAs MyFalse x = MyFalse;
10.09/4.17	asAs MyTrue x = x;
10.09/4.17	
10.09/4.17	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	esEsMyBool MyFalse MyFalse = MyTrue;
10.09/4.17	esEsMyBool MyFalse MyTrue = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyFalse = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyTrue = MyTrue;
10.09/4.17	
10.09/4.17	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.09/4.17	foldr f z Nil = z;
10.09/4.17	foldr f z (Cons x xs) = f x (foldr f z xs);
10.09/4.17	
10.09/4.17	fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	fsEsMyBool x y = not (esEsMyBool x y);
10.09/4.17	
10.09/4.17	map :: (a  ->  b)  ->  List a  ->  List b;
10.09/4.17	map f Nil = Nil;
10.09/4.17	map f (Cons x xs) = Cons (f x) (map f xs);
10.09/4.17	
10.09/4.17	not :: MyBool  ->  MyBool;
10.09/4.17	not MyTrue = MyFalse;
10.09/4.17	not MyFalse = MyTrue;
10.09/4.17	
10.09/4.17	notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
10.09/4.17	notElemMyBool = pt all fsEsMyBool;
10.09/4.17	
10.09/4.17	pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
10.09/4.17	pt f g x = f (g x);
10.09/4.17	
10.09/4.17	}
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(3) COR (EQUIVALENT)
10.09/4.17	Cond Reductions:
10.09/4.17	The following Function with conditions
10.09/4.17	"undefined |Falseundefined;
10.09/4.17	"
10.09/4.17	is transformed to
10.09/4.17	"undefined  = undefined1;
10.09/4.17	"
10.09/4.17	"undefined0 True = undefined;
10.09/4.17	"
10.09/4.17	"undefined1  = undefined0 False;
10.09/4.17	"
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(4)
10.09/4.17	Obligation:
10.09/4.17	mainModule Main 
10.09/4.17	module Main where {
10.09/4.17	import qualified Prelude;
10.09/4.17	data List a = Cons a (List a)  | Nil ;
10.09/4.17	
10.09/4.17	data MyBool = MyTrue  | MyFalse ;
10.09/4.17	
10.09/4.17	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.09/4.17	all p = pt and (map p);
10.09/4.17	
10.09/4.17	and :: List MyBool  ->  MyBool;
10.09/4.17	and = foldr asAs MyTrue;
10.09/4.17	
10.09/4.17	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	asAs MyFalse x = MyFalse;
10.09/4.17	asAs MyTrue x = x;
10.09/4.17	
10.09/4.17	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	esEsMyBool MyFalse MyFalse = MyTrue;
10.09/4.17	esEsMyBool MyFalse MyTrue = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyFalse = MyFalse;
10.09/4.17	esEsMyBool MyTrue MyTrue = MyTrue;
10.09/4.17	
10.09/4.17	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
10.09/4.17	foldr f z Nil = z;
10.09/4.17	foldr f z (Cons x xs) = f x (foldr f z xs);
10.09/4.17	
10.09/4.17	fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
10.09/4.17	fsEsMyBool x y = not (esEsMyBool x y);
10.09/4.17	
10.09/4.17	map :: (a  ->  b)  ->  List a  ->  List b;
10.09/4.17	map f Nil = Nil;
10.09/4.17	map f (Cons x xs) = Cons (f x) (map f xs);
10.09/4.17	
10.09/4.17	not :: MyBool  ->  MyBool;
10.09/4.17	not MyTrue = MyFalse;
10.09/4.17	not MyFalse = MyTrue;
10.09/4.17	
10.09/4.17	notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
10.09/4.17	notElemMyBool = pt all fsEsMyBool;
10.09/4.17	
10.09/4.17	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
10.09/4.17	pt f g x = f (g x);
10.09/4.17	
10.09/4.17	}
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(5) Narrow (SOUND)
10.09/4.17	Haskell To QDPs
10.09/4.17	
10.09/4.17	digraph dp_graph {
10.09/4.17	node [outthreshold=100, inthreshold=100];1[label="notElemMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.09/4.17	3[label="notElemMyBool vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.09/4.17	4[label="notElemMyBool vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.09/4.17	5[label="pt all fsEsMyBool vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.09/4.17	6[label="all (fsEsMyBool vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.09/4.17	7[label="pt and (map (fsEsMyBool vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.09/4.17	8[label="and (map (fsEsMyBool vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.09/4.17	9[label="foldr asAs MyTrue (map (fsEsMyBool vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 34[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	34 -> 10[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	35[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 35[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	35 -> 11[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	10[label="foldr asAs MyTrue (map (fsEsMyBool vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.09/4.17	11[label="foldr asAs MyTrue (map (fsEsMyBool vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.09/4.17	12[label="foldr asAs MyTrue (Cons (fsEsMyBool vx3 vx40) (map (fsEsMyBool vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.09/4.17	13[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.09/4.17	14 -> 16[label="",style="dashed", color="red", weight=0];
10.09/4.17	14[label="asAs (fsEsMyBool vx3 vx40) (foldr asAs MyTrue (map (fsEsMyBool vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
10.09/4.17	15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
10.09/4.17	17[label="foldr asAs MyTrue (map (fsEsMyBool vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
10.09/4.17	16[label="asAs (fsEsMyBool vx3 vx40) vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
10.09/4.17	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsMyBool vx3 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];36[label="vx3/MyTrue",fontsize=10,color="white",style="solid",shape="box"];19 -> 36[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	36 -> 20[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	37[label="vx3/MyFalse",fontsize=10,color="white",style="solid",shape="box"];19 -> 37[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	37 -> 21[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	20[label="asAs (not (esEsMyBool MyTrue vx40)) vx5",fontsize=16,color="burlywood",shape="box"];38[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];20 -> 38[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	38 -> 22[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	39[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];20 -> 39[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	39 -> 23[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	21[label="asAs (not (esEsMyBool MyFalse vx40)) vx5",fontsize=16,color="burlywood",shape="box"];40[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];21 -> 40[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	40 -> 24[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	41[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];21 -> 41[label="",style="solid", color="burlywood", weight=9];
10.09/4.17	41 -> 25[label="",style="solid", color="burlywood", weight=3];
10.09/4.17	22[label="asAs (not (esEsMyBool MyTrue MyTrue)) vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.09/4.17	23[label="asAs (not (esEsMyBool MyTrue MyFalse)) vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.09/4.17	24[label="asAs (not (esEsMyBool MyFalse MyTrue)) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.09/4.17	25[label="asAs (not (esEsMyBool MyFalse MyFalse)) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.09/4.17	26[label="asAs (not MyTrue) vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
10.09/4.17	27[label="asAs (not MyFalse) vx5",fontsize=16,color="black",shape="triangle"];27 -> 31[label="",style="solid", color="black", weight=3];
10.09/4.17	28 -> 27[label="",style="dashed", color="red", weight=0];
10.09/4.17	28[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];29 -> 26[label="",style="dashed", color="red", weight=0];
10.09/4.17	29[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];30[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
10.09/4.17	31[label="asAs MyTrue vx5",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
10.09/4.17	32[label="MyFalse",fontsize=16,color="green",shape="box"];33[label="vx5",fontsize=16,color="green",shape="box"];}
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(6)
10.09/4.17	Obligation:
10.09/4.17	Q DP problem:
10.09/4.17	The TRS P consists of the following rules:
10.09/4.17	
10.09/4.17	   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
10.09/4.17	
10.09/4.17	R is empty.
10.09/4.17	Q is empty.
10.09/4.17	We have to consider all minimal (P,Q,R)-chains.
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(7) QDPSizeChangeProof (EQUIVALENT)
10.09/4.17	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. 
10.09/4.17	
10.09/4.17	From the DPs we obtained the following set of size-change graphs:
10.09/4.17	*new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
10.09/4.17	The graph contains the following edges 1 >= 1, 2 > 2
10.09/4.17	
10.09/4.17	
10.09/4.17	----------------------------------------
10.09/4.17	
10.09/4.17	(8)
10.09/4.17	YES
10.29/4.28	EOF