7.87/3.53	YES
9.65/4.02	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.65/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.65/4.02	
9.65/4.02	
9.65/4.02	H-Termination with start terms of the given HASKELL could be proven:
9.65/4.02	
9.65/4.02	(0) HASKELL
9.65/4.02	(1) BR [EQUIVALENT, 0 ms]
9.65/4.02	(2) HASKELL
9.65/4.02	(3) COR [EQUIVALENT, 0 ms]
9.65/4.02	(4) HASKELL
9.65/4.02	(5) Narrow [SOUND, 0 ms]
9.65/4.02	(6) QDP
9.65/4.02	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.65/4.02	(8) YES
9.65/4.02	
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(0)
9.65/4.02	Obligation:
9.65/4.02	mainModule Main 
9.65/4.02	module Main where {
9.65/4.02	import qualified Prelude;
9.65/4.02	data List a = Cons a (List a)  | Nil ;
9.65/4.02	
9.65/4.02	data MyBool = MyTrue  | MyFalse ;
9.65/4.02	
9.65/4.02	data Ordering = LT  | EQ  | GT ;
9.65/4.02	
9.65/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.65/4.02	all p = pt and (map p);
9.65/4.02	
9.65/4.02	and :: List MyBool  ->  MyBool;
9.65/4.02	and = foldr asAs MyTrue;
9.65/4.02	
9.65/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.65/4.02	asAs MyFalse x = MyFalse;
9.65/4.02	asAs MyTrue x = x;
9.65/4.02	
9.65/4.02	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	esEsOrdering LT LT = MyTrue;
9.65/4.02	esEsOrdering LT EQ = MyFalse;
9.65/4.02	esEsOrdering LT GT = MyFalse;
9.65/4.02	esEsOrdering EQ LT = MyFalse;
9.65/4.02	esEsOrdering EQ EQ = MyTrue;
9.65/4.02	esEsOrdering EQ GT = MyFalse;
9.65/4.02	esEsOrdering GT LT = MyFalse;
9.65/4.02	esEsOrdering GT EQ = MyFalse;
9.65/4.02	esEsOrdering GT GT = MyTrue;
9.65/4.02	
9.65/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.65/4.02	foldr f z Nil = z;
9.65/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.65/4.02	
9.65/4.02	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	fsEsOrdering x y = not (esEsOrdering x y);
9.65/4.02	
9.65/4.02	map :: (a  ->  b)  ->  List a  ->  List b;
9.65/4.02	map f Nil = Nil;
9.65/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.65/4.02	
9.65/4.02	not :: MyBool  ->  MyBool;
9.65/4.02	not MyTrue = MyFalse;
9.65/4.02	not MyFalse = MyTrue;
9.65/4.02	
9.65/4.02	notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.65/4.02	notElemOrdering = pt all fsEsOrdering;
9.65/4.02	
9.65/4.02	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.65/4.02	pt f g x = f (g x);
9.65/4.02	
9.65/4.02	}
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(1) BR (EQUIVALENT)
9.65/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(2)
9.65/4.02	Obligation:
9.65/4.02	mainModule Main 
9.65/4.02	module Main where {
9.65/4.02	import qualified Prelude;
9.65/4.02	data List a = Cons a (List a)  | Nil ;
9.65/4.02	
9.65/4.02	data MyBool = MyTrue  | MyFalse ;
9.65/4.02	
9.65/4.02	data Ordering = LT  | EQ  | GT ;
9.65/4.02	
9.65/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.65/4.02	all p = pt and (map p);
9.65/4.02	
9.65/4.02	and :: List MyBool  ->  MyBool;
9.65/4.02	and = foldr asAs MyTrue;
9.65/4.02	
9.65/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.65/4.02	asAs MyFalse x = MyFalse;
9.65/4.02	asAs MyTrue x = x;
9.65/4.02	
9.65/4.02	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	esEsOrdering LT LT = MyTrue;
9.65/4.02	esEsOrdering LT EQ = MyFalse;
9.65/4.02	esEsOrdering LT GT = MyFalse;
9.65/4.02	esEsOrdering EQ LT = MyFalse;
9.65/4.02	esEsOrdering EQ EQ = MyTrue;
9.65/4.02	esEsOrdering EQ GT = MyFalse;
9.65/4.02	esEsOrdering GT LT = MyFalse;
9.65/4.02	esEsOrdering GT EQ = MyFalse;
9.65/4.02	esEsOrdering GT GT = MyTrue;
9.65/4.02	
9.65/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.65/4.02	foldr f z Nil = z;
9.65/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.65/4.02	
9.65/4.02	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	fsEsOrdering x y = not (esEsOrdering x y);
9.65/4.02	
9.65/4.02	map :: (a  ->  b)  ->  List a  ->  List b;
9.65/4.02	map f Nil = Nil;
9.65/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.65/4.02	
9.65/4.02	not :: MyBool  ->  MyBool;
9.65/4.02	not MyTrue = MyFalse;
9.65/4.02	not MyFalse = MyTrue;
9.65/4.02	
9.65/4.02	notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.65/4.02	notElemOrdering = pt all fsEsOrdering;
9.65/4.02	
9.65/4.02	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.65/4.02	pt f g x = f (g x);
9.65/4.02	
9.65/4.02	}
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(3) COR (EQUIVALENT)
9.65/4.02	Cond Reductions:
9.65/4.02	The following Function with conditions
9.65/4.02	"undefined |Falseundefined;
9.65/4.02	"
9.65/4.02	is transformed to
9.65/4.02	"undefined  = undefined1;
9.65/4.02	"
9.65/4.02	"undefined0 True = undefined;
9.65/4.02	"
9.65/4.02	"undefined1  = undefined0 False;
9.65/4.02	"
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(4)
9.65/4.02	Obligation:
9.65/4.02	mainModule Main 
9.65/4.02	module Main where {
9.65/4.02	import qualified Prelude;
9.65/4.02	data List a = Cons a (List a)  | Nil ;
9.65/4.02	
9.65/4.02	data MyBool = MyTrue  | MyFalse ;
9.65/4.02	
9.65/4.02	data Ordering = LT  | EQ  | GT ;
9.65/4.02	
9.65/4.02	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.65/4.02	all p = pt and (map p);
9.65/4.02	
9.65/4.02	and :: List MyBool  ->  MyBool;
9.65/4.02	and = foldr asAs MyTrue;
9.65/4.02	
9.65/4.02	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.65/4.02	asAs MyFalse x = MyFalse;
9.65/4.02	asAs MyTrue x = x;
9.65/4.02	
9.65/4.02	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	esEsOrdering LT LT = MyTrue;
9.65/4.02	esEsOrdering LT EQ = MyFalse;
9.65/4.02	esEsOrdering LT GT = MyFalse;
9.65/4.02	esEsOrdering EQ LT = MyFalse;
9.65/4.02	esEsOrdering EQ EQ = MyTrue;
9.65/4.02	esEsOrdering EQ GT = MyFalse;
9.65/4.02	esEsOrdering GT LT = MyFalse;
9.65/4.02	esEsOrdering GT EQ = MyFalse;
9.65/4.02	esEsOrdering GT GT = MyTrue;
9.65/4.02	
9.65/4.02	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.65/4.02	foldr f z Nil = z;
9.65/4.02	foldr f z (Cons x xs) = f x (foldr f z xs);
9.65/4.02	
9.65/4.02	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.65/4.02	fsEsOrdering x y = not (esEsOrdering x y);
9.65/4.02	
9.65/4.02	map :: (a  ->  b)  ->  List a  ->  List b;
9.65/4.02	map f Nil = Nil;
9.65/4.02	map f (Cons x xs) = Cons (f x) (map f xs);
9.65/4.02	
9.65/4.02	not :: MyBool  ->  MyBool;
9.65/4.02	not MyTrue = MyFalse;
9.65/4.02	not MyFalse = MyTrue;
9.65/4.02	
9.65/4.02	notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.65/4.02	notElemOrdering = pt all fsEsOrdering;
9.65/4.02	
9.65/4.02	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.65/4.02	pt f g x = f (g x);
9.65/4.02	
9.65/4.02	}
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(5) Narrow (SOUND)
9.65/4.02	Haskell To QDPs
9.65/4.02	
9.65/4.02	digraph dp_graph {
9.65/4.02	node [outthreshold=100, inthreshold=100];1[label="notElemOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.65/4.02	3[label="notElemOrdering vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.65/4.02	4[label="notElemOrdering vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.65/4.02	5[label="pt all fsEsOrdering vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.65/4.02	6[label="all (fsEsOrdering vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.65/4.02	7[label="pt and (map (fsEsOrdering vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.65/4.02	8[label="and (map (fsEsOrdering vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.65/4.02	9[label="foldr asAs MyTrue (map (fsEsOrdering vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];45[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 45[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	45 -> 10[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	46[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 46[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	46 -> 11[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	10[label="foldr asAs MyTrue (map (fsEsOrdering vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.65/4.02	11[label="foldr asAs MyTrue (map (fsEsOrdering vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.65/4.02	12[label="foldr asAs MyTrue (Cons (fsEsOrdering vx3 vx40) (map (fsEsOrdering vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.65/4.02	13[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.65/4.02	14 -> 16[label="",style="dashed", color="red", weight=0];
9.65/4.02	14[label="asAs (fsEsOrdering vx3 vx40) (foldr asAs MyTrue (map (fsEsOrdering vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.65/4.02	15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.65/4.02	17[label="foldr asAs MyTrue (map (fsEsOrdering vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.65/4.02	16[label="asAs (fsEsOrdering vx3 vx40) vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.65/4.02	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsOrdering vx3 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];47[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	47 -> 20[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	48[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 48[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	48 -> 21[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	49[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 49[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	49 -> 22[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	20[label="asAs (not (esEsOrdering LT vx40)) vx5",fontsize=16,color="burlywood",shape="box"];50[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];20 -> 50[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	50 -> 23[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	51[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 51[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	51 -> 24[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	52[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 52[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	52 -> 25[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	21[label="asAs (not (esEsOrdering EQ vx40)) vx5",fontsize=16,color="burlywood",shape="box"];53[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];21 -> 53[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	53 -> 26[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	54[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];21 -> 54[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	54 -> 27[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	55[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];21 -> 55[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	55 -> 28[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	22[label="asAs (not (esEsOrdering GT vx40)) vx5",fontsize=16,color="burlywood",shape="box"];56[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];22 -> 56[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	56 -> 29[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	57[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];22 -> 57[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	57 -> 30[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	58[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];22 -> 58[label="",style="solid", color="burlywood", weight=9];
9.65/4.02	58 -> 31[label="",style="solid", color="burlywood", weight=3];
9.65/4.02	23[label="asAs (not (esEsOrdering LT LT)) vx5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.65/4.02	24[label="asAs (not (esEsOrdering LT EQ)) vx5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.65/4.02	25[label="asAs (not (esEsOrdering LT GT)) vx5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.65/4.02	26[label="asAs (not (esEsOrdering EQ LT)) vx5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.65/4.02	27[label="asAs (not (esEsOrdering EQ EQ)) vx5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.65/4.02	28[label="asAs (not (esEsOrdering EQ GT)) vx5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.65/4.02	29[label="asAs (not (esEsOrdering GT LT)) vx5",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.65/4.02	30[label="asAs (not (esEsOrdering GT EQ)) vx5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.65/4.02	31[label="asAs (not (esEsOrdering GT GT)) vx5",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
9.65/4.02	32[label="asAs (not MyTrue) vx5",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
9.65/4.02	33[label="asAs (not MyFalse) vx5",fontsize=16,color="black",shape="triangle"];33 -> 42[label="",style="solid", color="black", weight=3];
9.65/4.02	34 -> 33[label="",style="dashed", color="red", weight=0];
9.65/4.02	34[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];35 -> 33[label="",style="dashed", color="red", weight=0];
9.65/4.02	35[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];36 -> 32[label="",style="dashed", color="red", weight=0];
9.65/4.02	36[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];37 -> 33[label="",style="dashed", color="red", weight=0];
9.65/4.02	37[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];38 -> 33[label="",style="dashed", color="red", weight=0];
9.65/4.02	38[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];39 -> 33[label="",style="dashed", color="red", weight=0];
9.65/4.02	39[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];40 -> 32[label="",style="dashed", color="red", weight=0];
9.65/4.02	40[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];41[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
9.65/4.02	42[label="asAs MyTrue vx5",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
9.65/4.02	43[label="MyFalse",fontsize=16,color="green",shape="box"];44[label="vx5",fontsize=16,color="green",shape="box"];}
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(6)
9.65/4.02	Obligation:
9.65/4.02	Q DP problem:
9.65/4.02	The TRS P consists of the following rules:
9.65/4.02	
9.65/4.02	   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.65/4.02	
9.65/4.02	R is empty.
9.65/4.02	Q is empty.
9.65/4.02	We have to consider all minimal (P,Q,R)-chains.
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(7) QDPSizeChangeProof (EQUIVALENT)
9.65/4.02	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. 
9.65/4.02	
9.65/4.02	From the DPs we obtained the following set of size-change graphs:
9.65/4.02	*new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.65/4.02	The graph contains the following edges 1 >= 1, 2 > 2
9.65/4.02	
9.65/4.02	
9.65/4.02	----------------------------------------
9.65/4.02	
9.65/4.02	(8)
9.65/4.02	YES
9.70/4.06	EOF